Theorem dchrptlem1

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

	Ref	Expression
Hypotheses	dchrpt.g	⊢ 𝐺 = ( DChr ‘ 𝑁 )
	dchrpt.z	⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
	dchrpt.d	⊢ 𝐷 = ( Base ‘ 𝐺 )
	dchrpt.b	⊢ 𝐵 = ( Base ‘ 𝑍 )
	dchrpt.1	⊢ 1 = ( 1r ‘ 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 → 𝐻 dom DProd 𝑆 )
	dchrpt.3	⊢ ( 𝜑 → ( 𝐻 DProd 𝑆 ) = 𝑈 )
	dchrpt.p	⊢ 𝑃 = ( 𝐻 dProj 𝑆 )
	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	dchrptlem1	⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) → ( 𝑋 ‘ 𝐶 ) = ( 𝑇 ↑ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		dchrpt.g	⊢ 𝐺 = ( DChr ‘ 𝑁 )
2		dchrpt.z	⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 )
3		dchrpt.d	⊢ 𝐷 = ( Base ‘ 𝐺 )
4		dchrpt.b	⊢ 𝐵 = ( Base ‘ 𝑍 )
5		dchrpt.1	⊢ 1 = ( 1r ‘ 𝑍 )
6		dchrpt.n	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
7		dchrpt.n1	⊢ ( 𝜑 → 𝐴 ≠ 1 )
8		dchrpt.u	⊢ 𝑈 = ( Unit ‘ 𝑍 )
9		dchrpt.h	⊢ 𝐻 = ( ( mulGrp ‘ 𝑍 ) ↾s 𝑈 )
10		dchrpt.m	⊢ · = ( .g ‘ 𝐻 )
11		dchrpt.s	⊢ 𝑆 = ( 𝑘 ∈ dom 𝑊 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑊 ‘ 𝑘 ) ) ) )
12		dchrpt.au	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
13		dchrpt.w	⊢ ( 𝜑 → 𝑊 ∈ Word 𝑈 )
14		dchrpt.2	⊢ ( 𝜑 → 𝐻 dom DProd 𝑆 )
15		dchrpt.3	⊢ ( 𝜑 → ( 𝐻 DProd 𝑆 ) = 𝑈 )
16		dchrpt.p	⊢ 𝑃 = ( 𝐻 dProj 𝑆 )
17		dchrpt.o	⊢ 𝑂 = ( od ‘ 𝐻 )
18		dchrpt.t	⊢ 𝑇 = ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) )
19		dchrpt.i	⊢ ( 𝜑 → 𝐼 ∈ dom 𝑊 )
20		dchrpt.4	⊢ ( 𝜑 → ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐴 ) ≠ 1 )
21		dchrpt.5	⊢ 𝑋 = ( 𝑢 ∈ 𝑈 ↦ ( ℩ ℎ ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝑢 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) )
22		fveqeq2	⊢ ( 𝑢 = 𝐶 → ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝑢 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ↔ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ) )
23	22	anbi1d	⊢ ( 𝑢 = 𝐶 → ( ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝑢 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ↔ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) )
24	23	rexbidv	⊢ ( 𝑢 = 𝐶 → ( ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝑢 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) )
25	24	iotabidv	⊢ ( 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 ) ) ) )
26		iotaex	⊢ ( ℩ ℎ ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝑢 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) ∈ V
27	25 21 26	fvmpt3i	⊢ ( 𝐶 ∈ 𝑈 → ( 𝑋 ‘ 𝐶 ) = ( ℩ ℎ ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) )
28	27	ad2antlr	⊢ ( ( ( 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 ) ) ) )
29		ovex	⊢ ( 𝑇 ↑ 𝑀 ) ∈ V
30		simpr	⊢ ( ( ( ( ( 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 ) ) )
31		simpllr	⊢ ( ( ( ( ( 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 ) ) ) )
32	31	simprd	⊢ ( ( ( ( ( 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 ) ) )
33	30 32	eqtr3d	⊢ ( ( ( ( ( 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 ) ) )
34		simp-4l	⊢ ( ( ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ) → 𝜑 )
35		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ) → 𝑚 ∈ ℤ )
36	31	simpld	⊢ ( ( ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ) → 𝑀 ∈ ℤ )
37	6	nnnn0d	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
38	2	zncrng	⊢ ( 𝑁 ∈ ℕ0 → 𝑍 ∈ CRing )
39		crngring	⊢ ( 𝑍 ∈ CRing → 𝑍 ∈ Ring )
40	8 9	unitgrp	⊢ ( 𝑍 ∈ Ring → 𝐻 ∈ Grp )
41	37 38 39 40	4syl	⊢ ( 𝜑 → 𝐻 ∈ Grp )
42	41	adantr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → 𝐻 ∈ Grp )
