Step |
Hyp |
Ref |
Expression |
1 |
|
pgpfac.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
pgpfac.c |
⊢ 𝐶 = { 𝑟 ∈ ( SubGrp ‘ 𝐺 ) ∣ ( 𝐺 ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } |
3 |
|
pgpfac.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
4 |
|
pgpfac.p |
⊢ ( 𝜑 → 𝑃 pGrp 𝐺 ) |
5 |
|
pgpfac.f |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
6 |
|
pgpfac.u |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
7 |
|
pgpfac.a |
⊢ ( 𝜑 → ∀ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( 𝑡 ⊊ 𝑈 → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑡 ) ) ) |
8 |
|
pgpfac.h |
⊢ 𝐻 = ( 𝐺 ↾s 𝑈 ) |
9 |
|
pgpfac.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐻 ) ) |
10 |
|
pgpfac.o |
⊢ 𝑂 = ( od ‘ 𝐻 ) |
11 |
|
pgpfac.e |
⊢ 𝐸 = ( gEx ‘ 𝐻 ) |
12 |
|
pgpfac.0 |
⊢ 0 = ( 0g ‘ 𝐻 ) |
13 |
|
pgpfac.l |
⊢ ⊕ = ( LSSum ‘ 𝐻 ) |
14 |
|
pgpfac.1 |
⊢ ( 𝜑 → 𝐸 ≠ 1 ) |
15 |
|
pgpfac.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑈 ) |
16 |
|
pgpfac.oe |
⊢ ( 𝜑 → ( 𝑂 ‘ 𝑋 ) = 𝐸 ) |
17 |
|
pgpfac.w |
⊢ ( 𝜑 → 𝑊 ∈ ( SubGrp ‘ 𝐻 ) ) |
18 |
|
pgpfac.i |
⊢ ( 𝜑 → ( ( 𝐾 ‘ { 𝑋 } ) ∩ 𝑊 ) = { 0 } ) |
19 |
|
pgpfac.s |
⊢ ( 𝜑 → ( ( 𝐾 ‘ { 𝑋 } ) ⊕ 𝑊 ) = 𝑈 ) |
20 |
|
pgpfac.2 |
⊢ ( 𝜑 → 𝑆 ∈ Word 𝐶 ) |
21 |
|
pgpfac.4 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
22 |
|
pgpfac.5 |
⊢ ( 𝜑 → ( 𝐺 DProd 𝑆 ) = 𝑊 ) |
23 |
|
pgpfac.t |
⊢ 𝑇 = ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) |
24 |
8
|
subggrp |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝐻 ∈ Grp ) |
25 |
6 24
|
syl |
⊢ ( 𝜑 → 𝐻 ∈ Grp ) |
26 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
27 |
26
|
subgacs |
⊢ ( 𝐻 ∈ Grp → ( SubGrp ‘ 𝐻 ) ∈ ( ACS ‘ ( Base ‘ 𝐻 ) ) ) |
28 |
25 27
|
syl |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐻 ) ∈ ( ACS ‘ ( Base ‘ 𝐻 ) ) ) |
29 |
28
|
acsmred |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ) |
30 |
8
|
subgbas |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 = ( Base ‘ 𝐻 ) ) |
31 |
6 30
|
syl |
⊢ ( 𝜑 → 𝑈 = ( Base ‘ 𝐻 ) ) |
32 |
15 31
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ 𝐻 ) ) |
33 |
9
|
mrcsncl |
⊢ ( ( ( SubGrp ‘ 𝐻 ) ∈ ( Moore ‘ ( Base ‘ 𝐻 ) ) ∧ 𝑋 ∈ ( Base ‘ 𝐻 ) ) → ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐻 ) ) |
34 |
29 32 33
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐻 ) ) |
35 |
8
|
subsubg |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) ) |
36 |
6 35
|
syl |
⊢ ( 𝜑 → ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐻 ) ↔ ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) ) |
37 |
34 36
|
mpbid |
⊢ ( 𝜑 → ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ) ) |
38 |
37
|
simpld |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
39 |
8
|
oveq1i |
⊢ ( 𝐻 ↾s ( 𝐾 ‘ { 𝑋 } ) ) = ( ( 𝐺 ↾s 𝑈 ) ↾s ( 𝐾 ‘ { 𝑋 } ) ) |
40 |
37
|
simprd |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ) |
41 |
|
ressabs |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ) → ( ( 𝐺 ↾s 𝑈 ) ↾s ( 𝐾 ‘ { 𝑋 } ) ) = ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
42 |
6 40 41
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐺 ↾s 𝑈 ) ↾s ( 𝐾 ‘ { 𝑋 } ) ) = ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
43 |
39 42
|
syl5eq |
⊢ ( 𝜑 → ( 𝐻 ↾s ( 𝐾 ‘ { 𝑋 } ) ) = ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
44 |
26 9
|
cycsubgcyg2 |
⊢ ( ( 𝐻 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝐻 ) ) → ( 𝐻 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ CycGrp ) |
45 |
25 32 44
|
syl2anc |
⊢ ( 𝜑 → ( 𝐻 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ CycGrp ) |
46 |
43 45
|
eqeltrrd |
⊢ ( 𝜑 → ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ CycGrp ) |
47 |
|
pgpprm |
⊢ ( 𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ ) |
48 |
4 47
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
49 |
|
subgpgp |
