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
|
ffvelrnd |
⊢ ( 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
|
mulid2d |
⊢ ( ( ( 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 ) ) |