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Theorem dchrptlem1 27308

Description: Lemma for dchrpt 27311. (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 6915	. . . . . . 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 3179	. . . . 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 6545	. . . 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 6534	. . . 4 ⊢ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) ∈ V
7	4, 5, 6	fvmpt3i 7021	. . 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 7464	. . 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 776	. . . . . . . . . . . . 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 2779	. . . . . . . . . . 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 783	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → 𝜑)
15		simplr 769	. . . . . . . . . . . 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 12587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		dchrpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20	19	zncrng 21563	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
21		crngring 20242	. . . . . . . . . . . . . . . 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 20383	. . . . . . . . . . . . . . . 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 14557	. . . . . . . . . . . . . . . . 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 6746	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
32	30, 31	eleqtrd 2843	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
33	29, 32	ffvelcdmd 7105	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
34	33	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊‘𝐼) ∈ 𝑈)
35		simprl 771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑚 ∈ ℤ)
36		simprr 773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
37	22, 23	unitgrpbas 20382	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Base‘𝐻)
38		dchrpt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐻)
39		dchrpt.m	. . . . . . . . . . . . . . 15 ⊢ · = (._g‘𝐻)
40		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
41	37, 38, 39, 40	odcong 19567	. . . . . . . . . . . . . 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 12380	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
45		2re 12340	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
46		dchrpt.b	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐵 = (Base‘𝑍)
47	19, 46	znfi 21578	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
48	17, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ Fin)
49	46, 22	unitss 20376	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 ⊆ 𝐵
50		ssfi 9213	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
51	48, 49, 50	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈 ∈ Fin)
52	37, 38	odcl2 19583	. . . . . . . . . . . . . . . . . . . . . 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 12307	. . . . . . . . . . . . . . . . . . . 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 11289	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
58		cxpcl 26716	. . . . . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . . . 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 12382	. . . . . . . . . . . . . . . . . . 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 26755	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
65	43	neeq1i 3005	. . . . . . . . . . . . . . . . 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 12659	. . . . . . . . . . . . . . . . 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 14147	. . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . 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 12723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℂ)
74	69	zcnd 12723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℂ)
75	73, 74	npcand 11624	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑚 − 𝑀) + 𝑀) = 𝑚)
76	75	oveq2d 7447	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = (𝑇↑𝑚))
77	43	oveq1i 7441	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇↑(𝑚 − 𝑀)) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀))
78		root1eq1 26798	. . . . . . . . . . . . . . . . . . . 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 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑(𝑚 − 𝑀)) = 1)
82	81	oveq1d 7446	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (1 · (𝑇↑𝑀)))
83	60, 66, 69	expclzd 14191	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑀) ∈ ℂ)
84	83	mullidd 11279	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (1 · (𝑇↑𝑀)) = (𝑇↑𝑀))
85	82, 84	eqtrd 2777	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (𝑇↑𝑀))
86	71, 76, 85	3eqtr3d 2785	. . . . . . . . . . . . . 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 837	. . . . . . . . . . 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 2748	. . . . . . . . 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 3155	. . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
96	95	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
97		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑇↑𝑚) = (𝑇↑𝑀))
98	97	eqeq2d 2748	. . . . . . . . . 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 3622	. . . . . . . 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 6544	. . 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 2777	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 2940 ∃wrex 3070 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ℩cio 6512 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ℤcz 12613 ..^cfzo 13694 ↑cexp 14102 ♯chash 14369 Word cword 14552 ∥ cdvds 16290 Basecbs 17247 ↾_s cress 17274 0_gc0g 17484 Grpcgrp 18951 ._gcmg 19085 odcod 19542 DProd cdprd 20013 dProjcdpj 20014 mulGrpcmgp 20137 1_rcur 20178 Ringcrg 20230 CRingccrg 20231 Unitcui 20355 ℤ/nℤczn 21513 ↑_𝑐ccxp 26597 DChrcdchr 27276

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-word 14553 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-dvds 16291 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-qus 17554 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 df-cntz 19335 df-od 19546 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-lsp 20970 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-rsp 21219 df-2idl 21260 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-zn 21517 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599

This theorem is referenced by: dchrptlem2 27309

W3C validator