⊢ ( ( 𝑃 pGrp 𝐺 ∧ ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑃 pGrp ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
50 |
4 38 49
|
syl2anc |
⊢ ( 𝜑 → 𝑃 pGrp ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
51 |
|
brelrng |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ CycGrp ∧ 𝑃 pGrp ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) → ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ ran pGrp ) |
52 |
48 46 50 51
|
syl3anc |
⊢ ( 𝜑 → ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ ran pGrp ) |
53 |
46 52
|
elind |
⊢ ( 𝜑 → ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ ( CycGrp ∩ ran pGrp ) ) |
54 |
|
oveq2 |
⊢ ( 𝑟 = ( 𝐾 ‘ { 𝑋 } ) → ( 𝐺 ↾s 𝑟 ) = ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ) |
55 |
54
|
eleq1d |
⊢ ( 𝑟 = ( 𝐾 ‘ { 𝑋 } ) → ( ( 𝐺 ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) ↔ ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
56 |
55 2
|
elrab2 |
⊢ ( ( 𝐾 ‘ { 𝑋 } ) ∈ 𝐶 ↔ ( ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 ↾s ( 𝐾 ‘ { 𝑋 } ) ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
57 |
38 53 56
|
sylanbrc |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝑋 } ) ∈ 𝐶 ) |
58 |
23 20 57
|
cats1cld |
⊢ ( 𝜑 → 𝑇 ∈ Word 𝐶 ) |
59 |
|
wrdf |
⊢ ( 𝑇 ∈ Word 𝐶 → 𝑇 : ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⟶ 𝐶 ) |
60 |
58 59
|
syl |
⊢ ( 𝜑 → 𝑇 : ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⟶ 𝐶 ) |
61 |
2
|
ssrab3 |
⊢ 𝐶 ⊆ ( SubGrp ‘ 𝐺 ) |
62 |
|
fss |
⊢ ( ( 𝑇 : ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⟶ 𝐶 ∧ 𝐶 ⊆ ( SubGrp ‘ 𝐺 ) ) → 𝑇 : ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⟶ ( SubGrp ‘ 𝐺 ) ) |
63 |
60 61 62
|
sylancl |
⊢ ( 𝜑 → 𝑇 : ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ⟶ ( SubGrp ‘ 𝐺 ) ) |
64 |
|
lencl |
⊢ ( 𝑆 ∈ Word 𝐶 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
65 |
20 64
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ℕ0 ) |
66 |
65
|
nn0zd |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ℤ ) |
67 |
|
fzosn |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ℤ → ( ( ♯ ‘ 𝑆 ) ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) = { ( ♯ ‘ 𝑆 ) } ) |
68 |
66 67
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑆 ) ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) = { ( ♯ ‘ 𝑆 ) } ) |
69 |
68
|
ineq2d |
⊢ ( 𝜑 → ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∩ ( ( ♯ ‘ 𝑆 ) ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) ) = ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∩ { ( ♯ ‘ 𝑆 ) } ) ) |
70 |
|
fzodisj |
⊢ ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∩ ( ( ♯ ‘ 𝑆 ) ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) ) = ∅ |
71 |
69 70
|
eqtr3di |
⊢ ( 𝜑 → ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∩ { ( ♯ ‘ 𝑆 ) } ) = ∅ ) |
72 |
23
|
fveq2i |
⊢ ( ♯ ‘ 𝑇 ) = ( ♯ ‘ ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) |
73 |
57
|
s1cld |
⊢ ( 𝜑 → 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ∈ Word 𝐶 ) |
74 |
|
ccatlen |
⊢ ( ( 𝑆 ∈ Word 𝐶 ∧ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ∈ Word 𝐶 ) → ( ♯ ‘ ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ) |
75 |
20 73 74
|
syl2anc |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ) |
76 |
72 75
|
syl5eq |
⊢ ( 𝜑 → ( ♯ ‘ 𝑇 ) = ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ) |
77 |
|
s1len |
⊢ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) = 1 |
78 |
77
|
oveq2i |
⊢ ( ( ♯ ‘ 𝑆 ) + ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) = ( ( ♯ ‘ 𝑆 ) + 1 ) |
79 |
76 78
|
eqtrdi |
⊢ ( 𝜑 → ( ♯ ‘ 𝑇 ) = ( ( ♯ ‘ 𝑆 ) + 1 ) ) |
80 |
79
|
oveq2d |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( 0 ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) ) |
81 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
82 |
65 81
|
