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Theorem dchrptlem1 27227

Description: Lemma for dchrpt 27230. (Contributed by Mario Carneiro, 28-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrpt.g	⊢ 𝐺 = (DChr‘𝑁)
dchrpt.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrpt.d	⊢ 𝐷 = (Base‘𝐺)
dchrpt.b	⊢ 𝐵 = (Base‘𝑍)
dchrpt.1	⊢ 1 = (1_r‘𝑍)
dchrpt.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
dchrpt.n1	⊢ (𝜑 → 𝐴 ≠ 1 )
dchrpt.u	⊢ 𝑈 = (Unit‘𝑍)
dchrpt.h	⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
dchrpt.m	⊢ · = (._g‘𝐻)
dchrpt.s	⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘))))
dchrpt.au	⊢ (𝜑 → 𝐴 ∈ 𝑈)
dchrpt.w	⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
dchrpt.2	⊢ (𝜑 → 𝐻dom DProd 𝑆)
dchrpt.3	⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈)
dchrpt.p	⊢ 𝑃 = (𝐻dProj𝑆)
dchrpt.o	⊢ 𝑂 = (od‘𝐻)
dchrpt.t	⊢ 𝑇 = (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))
dchrpt.i	⊢ (𝜑 → 𝐼 ∈ dom 𝑊)
dchrpt.4	⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 )
dchrpt.5	⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))

**Assertion**
Ref	Expression
dchrptlem1	⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (𝑋‘𝐶) = (𝑇↑𝑀))

Distinct variable groups: ℎ,𝑘,𝑚,𝑛, 1 𝑢,ℎ,𝐴,𝑘,𝑚,𝑛 ℎ,𝐼,𝑘,𝑚,𝑢 𝐶,ℎ,𝑚,𝑢 ℎ,𝐻,𝑘,𝑚,𝑛,𝑢 ℎ,𝑊,𝑘,𝑚,𝑛,𝑢 · ,ℎ,𝑘,𝑚,𝑛,𝑢 𝑃,ℎ,𝑚,𝑢 𝑆,ℎ,𝑘,𝑚,𝑛,𝑢 ℎ,𝑍,𝑘,𝑚,𝑛,𝑢 ℎ,𝑀,𝑚 𝜑,ℎ,𝑘,𝑚,𝑛 𝑇,ℎ,𝑚,𝑢 𝑈,ℎ,𝑚,𝑢 Allowed substitution hints: 𝜑(𝑢) 𝐵(𝑢,ℎ,𝑘,𝑚,𝑛) 𝐶(𝑘,𝑛) 𝐷(𝑢,ℎ,𝑘,𝑚,𝑛) 𝑃(𝑘,𝑛) 𝑇(𝑘,𝑛) 𝑈(𝑘,𝑛) 1 (𝑢) 𝐺(𝑢,ℎ,𝑘,𝑚,𝑛) 𝐼(𝑛) 𝑀(𝑢,𝑘,𝑛) 𝑁(𝑢,ℎ,𝑘,𝑚,𝑛) 𝑂(𝑢,ℎ,𝑘,𝑚,𝑛) 𝑋(𝑢,ℎ,𝑘,𝑚,𝑛)