43		wrdf	⊢ ( 𝑊 ∈ Word 𝑈 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑈 )
44	13 43	syl	⊢ ( 𝜑 → 𝑊 : ( 0 ..^ ( ♯ ‘ 𝑊 ) ) ⟶ 𝑈 )
45	44	fdmd	⊢ ( 𝜑 → dom 𝑊 = ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
46	19 45	eleqtrd	⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ ( ♯ ‘ 𝑊 ) ) )
47	44 46	ffvelcdmd	⊢ ( 𝜑 → ( 𝑊 ‘ 𝐼 ) ∈ 𝑈 )
48	47	adantr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( 𝑊 ‘ 𝐼 ) ∈ 𝑈 )
49		simprl	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → 𝑚 ∈ ℤ )
50		simprr	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → 𝑀 ∈ ℤ )
51	8 9	unitgrpbas	⊢ 𝑈 = ( Base ‘ 𝐻 )
52		eqid	⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 )
53	51 17 10 52	odcong	⊢ ( ( 𝐻 ∈ Grp ∧ ( 𝑊 ‘ 𝐼 ) ∈ 𝑈 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ↔ ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) )
54	42 48 49 50 53	syl112anc	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ↔ ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) )
55		neg1cn	⊢ - 1 ∈ ℂ
56		2re	⊢ 2 ∈ ℝ
57	2 4	znfi	⊢ ( 𝑁 ∈ ℕ → 𝐵 ∈ Fin )
58	6 57	syl	⊢ ( 𝜑 → 𝐵 ∈ Fin )
59	4 8	unitss	⊢ 𝑈 ⊆ 𝐵
60		ssfi	⊢ ( ( 𝐵 ∈ Fin ∧ 𝑈 ⊆ 𝐵 ) → 𝑈 ∈ Fin )
61	58 59 60	sylancl	⊢ ( 𝜑 → 𝑈 ∈ Fin )
62	51 17	odcl2	⊢ ( ( 𝐻 ∈ Grp ∧ 𝑈 ∈ Fin ∧ ( 𝑊 ‘ 𝐼 ) ∈ 𝑈 ) → ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∈ ℕ )
63	41 61 47 62	syl3anc	⊢ ( 𝜑 → ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∈ ℕ )
64	63	ad2antrr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∈ ℕ )
65		nndivre	⊢ ( ( 2 ∈ ℝ ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∈ ℕ ) → ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ∈ ℝ )
66	56 64 65	sylancr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ∈ ℝ )
67	66	recnd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ∈ ℂ )
68		cxpcl	⊢ ( ( - 1 ∈ ℂ ∧ ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ∈ ℂ ) → ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ∈ ℂ )
69	55 67 68	sylancr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ∈ ℂ )
70	18 69	eqeltrid	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑇 ∈ ℂ )
71	55	a1i	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → - 1 ∈ ℂ )
72		neg1ne0	⊢ - 1 ≠ 0
73	72	a1i	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → - 1 ≠ 0 )
74	71 73 67	cxpne0d	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ≠ 0 )
75	18	neeq1i	⊢ ( 𝑇 ≠ 0 ↔ ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ≠ 0 )
76	74 75	sylibr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑇 ≠ 0 )
77		zsubcl	⊢ ( ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑚 − 𝑀 ) ∈ ℤ )
78	77	ad2antlr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑚 − 𝑀 ) ∈ ℤ )
79	50	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑀 ∈ ℤ )
80		expaddz	⊢ ( ( ( 𝑇 ∈ ℂ ∧ 𝑇 ≠ 0 ) ∧ ( ( 𝑚 − 𝑀 ) ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( 𝑇 ↑ ( ( 𝑚 − 𝑀 ) + 𝑀 ) ) = ( ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) · ( 𝑇 ↑ 𝑀 ) ) )
81	70 76 78 79 80	syl22anc	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑇 ↑ ( ( 𝑚 − 𝑀 ) + 𝑀 ) ) = ( ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) · ( 𝑇 ↑ 𝑀 ) ) )
82	49	adantr	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑚 ∈ ℤ )
83	82	zcnd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑚 ∈ ℂ )
84	79	zcnd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → 𝑀 ∈ ℂ )
85	83 84	npcand	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( ( 𝑚 − 𝑀 ) + 𝑀 ) = 𝑚 )
86	85	oveq2d	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑇 ↑ ( ( 𝑚 − 𝑀 ) + 𝑀 ) ) = ( 𝑇 ↑ 𝑚 ) )
87	18	oveq1i	⊢ ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) = ( ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ↑ ( 𝑚 − 𝑀 ) )
