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Theorem dchrptlem1 27244

Description: Lemma for dchrpt 27247. (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 6895	. . . . . . 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 3166	. . . . 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 6525	. . . 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 6514	. . . 4 ⊢ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚))) ∈ V
7	4, 5, 6	fvmpt3i 7001	. . 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 7446	. . 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 2771	. . . . . . . . . . 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 12570	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
19		dchrpt.z	. . . . . . . . . . . . . . . . 17 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
20	19	zncrng 21517	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
21		crngring 20210	. . . . . . . . . . . . . . . 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 20351	. . . . . . . . . . . . . . . 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 14539	. . . . . . . . . . . . . . . . 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 6726	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
32	30, 31	eleqtrd 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
33	29, 32	ffvelcdmd 7085	. . . . . . . . . . . . . . 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 20350	. . . . . . . . . . . . . . 15 ⊢ 𝑈 = (Base‘𝐻)
38		dchrpt.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = (od‘𝐻)
39		dchrpt.m	. . . . . . . . . . . . . . 15 ⊢ · = (._g‘𝐻)
40		eqid 2734	. . . . . . . . . . . . . . 15 ⊢ (0_g‘𝐻) = (0_g‘𝐻)
41	37, 38, 39, 40	odcong 19535	. . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 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 12362	. . . . . . . . . . . . . . . . . 18 ⊢ -1 ∈ ℂ
45		2re 12322	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ ℝ
46		dchrpt.b	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐵 = (Base‘𝑍)
47	19, 46	znfi 21532	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin)
48	17, 47	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ Fin)
49	46, 22	unitss 20344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑈 ⊆ 𝐵
50		ssfi 9195	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵) → 𝑈 ∈ Fin)
51	48, 49, 50	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑈 ∈ Fin)
52	37, 38	odcl2 19551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ (𝑊‘𝐼) ∈ 𝑈) → (𝑂‘(𝑊‘𝐼)) ∈ ℕ)
53	25, 51, 33, 52	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . 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 12289	. . . . . . . . . . . . . . . . . . . 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 11271	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (2 / (𝑂‘(𝑊‘𝐼))) ∈ ℂ)
58		cxpcl 26652	. . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . 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 12364	. . . . . . . . . . . . . . . . . . 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 26691	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼)))) ≠ 0)
65	43	neeq1i 2995	. . . . . . . . . . . . . . . . 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 12642	. . . . . . . . . . . . . . . . 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 14129	. . . . . . . . . . . . . . . 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 12706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑚 ∈ ℂ)
74	69	zcnd 12706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → 𝑀 ∈ ℂ)
75	73, 74	npcand 11606	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑚 − 𝑀) + 𝑀) = 𝑚)
76	75	oveq2d 7429	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑((𝑚 − 𝑀) + 𝑀)) = (𝑇↑𝑚))
77	43	oveq1i 7423	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇↑(𝑚 − 𝑀)) = ((-1↑_𝑐(2 / (𝑂‘(𝑊‘𝐼))))↑(𝑚 − 𝑀))
78		root1eq1 26734	. . . . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑(𝑚 − 𝑀)) = 1)
82	81	oveq1d 7428	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (1 · (𝑇↑𝑀)))
83	60, 66, 69	expclzd 14173	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (𝑇↑𝑀) ∈ ℂ)
84	83	mullidd 11261	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → (1 · (𝑇↑𝑀)) = (𝑇↑𝑀))
85	82, 84	eqtrd 2769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ)) ∧ (𝑂‘(𝑊‘𝐼)) ∥ (𝑚 − 𝑀)) → ((𝑇↑(𝑚 − 𝑀)) · (𝑇↑𝑀)) = (𝑇↑𝑀))
86	71, 76, 85	3eqtr3d 2777	. . . . . . . . . . . . . 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 2745	. . . . . . . . 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 3142	. . . . . 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 7420	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑚 · (𝑊‘𝐼)) = (𝑀 · (𝑊‘𝐼)))
96	95	eqeq2d 2745	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (((𝑃‘𝐼)‘𝐶) = (𝑚 · (𝑊‘𝐼)) ↔ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼))))
97		oveq2 7421	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (𝑇↑𝑚) = (𝑇↑𝑀))
98	97	eqeq2d 2745	. . . . . . . . . 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 3605	. . . . . . . 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 6524	. . 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 2769	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 1539 ∈ wcel 2107 ≠ wne 2931 ∃wrex 3059 Vcvv 3463 ⊆ wss 3931 class class class wbr 5123 ↦ cmpt 5205 dom cdm 5665 ran crn 5666 ℩cio 6492 ⟶wf 6537 ‘cfv 6541 (class class class)co 7413 Fincfn 8967 ℂcc 11135 ℝcr 11136 0cc0 11137 1c1 11138 + caddc 11140 · cmul 11142 − cmin 11474 -cneg 11475 / cdiv 11902 ℕcn 12248 2c2 12303 ℕ₀cn0 12509 ℤcz 12596 ..^cfzo 13676 ↑cexp 14084 ♯chash 14351 Word cword 14534 ∥ cdvds 16272 Basecbs 17229 ↾_s cress 17252 0_gc0g 17455 Grpcgrp 18920 ._gcmg 19054 odcod 19510 DProd cdprd 19981 dProjcdpj 19982 mulGrpcmgp 20105 1_rcur 20146 Ringcrg 20198 CRingccrg 20199 Unitcui 20323 ℤ/nℤczn 21475 ↑_𝑐ccxp 26533 DChrcdchr 27212