**Proof of Theorem dchrptlem1**
Step	Hyp	Ref	Expression
1		fveqeq2 6885	. . . . . . 7 ⊢ (𝑢 = 𝐶 → (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))))
2	1	anbi1d 631	. . . . . 6 ⊢ (𝑢 = 𝐶 → ((((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
3	2	rexbidv 3164	. . . . 5 ⊢ (𝑢 = 𝐶 → (∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
4	3	iotabidv 6515	. . . 4 ⊢ (𝑢 = 𝐶 → (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) = (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
5		dchrpt.5	. . . 4 ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
6		iotaex 6504	. . . 4 ⊢ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) ∈ V
7	4, 5, 6	fvmpt3i 6991	. . 3 ⊢ (𝐶 ∈ 𝑈 → (𝑋‘𝐶) = (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
8	7	ad2antlr 727	. 2 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (𝑋‘𝐶) = (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
9		ovex 7438	. . 3 ⊢ (𝑇↑𝑀) ∈ V
10		simpr 484	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)))
11		simpllr 775	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
12	11	simprd 495	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))
13	10, 12	eqtr3d 2772	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
14		simp-4l 782	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → 𝜑)
15		simplr 768	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → 𝑚 ∈ ℤ)
16	11	simpld 494	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → 𝑀 ∈ ℤ)
17		dchrpt.n	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ ℕ)
18	17	nnnn0d 12562	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		dchrpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20	19	zncrng 21505	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
21		crngring 20205	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
22		dchrpt.u	. . . . . . . . . . . . . . . . 17 ⊢ 𝑈 = (Unit‘𝑍)
23		dchrpt.h	. . . . . . . . . . . . . . . . 17 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
24	22, 23	unitgrp 20343	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
25	18, 20, 21, 24	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 ∈ Grp)
26	25	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐻 ∈ Grp)
27		dchrpt.w	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
28		wrdf 14536	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word 𝑈 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
29	27, 28	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑈)
30		dchrpt.i	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐼 ∈ dom 𝑊)
31	29	fdmd 6716	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
32	30, 31	eleqtrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
33	29, 32	ffvelcdmd 7075	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
34	33	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊‘𝐼) ∈ 𝑈)
35		simprl 770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑚 ∈ ℤ)
36		simprr 772	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
37	22, 23	unitgrpbas 20342	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Base‘𝐻)
38		dchrpt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐻)
39		dchrpt.m	. . . . . . . . . . . . . . 15 ⊢ · = (._g‘𝐻)
40		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
41	37, 38, 39, 40	odcong 19530	. . . . . . . . . . . . . 14 ⊢ ((𝐻 ∈ Grp ∧ (𝑊‘𝐼) ∈ 𝑈 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀) ↔ (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼))))
42	26, 34, 35, 36, 41	syl112anc 1376	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀) ↔ (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼))))
43		dchrpt.t	. . . . . . . . . . . . . . . . 17 ⊢ 𝑇 = (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))
44		neg1cn 12354	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
45		2re 12314	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
46		dchrpt.b	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐵 = (Base‘𝑍)
47	19, 46	znfi 21520	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
48	17, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ Fin)
49	46, 22	unitss 20336	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 ⊆ 𝐵
50		ssfi 9187	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
51	48, 49, 50	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈 ∈ Fin)
52	37, 38	odcl2 19546	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
53	25, 51, 33, 52	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
54	53	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
55		nndivre 12281	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
56	45, 54, 55	sylancr 587	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
57	56	recnd 11263	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
58		cxpcl 26635	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
59	44, 57, 58	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
60	43, 59	eqeltrid 2838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑇 ∈ ℂ)
61	44	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → -1 ∈ ℂ)
62		neg1ne0 12356	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
63	62	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → -1 ≠ 0)
64	61, 63, 57	cxpne0d 26674	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
65	43	neeq1i 2996	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
66	64, 65	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑇 ≠ 0)
67		zsubcl 12634	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑚 − 𝑀) ∈ ℤ)
68	67	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑚 − 𝑀) ∈ ℤ)