eleqtrdi |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 0 ) ) |
83 |
|
fzosplitsn |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ( ℤ≥ ‘ 0 ) → ( 0 ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) = ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) ) |
84 |
82 83
|
syl |
⊢ ( 𝜑 → ( 0 ..^ ( ( ♯ ‘ 𝑆 ) + 1 ) ) = ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) ) |
85 |
80 84
|
eqtrd |
⊢ ( 𝜑 → ( 0 ..^ ( ♯ ‘ 𝑇 ) ) = ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) ) |
86 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
87 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
88 |
|
cats1un |
⊢ ( ( 𝑆 ∈ Word 𝐶 ∧ ( 𝐾 ‘ { 𝑋 } ) ∈ 𝐶 ) → ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) = ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ) |
89 |
20 57 88
|
syl2anc |
⊢ ( 𝜑 → ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) = ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ) |
90 |
23 89
|
syl5eq |
⊢ ( 𝜑 → 𝑇 = ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ) |
91 |
90
|
reseq1d |
⊢ ( 𝜑 → ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) = ( ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) |
92 |
|
wrdfn |
⊢ ( 𝑆 ∈ Word 𝐶 → 𝑆 Fn ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
93 |
20 92
|
syl |
⊢ ( 𝜑 → 𝑆 Fn ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) |
94 |
|
fzonel |
⊢ ¬ ( ♯ ‘ 𝑆 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) |
95 |
|
fsnunres |
⊢ ( ( 𝑆 Fn ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∧ ¬ ( ♯ ‘ 𝑆 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) → ( ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) = 𝑆 ) |
96 |
93 94 95
|
sylancl |
⊢ ( 𝜑 → ( ( 𝑆 ∪ { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) = 𝑆 ) |
97 |
91 96
|
eqtrd |
⊢ ( 𝜑 → ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) = 𝑆 ) |
98 |
21 97
|
breqtrrd |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) |
99 |
|
fvex |
⊢ ( ♯ ‘ 𝑆 ) ∈ V |
100 |
|
dprdsn |
⊢ ( ( ( ♯ ‘ 𝑆 ) ∈ V ∧ ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝐺 dom DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ∧ ( 𝐺 DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) = ( 𝐾 ‘ { 𝑋 } ) ) ) |
101 |
99 38 100
|
sylancr |
⊢ ( 𝜑 → ( 𝐺 dom DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ∧ ( 𝐺 DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) = ( 𝐾 ‘ { 𝑋 } ) ) ) |
102 |
101
|
simpld |
⊢ ( 𝜑 → 𝐺 dom DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) |
103 |
|
wrdfn |
⊢ ( 𝑇 ∈ Word 𝐶 → 𝑇 Fn ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) |
104 |
58 103
|
syl |
⊢ ( 𝜑 → 𝑇 Fn ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) |
105 |
|
ssun2 |
⊢ { ( ♯ ‘ 𝑆 ) } ⊆ ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) |
106 |
99
|
snss |
⊢ ( ( ♯ ‘ 𝑆 ) ∈ ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) ↔ { ( ♯ ‘ 𝑆 ) } ⊆ ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) ) |
107 |
105 106
|
mpbir |
⊢ ( ♯ ‘ 𝑆 ) ∈ ( ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ∪ { ( ♯ ‘ 𝑆 ) } ) |
108 |
107 85
|
eleqtrrid |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) |
109 |
|
fnressn |
⊢ ( ( 𝑇 Fn ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ∧ ( ♯ ‘ 𝑆 ) ∈ ( 0 ..^ ( ♯ ‘ 𝑇 ) ) ) → ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) = { 〈 ( ♯ ‘ 𝑆 ) , ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) 〉 } ) |
110 |
104 108 109
|
syl2anc |
⊢ ( 𝜑 → ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) = { 〈 ( ♯ ‘ 𝑆 ) , ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) 〉 } ) |
111 |
23
|
fveq1i |
⊢ ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) = ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( ♯ ‘ 𝑆 ) ) |
112 |
65
|
nn0cnd |
⊢ ( 𝜑 → ( ♯ ‘ 𝑆 ) ∈ ℂ ) |
113 |
112
|
addid2d |
⊢ ( 𝜑 → ( 0 + ( ♯ ‘ 𝑆 ) ) = ( ♯ ‘ 𝑆 ) ) |
114 |