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-inf2 9663 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215 ax-addf 11216 ax-mulf 11217

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-supp 8168 df-tpos 8233 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-oadd 8492 df-omul 8493 df-er 8727 df-ec 8729 df-qs 8733 df-map 8850 df-pm 8851 df-ixp 8920 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-fsupp 9384 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9532 df-card 9961 df-acn 9964 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12510 df-z 12597 df-dec 12717 df-uz 12861 df-q 12973 df-rp 13017 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-ioo 13373 df-ioc 13374 df-ico 13375 df-icc 13376 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14295 df-bc 14324 df-hash 14352 df-word 14535 df-shft 15088 df-cj 15120 df-re 15121 df-im 15122 df-sqrt 15256 df-abs 15257 df-limsup 15489 df-clim 15506 df-rlim 15507 df-sum 15705 df-ef 16085 df-sin 16087 df-cos 16088 df-pi 16090 df-dvds 16273 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17230 df-ress 17253 df-plusg 17286 df-mulr 17287 df-starv 17288 df-sca 17289 df-vsca 17290 df-ip 17291 df-tset 17292 df-ple 17293 df-ds 17295 df-unif 17296 df-hom 17297 df-cco 17298 df-rest 17438 df-topn 17439 df-0g 17457 df-gsum 17458 df-topgen 17459 df-pt 17460 df-prds 17463 df-xrs 17518 df-qtop 17523 df-imas 17524 df-qus 17525 df-xps 17526 df-mre 17600 df-mrc 17601 df-acs 17603 df-mgm 18622 df-sgrp 18701 df-mnd 18717 df-mhm 18765 df-submnd 18766 df-grp 18923 df-minusg 18924 df-sbg 18925 df-mulg 19055 df-subg 19110 df-nsg 19111 df-eqg 19112 df-ghm 19200 df-cntz 19304 df-od 19514 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20325 df-unit 20326 df-rhm 20440 df-subrng 20514 df-subrg 20538 df-lmod 20828 df-lss 20898 df-lsp 20938 df-sra 21140 df-rgmod 21141 df-lidl 21180 df-rsp 21181 df-2idl 21222 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-zring 21420 df-zrh 21476 df-zn 21479 df-top 22848 df-topon 22865 df-topsp 22887 df-bases 22900 df-cld 22973 df-ntr 22974 df-cls 22975 df-nei 23052 df-lp 23090 df-perf 23091 df-cn 23181 df-cnp 23182 df-haus 23269 df-tx 23516 df-hmeo 23709 df-fil 23800 df-fm 23892 df-flim 23893 df-flf 23894 df-xms 24275 df-ms 24276 df-tms 24277 df-cncf 24840 df-limc 25837 df-dv 25838 df-log 26534 df-cxp 26535

This theorem is referenced by: dchrptlem2 27245

W3C validator