69	36	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℤ)
70		expaddz 14124	. . . . . . . . . . . . . . . 16 ⊢ (((𝑇 ∈ ℂ ∧ 𝑇 ≠ 0) ∧ ((𝑚 − 𝑀) ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)))
71	60, 66, 68, 69, 70	syl22anc 838	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)))
72	35	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℤ)
73	72	zcnd 12698	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℂ)
74	69	zcnd 12698	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℂ)
75	73, 74	npcand 11598	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑚 − 𝑀) + 𝑀) = 𝑚)
76	75	oveq2d 7421	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = (𝑇↑𝑚))
77	43	oveq1i 7415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇↑(𝑚 − 𝑀)) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀))
78		root1eq1 26717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑂‘(𝑊‘𝐼)) ∈ ℕ ∧ (𝑚 − 𝑀) ∈ ℤ) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)))
79	53, 67, 78	syl2an 596	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)))
80	79	biimpar 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1)
81	77, 80	eqtrid 2782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑(𝑚 − 𝑀)) = 1)
82	81	oveq1d 7420	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (1 · (𝑇↑𝑀)))
83	60, 66, 69	expclzd 14169	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑀) ∈ ℂ)
84	83	mullidd 11253	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (1 · (𝑇↑𝑀)) = (𝑇↑𝑀))
85	82, 84	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (𝑇↑𝑀))
86	71, 76, 85	3eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑚) = (𝑇↑𝑀))
87	86	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀) → (𝑇↑𝑚) = (𝑇↑𝑀)))
88	42, 87	sylbird 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)) → (𝑇↑𝑚) = (𝑇↑𝑀)))
89	14, 15, 16, 88	syl12anc 836	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → ((𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)) → (𝑇↑𝑚) = (𝑇↑𝑀)))
90	13, 89	mpd 15	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → (𝑇↑𝑚) = (𝑇↑𝑀))
91	90	eqeq2d 2746	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → (ℎ = (𝑇↑𝑚) ↔ ℎ = (𝑇↑𝑀)))
92	91	biimpd 229	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → (ℎ = (𝑇↑𝑚) → ℎ = (𝑇↑𝑀)))
93	92	expimpd 453	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) → ((((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) → ℎ = (𝑇↑𝑀)))
94	93	rexlimdva 3141	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) → ℎ = (𝑇↑𝑀)))
95		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
96	95	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
97		oveq2 7413	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑇↑𝑚) = (𝑇↑𝑀))
98	97	eqeq2d 2746	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℎ = (𝑇↑𝑚) ↔ ℎ = (𝑇↑𝑀)))
99	96, 98	anbi12d 632	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑀))))
100	99	rspcev 3601	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ (((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑀))) → ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)))
101	100	expr 456	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))) → (ℎ = (𝑇↑𝑀) → ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
102	101	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (ℎ = (𝑇↑𝑀) → ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
103	94, 102	impbid 212	. . . . 5 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ ℎ = (𝑇↑𝑀)))
104	103	adantr 480	. . . 4 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ (𝑇↑𝑀) ∈ V) → (∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ ℎ = (𝑇↑𝑀)))
105	104	iota5 6514	. . 3 ⊢ ((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ (𝑇↑𝑀) ∈ V) → (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) = (𝑇↑𝑀))
106	9, 105	mpan2 691	. 2 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) = (𝑇↑𝑀))
107	8, 106	eqtrd 2770	1 ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (𝑋‘𝐶) = (𝑇↑𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 Vcvv 3459 ⊆ wss 3926 class class class wbr 5119 ↦ cmpt 5201 dom cdm 5654 ran crn 5655 ℩cio 6482 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 Fincfn 8959 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 − cmin 11466 -cneg 11467 / cdiv 11894 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ..^cfzo 13671 ↑cexp 14079 ♯chash 14348 Word cword 14531 ∥ cdvds 16272 Basecbs 17228 ↾_s cress 17251 0_gc0g 17453 Grpcgrp 18916 ._gcmg 19050 odcod 19505 DProd cdprd 19976 dProjcdpj 19977 mulGrpcmgp 20100 1_rcur 20141 Ringcrg 20193 CRingccrg 20194 Unitcui 20315 ℤ/nℤczn 21463 ↑_𝑐ccxp 26516 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-ec 8721 df-qs 8725 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-word 14532 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-dvds 16273 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-qus 17523 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-mhm 18761 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-mulg 19051 df-subg 19106 df-nsg 19107 df-eqg 19108 df-ghm 19196 df-cntz 19300 df-od 19509 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-rhm 20432 df-subrng 20506 df-subrg 20530 df-lmod 20819 df-lss 20889 df-lsp 20929 df-sra 21131 df-rgmod 21132 df-lidl 21169 df-rsp 21170 df-2idl 21211 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-zring 21408 df-zrh 21464 df-zn 21467 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-cxp 26518

This theorem is referenced by: dchrptlem2 27228

W3C validator