climmulf - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > climmulf		Structured version Visualization version GIF version

Theorem climmulf 44992

Description: A version of climmul 15610 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
climmulf.1	⊢ Ⅎ𝑘𝜑
climmulf.2	⊢ Ⅎ𝑘𝐹
climmulf.3	⊢ Ⅎ𝑘𝐺
climmulf.4	⊢ Ⅎ𝑘𝐻
climmulf.5	⊢ 𝑍 = (ℤ_≥‘𝑀)
climmulf.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
climmulf.7	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climmulf.8	⊢ (𝜑 → 𝐻 ∈ 𝑋)
climmulf.9	⊢ (𝜑 → 𝐺 ⇝ 𝐵)
climmulf.10	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
climmulf.11	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
climmulf.12	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))

**Assertion**
Ref	Expression
climmulf	⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))

Distinct variable group: 𝑘,𝑍 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘) 𝐵(𝑘) 𝐹(𝑘) 𝐺(𝑘) 𝐻(𝑘) 𝑀(𝑘) 𝑋(𝑘)

Proof of Theorem climmulf Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climmulf.5	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		climmulf.6	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climmulf.7	. 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
4		climmulf.8	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
5		climmulf.9	. 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
6		climmulf.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
7		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘𝑗
8	7	nfel1 2916	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
9	6, 8	nfan 1895	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
10		climmulf.2	. . . . . 6 ⊢ Ⅎ𝑘𝐹
11	10, 7	nffv 6907	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
12	11	nfel1 2916	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ
13	9, 12	nfim 1892	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
14		eleq1w 2812	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
15	14	anbi2d 629	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
16		fveq2 6897	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
17	16	eleq1d 2814	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ))
18	15, 17	imbi12d 344	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)))
19		climmulf.10	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
20	13, 18, 19	chvarfv 2229	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
21		climmulf.3	. . . . . 6 ⊢ Ⅎ𝑘𝐺
22	21, 7	nffv 6907	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
23	22	nfel1 2916	. . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ
24	9, 23	nfim 1892	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
25		fveq2 6897	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
26	25	eleq1d 2814	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ))
27	15, 26	imbi12d 344	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)))
28		climmulf.11	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
29	24, 27, 28	chvarfv 2229	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
30		climmulf.4	. . . . . 6 ⊢ Ⅎ𝑘𝐻
31	30, 7	nffv 6907	. . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗)
32		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑘 ·
33	11, 32, 22	nfov 7450	. . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) · (𝐺‘𝑗))
34	31, 33	nfeq 2913	. . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))
35	9, 34	nfim 1892	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
36		fveq2 6897	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
37	16, 25	oveq12d 7438	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) · (𝐺‘𝑘)) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
38	36, 37	eqeq12d 2744	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))))
39	15, 38	imbi12d 344	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))))
40		climmulf.12	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))
41	35, 39, 40	chvarfv 2229	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
42	1, 2, 3, 4, 5, 20, 29, 41	climmul 15610	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 Ⅎwnfc 2879 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 ℂcc 11137 · cmul 11144 ℤcz 12589 ℤ_≥cuz 12853 ⇝ cli 15461

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-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

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-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-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-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465

This theorem is referenced by: climneg 44998 climdivf 45000 stirlinglem15 45476 etransclem48 45670

W3C validator

	
		OSZAR »