bfplem1 - Mathbox for Jeff Madsen

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Theorem bfplem1 37295

Description: Lemma for bfp 37297. The sequence 𝐺, which simply starts from any point in the space and iterates 𝐹, satisfies the property that the distance from 𝐺(𝑛) to 𝐺(𝑛 + 1) decreases by at least 𝐾 after each step. Thus, the total distance from any 𝐺(𝑖) to 𝐺(𝑗) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point ((⇝_𝑡‘𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)

**Hypotheses**
Ref	Expression
bfp.2	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bfp.3	⊢ (𝜑 → 𝑋 ≠ ∅)
bfp.4	⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
bfp.5	⊢ (𝜑 → 𝐾 < 1)
bfp.6	⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
bfp.7	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
bfp.8	⊢ 𝐽 = (MetOpen‘𝐷)
bfp.9	⊢ (𝜑 → 𝐴 ∈ 𝑋)
bfp.10	⊢ 𝐺 = seq1((𝐹 ∘ 1^st ), (ℕ × {𝐴}))

**Assertion**
Ref	Expression
bfplem1	⊢ (𝜑 → 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺))

Distinct variable groups: 𝑥,𝑦,𝐷 𝑥,𝐺,𝑦 𝑥,𝐽,𝑦 𝜑,𝑥,𝑦 𝑥,𝐹,𝑦 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐴(𝑥,𝑦)

