metakunt16 - Mathbox for metakunt

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Theorem metakunt16 41672

Description: Construction of another permutation. (Contributed by metakunt, 25-May-2024.)

**Hypotheses**
Ref	Expression
metakunt16.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
metakunt16.2	⊢ (𝜑 → 𝐼 ∈ ℕ)
metakunt16.3	⊢ (𝜑 → 𝐼 ≤ 𝑀)
metakunt16.4	⊢ 𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))

**Assertion**
Ref	Expression
metakunt16	⊢ (𝜑 → 𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀 − 𝐼)))

Distinct variable groups: 𝑥,𝐼 𝑥,𝑀 𝜑,𝑥 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem metakunt16 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		metakunt16.4	. 2 ⊢ 𝐹 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼)))
2		metakunt16.2	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ ℕ)
3	2	nnzd 12615	. . . 4 ⊢ (𝜑 → 𝐼 ∈ ℤ)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℤ)
5		metakunt16.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
6	5	nnzd 12615	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑀 ∈ ℤ)
8		1zzd 12623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈ ℤ)
9	7, 8	zsubcld 12701	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑀 − 1) ∈ ℤ)
10	8, 4	zsubcld 12701	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ℤ)
11		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑥 ∈ (𝐼...(𝑀 − 1)))
12		elfz3 13543	. . . 4 ⊢ ((1 − 𝐼) ∈ ℤ → (1 − 𝐼) ∈ ((1 − 𝐼)...(1 − 𝐼)))
13	10, 12	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ((1 − 𝐼)...(1 − 𝐼)))
14	4	zcnd 12697	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℂ)
15		1cnd 11239	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈ ℂ)
16	14, 15	pncan3d 11604	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝐼 + (1 − 𝐼)) = 1)
17	16	eqcomd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 = (𝐼 + (1 − 𝐼)))
18	5	nncnd 12258	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℂ)
19	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑀 ∈ ℂ)
20	19, 15, 14	npncand 11625	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → ((𝑀 − 1) + (1 − 𝐼)) = (𝑀 − 𝐼))
21	20	eqcomd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑀 − 𝐼) = ((𝑀 − 1) + (1 − 𝐼)))
22	4, 9, 10, 10, 11, 13, 17, 21	fzadd2d 41448	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑥 + (1 − 𝐼)) ∈ (1...(𝑀 − 𝐼)))
23	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℤ)
24	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑀 ∈ ℤ)
25		1zzd 12623	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈ ℤ)
26	24, 25	zsubcld 12701	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑀 − 1) ∈ ℤ)
27		elfznn 13562	. . . . . 6 ⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 𝑦 ∈ ℕ)
28	27	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℕ)
29		nnz 12609	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
30	28, 29	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℤ)
31	25, 23	zsubcld 12701	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℤ)
32	30, 31	zsubcld 12701	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ∈ ℤ)
33	23	zred 12696	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℝ)
34	33	recnd 11272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℂ)
35		1cnd 11239	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈ ℂ)
36	34, 35	pncan3d 11604	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 + (1 − 𝐼)) = 1)
37	27	nnge1d 12290	. . . . . 6 ⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 1 ≤ 𝑦)
38	37	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ≤ 𝑦)
39	36, 38	eqbrtrd 5170	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 + (1 − 𝐼)) ≤ 𝑦)
40		1red 11245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 1 ∈ ℝ)
41	40, 33	resubcld 11672	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℝ)
