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Theorem dchrptlem1 25848

Description: Lemma for dchrpt 25851. (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 6654	. . . . . . 7 ⊢ (𝑢 = 𝐶 → (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))))
2	1	anbi1d 632	. . . . . 6 ⊢ (𝑢 = 𝐶 → ((((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
3	2	rexbidv 3256	. . . . 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 6308	. . . 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 6304	. . . 4 ⊢ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) ∈ V
7	4, 5, 6	fvmpt3i 6750	. . 3 ⊢ (𝐶 ∈ 𝑈 → (𝑋‘𝐶) = (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
8	7	ad2antlr 726	. 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 7168	. . 3 ⊢ (𝑇↑𝑀) ∈ V
10		simpr 488	. . . . . . . . . . . 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 499	. . . . . . . . . . . 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 2835	. . . . . . . . . . 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 498	. . . . . . . . . . . 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 11943	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		dchrpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20	19	zncrng 20236	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
21		crngring 19302	. . . . . . . . . . . . . . . 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 19413	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
25	18, 20, 21, 24	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 ∈ Grp)
26	25	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐻 ∈ Grp)
27		dchrpt.w	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
28		wrdf 13862	. . . . . . . . . . . . . . . . 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 6497	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
32	30, 31	eleqtrd 2892	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
33	29, 32	ffvelrnd 6829	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
34	33	adantr 484	. . . . . . . . . . . . . 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 19412	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Base‘𝐻)
38		dchrpt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐻)
39		dchrpt.m	. . . . . . . . . . . . . . 15 ⊢ · = (._g‘𝐻)
40		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
41	37, 38, 39, 40	odcong 18669	. . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . 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 11739	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
45		2re 11699	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
46		dchrpt.b	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐵 = (Base‘𝑍)
47	19, 46	znfi 20251	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
48	17, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ Fin)
49	46, 22	unitss 19406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 ⊆ 𝐵
50		ssfi 8722	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
51	48, 49, 50	sylancl 589	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈 ∈ Fin)
52	37, 38	odcl2 18684	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
53	25, 51, 33, 52	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
54	53	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
55		nndivre 11666	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
56	45, 54, 55	sylancr 590	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
57	56	recnd 10658	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
58		cxpcl 25265	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
59	44, 57, 58	sylancr 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
60	43, 59	eqeltrid 2894	. . . . . . . . . . . . . . . 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 11741	. . . . . . . . . . . . . . . . . . 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 25304	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
65	43	neeq1i 3051	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
66	64, 65	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑇 ≠ 0)
67		zsubcl 12012	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑚 − 𝑀) ∈ ℤ)
68	67	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑚 − 𝑀) ∈ ℤ)
69	36	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℤ)
70		expaddz 13469	. . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)))
72	35	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℤ)
73	72	zcnd 12076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℂ)
74	69	zcnd 12076	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℂ)
75	73, 74	npcand 10990	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑚 − 𝑀) + 𝑀) = 𝑚)
76	75	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = (𝑇↑𝑚))
77	43	oveq1i 7145	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇↑(𝑚 − 𝑀)) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀))
78		root1eq1 25344	. . . . . . . . . . . . . . . . . . . 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 598	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)))
80	79	biimpar 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1)
81	77, 80	syl5eq 2845	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑(𝑚 − 𝑀)) = 1)
82	81	oveq1d 7150	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (1 · (𝑇↑𝑀)))
83	60, 66, 69	expclzd 13511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑀) ∈ ℂ)
84	83	mulid2d 10648	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (1 · (𝑇↑𝑀)) = (𝑇↑𝑀))
85	82, 84	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (𝑇↑𝑀))
86	71, 76, 85	3eqtr3d 2841	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑚) = (𝑇↑𝑀))
87	86	ex 416	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀) → (𝑇↑𝑚) = (𝑇↑𝑀)))
88	42, 87	sylbird 263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)) → (𝑇↑𝑚) = (𝑇↑𝑀)))
89	14, 15, 16, 88	syl12anc 835	. . . . . . . . . . 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 2809	. . . . . . . . 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 232	. . . . . . . 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 457	. . . . . . 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 3243	. . . . . 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 7142	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
96	95	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
97		oveq2 7143	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑇↑𝑚) = (𝑇↑𝑀))
98	97	eqeq2d 2809	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℎ = (𝑇↑𝑚) ↔ ℎ = (𝑇↑𝑀)))
99	96, 98	anbi12d 633	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑀))))
100	99	rspcev 3571	. . . . . . . 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 460	. . . . . . 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 485	. . . . . 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 215	. . . . 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 484	. . . 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 6307	. . 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 690	. 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 2833	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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ℩cio 6281 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 / cdiv 11286 ℕcn 11625 2c2 11680 ℕ₀cn0 11885 ℤcz 11969 ..^cfzo 13028 ↑cexp 13425 ♯chash 13686 Word cword 13857 ∥ cdvds 15599 Basecbs 16475 ↾_s cress 16476 0_gc0g 16705 Grpcgrp 18095 ._gcmg 18216 odcod 18644 DProd cdprd 19108 dProjcdpj 19109 mulGrpcmgp 19232 1_rcur 19244 Ringcrg 19290 CRingccrg 19291 Unitcui 19385 ℤ/nℤczn 20196 ↑_𝑐ccxp 25147 DChrcdchr 25816

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-word 13858 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-dvds 15600 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-qus 16774 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-cntz 18439 df-od 18648 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-rnghom 19463 df-subrg 19526 df-lmod 19629 df-lss 19697 df-lsp 19737 df-sra 19937 df-rgmod 19938 df-lidl 19939 df-rsp 19940 df-2idl 19998 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-zring 20164 df-zrh 20197 df-zn 20200 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-log 25148 df-cxp 25149

This theorem is referenced by: dchrptlem2 25849

W3C validator