88		root1eq1	⊢ ( ( ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∈ ℕ ∧ ( 𝑚 − 𝑀 ) ∈ ℤ ) → ( ( ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ↑ ( 𝑚 − 𝑀 ) ) = 1 ↔ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) )
89	63 77 88	syl2an	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( ( ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ↑ ( 𝑚 − 𝑀 ) ) = 1 ↔ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) )
90	89	biimpar	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( ( - 1 ↑𝑐 ( 2 / ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ) ) ↑ ( 𝑚 − 𝑀 ) ) = 1 )
91	87 90	eqtrid	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) = 1 )
92	91	oveq1d	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) · ( 𝑇 ↑ 𝑀 ) ) = ( 1 · ( 𝑇 ↑ 𝑀 ) ) )
93	70 76 79	expclzd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑇 ↑ 𝑀 ) ∈ ℂ )
94	93	mullidd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 1 · ( 𝑇 ↑ 𝑀 ) ) = ( 𝑇 ↑ 𝑀 ) )
95	92 94	eqtrd	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( ( 𝑇 ↑ ( 𝑚 − 𝑀 ) ) · ( 𝑇 ↑ 𝑀 ) ) = ( 𝑇 ↑ 𝑀 ) )
96	81 86 95	3eqtr3d	⊢ ( ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) ∧ ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) ) → ( 𝑇 ↑ 𝑚 ) = ( 𝑇 ↑ 𝑀 ) )
97	96	ex	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( ( 𝑂 ‘ ( 𝑊 ‘ 𝐼 ) ) ∥ ( 𝑚 − 𝑀 ) → ( 𝑇 ↑ 𝑚 ) = ( 𝑇 ↑ 𝑀 ) ) )
98	54 97	sylbird	⊢ ( ( 𝜑 ∧ ( 𝑚 ∈ ℤ ∧ 𝑀 ∈ ℤ ) ) → ( ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) → ( 𝑇 ↑ 𝑚 ) = ( 𝑇 ↑ 𝑀 ) ) )
99	34 35 36 98	syl12anc	⊢ ( ( ( ( ( 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 ) ) )
100	33 99	mpd	⊢ ( ( ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) ∧ 𝑚 ∈ ℤ ) ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ) → ( 𝑇 ↑ 𝑚 ) = ( 𝑇 ↑ 𝑀 ) )
101	100	eqeq2d	⊢ ( ( ( ( ( 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 ) ) )
102	101	biimpd	⊢ ( ( ( ( ( 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 ) ) )
103	102	expimpd	⊢ ( ( ( ( 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	rexlimdva	⊢ ( ( ( 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 ) ) )
105		oveq1	⊢ ( 𝑚 = 𝑀 → ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) )
106	105	eqeq2d	⊢ ( 𝑚 = 𝑀 → ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ↔ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) )
107		oveq2	⊢ ( 𝑚 = 𝑀 → ( 𝑇 ↑ 𝑚 ) = ( 𝑇 ↑ 𝑀 ) )
108	107	eqeq2d	⊢ ( 𝑚 = 𝑀 → ( ℎ = ( 𝑇 ↑ 𝑚 ) ↔ ℎ = ( 𝑇 ↑ 𝑀 ) ) )
109	106 108	anbi12d	⊢ ( 𝑚 = 𝑀 → ( ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ↔ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑀 ) ) ) )
110	109	rspcev	⊢ ( ( 𝑀 ∈ ℤ ∧ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑀 ) ) ) → ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) )
111	110	expr	⊢ ( ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) → ( ℎ = ( 𝑇 ↑ 𝑀 ) → ∃ 𝑚 ∈ ℤ ( ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑚 · ( 𝑊 ‘ 𝐼 ) ) ∧ ℎ = ( 𝑇 ↑ 𝑚 ) ) ) )
112	111	adantl	⊢ ( ( ( 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 ) ) ) )
113	104 112	impbid	⊢ ( ( ( 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 ) ) )
114	113	adantr	⊢ ( ( ( ( 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 ) ) )
115	114	iota5	⊢ ( ( ( ( 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 ) )
116	29 115	mpan2	⊢ ( ( ( 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 ) )
117	28 116	eqtrd	⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ 𝑈 ) ∧ ( 𝑀 ∈ ℤ ∧ ( ( 𝑃 ‘ 𝐼 ) ‘ 𝐶 ) = ( 𝑀 · ( 𝑊 ‘ 𝐼 ) ) ) ) → ( 𝑋 ‘ 𝐶 ) = ( 𝑇 ↑ 𝑀 ) )