42	28	nnred 12257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ∈ ℝ)
43	33, 41, 42	3jca 1126	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝐼 ∈ ℝ ∧ (1 − 𝐼) ∈ ℝ ∧ 𝑦 ∈ ℝ))
44		leaddsub 11720	. . . . 5 ⊢ ((𝐼 ∈ ℝ ∧ (1 − 𝐼) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝐼 + (1 − 𝐼)) ≤ 𝑦 ↔ 𝐼 ≤ (𝑦 − (1 − 𝐼))))
45	43, 44	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝐼 + (1 − 𝐼)) ≤ 𝑦 ↔ 𝐼 ≤ (𝑦 − (1 − 𝐼))))
46	39, 45	mpbid 231	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ≤ (𝑦 − (1 − 𝐼)))
47		elfzle2 13537	. . . . . 6 ⊢ (𝑦 ∈ (1...(𝑀 − 𝐼)) → 𝑦 ≤ (𝑀 − 𝐼))
48	47	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ≤ (𝑀 − 𝐼))
49	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑀 ∈ ℂ)
50	23	zcnd 12697	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝐼 ∈ ℂ)
51	49, 35, 50	npncand 11625	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝑀 − 1) + (1 − 𝐼)) = (𝑀 − 𝐼))
52	48, 51	breqtrrd 5176	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → 𝑦 ≤ ((𝑀 − 1) + (1 − 𝐼)))
53	31	zred 12696	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (1 − 𝐼) ∈ ℝ)
54	26	zred 12696	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑀 − 1) ∈ ℝ)
55	42, 53, 54	lesubaddd 11841	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → ((𝑦 − (1 − 𝐼)) ≤ (𝑀 − 1) ↔ 𝑦 ≤ ((𝑀 − 1) + (1 − 𝐼))))
56	52, 55	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ≤ (𝑀 − 1))
57	23, 26, 32, 46, 56	elfzd 13524	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (1...(𝑀 − 𝐼))) → (𝑦 − (1 − 𝐼)) ∈ (𝐼...(𝑀 − 1)))
58		1cnd 11239	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 1 ∈ ℂ)
59	34	adantrl 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝐼 ∈ ℂ)
60	58, 59	subcld 11601	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (1 − 𝐼) ∈ ℂ)
61		elfzelz 13533	. . . . . . 7 ⊢ (𝑥 ∈ (𝐼...(𝑀 − 1)) → 𝑥 ∈ ℤ)
62	61	ad2antrl 727	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑥 ∈ ℤ)
63		zcn 12593	. . . . . 6 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ)
64	62, 63	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑥 ∈ ℂ)
65	28	adantrl 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑦 ∈ ℕ)
66		nncn 12250	. . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
67	65, 66	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → 𝑦 ∈ ℂ)
68	60, 64, 67	addrsub 11661	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ 𝑥 = (𝑦 − (1 − 𝐼))))
69	68	bicomd 222	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (𝑥 = (𝑦 − (1 − 𝐼)) ↔ ((1 − 𝐼) + 𝑥) = 𝑦))
70	60, 64	addcomd 11446	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → ((1 − 𝐼) + 𝑥) = (𝑥 + (1 − 𝐼)))
71	70	eqeq1d 2730	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ (𝑥 + (1 − 𝐼)) = 𝑦))
72		eqcom 2735	. . . . 5 ⊢ ((𝑥 + (1 − 𝐼)) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼)))
73	72	a1i 11	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → ((𝑥 + (1 − 𝐼)) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼))))
74	71, 73	bitrd 279	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (((1 − 𝐼) + 𝑥) = 𝑦 ↔ 𝑦 = (𝑥 + (1 − 𝐼))))
75	69, 74	bitrd 279	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐼...(𝑀 − 1)) ∧ 𝑦 ∈ (1...(𝑀 − 𝐼)))) → (𝑥 = (𝑦 − (1 − 𝐼)) ↔ 𝑦 = (𝑥 + (1 − 𝐼))))
76	1, 22, 57, 75	f1o2d 7675	1 ⊢ (𝜑 → 𝐹:(𝐼...(𝑀 − 1))–1-1-onto→(1...(𝑀 − 𝐼)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ↦ cmpt 5231 –1-1-onto→wf1o 6547 (class class class)co 7420 ℂcc 11136 ℝcr 11137 1c1 11139 + caddc 11141 ≤ cle 11279 − cmin 11474 ℕcn 12242 ℤcz 12588 ...cfz 13516

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 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

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-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-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517

This theorem is referenced by: metakunt25 41681

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