Proof of Theorem bfplem1 Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bfp.2	. . 3 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
2		cmetmet 25213	. . . . 5 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
4		nnuz 12895	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
5		bfp.10	. . . . 5 ⊢ 𝐺 = seq1((𝐹 ∘ 1^st ), (ℕ × {𝐴}))
6		1zzd 12623	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
7		bfp.9	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
8		bfp.6	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶𝑋)
9	4, 5, 6, 7, 8	algrf 16543	. . . 4 ⊢ (𝜑 → 𝐺:ℕ⟶𝑋)
10	8, 7	ffvelcdmd 7095	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑋)
11		metcl 24237	. . . . . 6 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑋) → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
12	3, 7, 10, 11	syl3anc 1369	. . . . 5 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ)
13		bfp.4	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
14	12, 13	rerpdivcld 13079	. . . 4 ⊢ (𝜑 → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
15		bfp.5	. . . 4 ⊢ (𝜑 → 𝐾 < 1)
16		fveq2 6897	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘𝑗) = (𝐺‘1))
17		fvoveq1 7443	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐺‘(𝑗 + 1)) = (𝐺‘(1 + 1)))
18	16, 17	oveq12d 7438	. . . . . . . 8 ⊢ (𝑗 = 1 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘1)𝐷(𝐺‘(1 + 1))))
19		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = 1 → (𝐾↑𝑗) = (𝐾↑1))
20	19	oveq2d 7436	. . . . . . . 8 ⊢ (𝑗 = 1 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
21	18, 20	breq12d 5161	. . . . . . 7 ⊢ (𝑗 = 1 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1))))
22	21	imbi2d 340	. . . . . 6 ⊢ (𝑗 = 1 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))))
23		fveq2 6897	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘))
24		fvoveq1 7443	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐺‘(𝑗 + 1)) = (𝐺‘(𝑘 + 1)))
25	23, 24	oveq12d 7438	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
26		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (𝐾↑𝑗) = (𝐾↑𝑘))
27	26	oveq2d 7436	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
28	25, 27	breq12d 5161	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))
29	28	imbi2d 340	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))))
30		fveq2 6897	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘𝑗) = (𝐺‘(𝑘 + 1)))
31		fvoveq1 7443	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐺‘(𝑗 + 1)) = (𝐺‘((𝑘 + 1) + 1)))
32	30, 31	oveq12d 7438	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))))
33		oveq2 7428	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐾↑𝑗) = (𝐾↑(𝑘 + 1)))
34	33	oveq2d 7436	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
35	32, 34	breq12d 5161	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
36	35	imbi2d 340	. . . . . 6 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
37	12	leidd 11810	. . . . . . 7 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ≤ (𝐴𝐷(𝐹‘𝐴)))
38	4, 5, 6, 7	algr0 16542	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) = 𝐴)
39		1nn 12253	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
40	4, 5, 6, 7, 8	algrp1 16544	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ∈ ℕ) → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
41	39, 40	mpan2 690	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1)))
42	38	fveq2d 6901	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘𝐴))
43	41, 42	eqtrd 2768	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘𝐴))
44	38, 43	oveq12d 7438	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) = (𝐴𝐷(𝐹‘𝐴)))
45	13	rpred 13048	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ ℝ)
46	45	recnd 11272	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℂ)
47	46	exp1d 14137	. . . . . . . . 9 ⊢ (𝜑 → (𝐾↑1) = 𝐾)
48	47	oveq2d 7436	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾))
49	12	recnd 11272	. . . . . . . . 9 ⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℂ)
50	13	rpne0d 13053	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ≠ 0)
51	49, 46, 50	divcan1d 12021	. . . . . . . 8 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾) = (𝐴𝐷(𝐹‘𝐴)))
52	48, 51	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (𝐴𝐷(𝐹‘𝐴)))
53	37, 44, 52	3brtr4d 5180	. . . . . 6 ⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))
54	9	ffvelcdmda 7094	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑋)
55		peano2nn 12254	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
56		ffvelcdm 7091	. . . . . . . . . . . . 13 ⊢ ((𝐺:ℕ⟶𝑋 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
57	9, 55, 56	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋)
58	54, 57	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋))
59		bfp.7	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
60	59	ralrimivva 3197	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
61	60	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))
62		fveq2 6897	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘)))
63	62	oveq1d 7435	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)))
64		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐷𝑦) = ((𝐺‘𝑘)𝐷𝑦))
65	64	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝐺‘𝑘) → (𝐾 · (𝑥𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷𝑦)))
66	63, 65	breq12d 5161	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐺‘𝑘) → (((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦))))
67		fveq2 6897	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐹‘𝑦) = (𝐹‘(𝐺‘(𝑘 + 1))))
68	67	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
69		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐺‘𝑘)𝐷𝑦) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))
70	69	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
71	68, 70	breq12d 5161	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))))
72	66, 71	rspc2v 3620	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))))
73	58, 61, 72	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
74	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (Met‘𝑋))
75	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:𝑋⟶𝑋)
76	75, 54	ffvelcdmd 7095	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘𝑘)) ∈ 𝑋)
77	75, 57	ffvelcdmd 7095	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋)
78		metcl 24237	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘(𝐺‘𝑘)) ∈ 𝑋 ∧ (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ)
79	74, 76, 77, 78	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ)
80	45	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℝ)
81		metcl 24237	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
82	74, 54, 57, 81	syl3anc 1369	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ)
83	80, 82	remulcld 11274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ)
84	14	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ)
85	55	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
86	85	nnnn0d 12562	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ₀)
87	80, 86	reexpcld 14159	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) ∈ ℝ)
88	84, 87	remulcld 11274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ)
89		letr 11338	. . . . . . . . . . 11 ⊢ ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
90	79, 83, 88, 89	syl3anc 1369	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
91	73, 90	mpand 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
92		nnnn0 12509	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℕ₀)
93		reexpcl 14075	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ ℝ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑𝑘) ∈ ℝ)
94	45, 92, 93	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℝ)
95	84, 94	remulcld 11274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ)
96	13	rpgt0d 13051	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝐾)
97	96	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐾)
98		lemul1 12096	. . . . . . . . . . 11 ⊢ ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ ∧ (𝐾 ∈ ℝ ∧ 0 < 𝐾)) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾)))
99	82, 95, 80, 97, 98	syl112anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾)))
100	82	recnd 11272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℂ)
101	46	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℂ)
102	100, 101	mulcomd 11265	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))
103	84	recnd 11272	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℂ)
104	94	recnd 11272	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℂ)
105	103, 104, 101	mulassd 11267	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
106		expp1 14065	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℕ₀) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
107	46, 92, 106	syl2an 595	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾))
108	107	oveq2d 7436	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾)))
109	105, 108	eqtr4d 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))
110	102, 109	breq12d 5161	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
111	99, 110	bitrd 279	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
112	4, 5, 6, 7, 8	algrp1 16544	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) = (𝐹‘(𝐺‘𝑘)))
113	4, 5, 6, 7, 8	algrp1 16544	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
114	55, 113	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1))))
115	112, 114	oveq12d 7438	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))))
116	115	breq1d 5158	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
117	91, 111, 116	3imtr4d 294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))
118	117	expcom 413	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → (𝜑 → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
119	118	a2d 29	. . . . . 6 ⊢ (𝑘 ∈ ℕ → ((𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))) → (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))))
120	22, 29, 36, 29, 53, 119	nnind 12260	. . . . 5 ⊢ (𝑘 ∈ ℕ → (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))
121	120	impcom 407	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))
122	3, 9, 14, 13, 15, 121	geomcau 37232	. . 3 ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷))
123		bfp.8	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
124	123	cmetcau 25216	. . 3 ⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐺 ∈ (Cau‘𝐷)) → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
125	1, 122, 124	syl2anc 583	. 2 ⊢ (𝜑 → 𝐺 ∈ dom (⇝_𝑡‘𝐽))
126		metxmet 24239	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
127	123	methaus 24428	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus)
128	3, 126, 127	3syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Haus)
129		lmfun 23284	. . 3 ⊢ (𝐽 ∈ Haus → Fun (⇝_𝑡‘𝐽))
130		funfvbrb 7060	. . 3 ⊢ (Fun (⇝_𝑡‘𝐽) → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
131	128, 129, 130	3syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ dom (⇝_𝑡‘𝐽) ↔ 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺)))
132	125, 131	mpbid 231	1 ⊢ (𝜑 → 𝐺(⇝_𝑡‘𝐽)((⇝_𝑡‘𝐽)‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∅c0 4323 {csn 4629 class class class wbr 5148 × cxp 5676 dom cdm 5678 ∘ ccom 5682 Fun wfun 6542 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 1^st c1st 7991 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 < clt 11278 ≤ cle 11279 / cdiv 11901 ℕcn 12242 ℕ₀cn0 12502 ℝ⁺crp 13006 seqcseq 13998 ↑cexp 14058 ∞Metcxmet 21263 Metcmet 21264 MetOpencmopn 21268 ⇝_𝑡clm 23129 Hauscha 23211 Cauccau 25180 CMetccmet 25181

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-rest 17403 df-topgen 17424 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-top 22795 df-topon 22812 df-bases 22848 df-ntr 22923 df-nei 23001 df-lm 23132 df-haus 23218 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-cfil 25182 df-cau 25183 df-cmet 25184

This theorem is referenced by: bfplem2 37296

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