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Theorem dchrptlem1 27217

Description: Lemma for dchrpt 27220. (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 6911	. . . . . . 7 ⊢ (𝑢 = 𝐶 → (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))))
2	1	anbi1d 629	. . . . . 6 ⊢ (𝑢 = 𝐶 → ((((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
3	2	rexbidv 3176	. . . . 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 6537	. . . 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 6526	. . . 4 ⊢ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) ∈ V
7	4, 5, 6	fvmpt3i 7015	. . 3 ⊢ (𝐶 ∈ 𝑈 → (𝑋‘𝐶) = (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))))
8	7	ad2antlr 725	. 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 7459	. . 3 ⊢ (𝑇↑𝑀) ∈ V
10		simpr 483	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 494	. . . . . . . . . . . 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 2770	. . . . . . . . . . 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 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) ∧ 𝑚 ∈ ℤ) ∧ ((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼))) → 𝜑)
15		simplr 767	. . . . . . . . . . . 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 493	. . . . . . . . . . . 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 12570	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		dchrpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20	19	zncrng 21485	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
21		crngring 20192	. . . . . . . . . . . . . . . 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 20329	. . . . . . . . . . . . . . . 16 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
25	18, 20, 21, 24	4syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐻 ∈ Grp)
26	25	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝐻 ∈ Grp)
27		dchrpt.w	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑊 ∈ Word 𝑈)
28		wrdf 14509	. . . . . . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
32	30, 31	eleqtrd 2831	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
33	29, 32	ffvelcdmd 7100	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑈)
34	33	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑊‘𝐼) ∈ 𝑈)
35		simprl 769	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑚 ∈ ℤ)
36		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → 𝑀 ∈ ℤ)
37	22, 23	unitgrpbas 20328	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Base‘𝐻)
38		dchrpt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐻)
39		dchrpt.m	. . . . . . . . . . . . . . 15 ⊢ · = (._g‘𝐻)
40		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
41	37, 38, 39, 40	odcong 19511	. . . . . . . . . . . . . 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 12364	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
45		2re 12324	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
46		dchrpt.b	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐵 = (Base‘𝑍)
47	19, 46	znfi 21500	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
48	17, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ Fin)
49	46, 22	unitss 20322	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 ⊆ 𝐵
50		ssfi 9204	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
51	48, 49, 50	sylancl 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈 ∈ Fin)
52	37, 38	odcl2 19527	. . . . . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
55		nndivre 12291	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ ℝ ∧ (𝑂‘(𝑊‘𝐼)) ∈ ℕ) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
56	45, 54, 55	sylancr 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℝ)
57	56	recnd 11280	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
58		cxpcl 26628	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1 ∈ ℂ ∧ (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
59	44, 57, 58	sylancr 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ∈ ℂ)
60	43, 59	eqeltrid 2833	. . . . . . . . . . . . . . . 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 12366	. . . . . . . . . . . . . . . . . . 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 26667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
65	43	neeq1i 3002	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ≠ 0 ↔ (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
66	64, 65	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑇 ≠ 0)
67		zsubcl 12642	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑚 − 𝑀) ∈ ℤ)
68	67	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑚 − 𝑀) ∈ ℤ)
69	36	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℤ)
70		expaddz 14111	. . . . . . . . . . . . . . . 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 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℤ)
73	72	zcnd 12705	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℂ)
74	69	zcnd 12705	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℂ)
75	73, 74	npcand 11613	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑚 − 𝑀) + 𝑀) = 𝑚)
76	75	oveq2d 7442	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = (𝑇↑𝑚))
77	43	oveq1i 7436	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇↑(𝑚 − 𝑀)) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀))
78		root1eq1 26710	. . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1 ↔ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)))
80	79	biimpar 476	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀)) = 1)
81	77, 80	eqtrid 2780	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑(𝑚 − 𝑀)) = 1)
82	81	oveq1d 7441	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (1 · (𝑇↑𝑀)))
83	60, 66, 69	expclzd 14155	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑀) ∈ ℂ)
84	83	mullidd 11270	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (1 · (𝑇↑𝑀)) = (𝑇↑𝑀))
85	82, 84	eqtrd 2768	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (𝑇↑𝑀))
86	71, 76, 85	3eqtr3d 2776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑚) = (𝑇↑𝑀))
87	86	ex 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → ((𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀) → (𝑇↑𝑚) = (𝑇↑𝑀)))
88	42, 87	sylbird 259	. . . . . . . . . . . 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 2739	. . . . . . . . 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 228	. . . . . . . 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 452	. . . . . . 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 3152	. . . . . 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 7433	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
96	95	eqeq2d 2739	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
97		oveq2 7434	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑇↑𝑚) = (𝑇↑𝑀))
98	97	eqeq2d 2739	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (ℎ = (𝑇↑𝑚) ↔ ℎ = (𝑇↑𝑀)))
99	96, 98	anbi12d 630	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → ((((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)) ↔ (((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑀))))
100	99	rspcev 3611	. . . . . . . 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 455	. . . . . . 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 480	. . . . . 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 211	. . . . 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 479	. . . 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 6536	. . 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 689	. 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 2768	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 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 Vcvv 3473 ⊆ wss 3949 class class class wbr 5152 ↦ cmpt 5235 dom cdm 5682 ran crn 5683 ℩cio 6503 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 Fincfn 8970 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 − cmin 11482 -cneg 11483 / cdiv 11909 ℕcn 12250 2c2 12305 ℕ₀cn0 12510 ℤcz 12596 ..^cfzo 13667 ↑cexp 14066 ♯chash 14329 Word cword 14504 ∥ cdvds 16238 Basecbs 17187 ↾_s cress 17216 0_gc0g 17428 Grpcgrp 18897 ._gcmg 19030 odcod 19486 DProd cdprd 19957 dProjcdpj 19958 mulGrpcmgp 20081 1_rcur 20128 Ringcrg 20180 CRingccrg 20181 Unitcui 20301 ℤ/nℤczn 21435 ↑_𝑐ccxp 26509 DChrcdchr 27185

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 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 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ioc 13369 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-word 14505 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 df-sin 16053 df-cos 16054 df-pi 16056 df-dvds 16239 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-qus 17498 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-submnd 18748 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-nsg 19086 df-eqg 19087 df-ghm 19175 df-cntz 19275 df-od 19490 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-lidl 21111 df-rsp 21112 df-2idl 21151 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-zn 21439 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-lp 23060 df-perf 23061 df-cn 23151 df-cnp 23152 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24818 df-limc 25815 df-dv 25816 df-log 26510 df-cxp 26511

This theorem is referenced by: dchrptlem2 27218

W3C validator