113
|
fveq2d |
⊢ ( 𝜑 → ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( 0 + ( ♯ ‘ 𝑆 ) ) ) = ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( ♯ ‘ 𝑆 ) ) ) |
115 |
111 114
|
eqtr4id |
⊢ ( 𝜑 → ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) = ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( 0 + ( ♯ ‘ 𝑆 ) ) ) ) |
116 |
|
1nn |
⊢ 1 ∈ ℕ |
117 |
77 116
|
eqeltri |
⊢ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ∈ ℕ |
118 |
|
lbfzo0 |
⊢ ( 0 ∈ ( 0 ..^ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ↔ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ∈ ℕ ) |
119 |
117 118
|
mpbir |
⊢ 0 ∈ ( 0 ..^ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) |
120 |
119
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ) |
121 |
|
ccatval3 |
⊢ ( ( 𝑆 ∈ Word 𝐶 ∧ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ∈ Word 𝐶 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ) ) → ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( 0 + ( ♯ ‘ 𝑆 ) ) ) = ( 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ‘ 0 ) ) |
122 |
20 73 120 121
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝑆 ++ 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ) ‘ ( 0 + ( ♯ ‘ 𝑆 ) ) ) = ( 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ‘ 0 ) ) |
123 |
|
fvex |
⊢ ( 𝐾 ‘ { 𝑋 } ) ∈ V |
124 |
|
s1fv |
⊢ ( ( 𝐾 ‘ { 𝑋 } ) ∈ V → ( 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ‘ 0 ) = ( 𝐾 ‘ { 𝑋 } ) ) |
125 |
123 124
|
mp1i |
⊢ ( 𝜑 → ( 〈“ ( 𝐾 ‘ { 𝑋 } ) ”〉 ‘ 0 ) = ( 𝐾 ‘ { 𝑋 } ) ) |
126 |
115 122 125
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) = ( 𝐾 ‘ { 𝑋 } ) ) |
127 |
126
|
opeq2d |
⊢ ( 𝜑 → 〈 ( ♯ ‘ 𝑆 ) , ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) 〉 = 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 ) |
128 |
127
|
sneqd |
⊢ ( 𝜑 → { 〈 ( ♯ ‘ 𝑆 ) , ( 𝑇 ‘ ( ♯ ‘ 𝑆 ) ) 〉 } = { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) |
129 |
110 128
|
eqtrd |
⊢ ( 𝜑 → ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) = { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) |
130 |
102 129
|
breqtrrd |
⊢ ( 𝜑 → 𝐺 dom DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) |
131 |
|
dprdsubg |
⊢ ( 𝐺 dom DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) → ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
132 |
98 131
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
133 |
|
dprdsubg |
⊢ ( 𝐺 dom DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) → ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
134 |
130 133
|
syl |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ∈ ( SubGrp ‘ 𝐺 ) ) |
135 |
86 3 132 134
|
ablcntzd |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) ) |
136 |
97
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) = ( 𝐺 DProd 𝑆 ) ) |
137 |
136 22
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) = 𝑊 ) |
138 |
129
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) = ( 𝐺 DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) ) |
139 |
101
|
simprd |
⊢ ( 𝜑 → ( 𝐺 DProd { 〈 ( ♯ ‘ 𝑆 ) , ( 𝐾 ‘ { 𝑋 } ) 〉 } ) = ( 𝐾 ‘ { 𝑋 } ) ) |
140 |
138 139
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) = ( 𝐾 ‘ { 𝑋 } ) ) |
141 |
137 140
|
ineq12d |
⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ∩ ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) = ( 𝑊 ∩ ( 𝐾 ‘ { 𝑋 } ) ) ) |
142 |
|
incom |
⊢ ( 𝑊 ∩ ( 𝐾 ‘ { 𝑋 } ) ) = ( ( 𝐾 ‘ { 𝑋 } ) ∩ 𝑊 ) |
143 |
141 142
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ∩ ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) = ( ( 𝐾 ‘ { 𝑋 } ) ∩ 𝑊 ) ) |
144 |
8 87
|
subg0 |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
145 |
6 144
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) ) |
146 |
145 12
|
eqtr4di |
⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) = 0 ) |
147 |
146
|
sneqd |
⊢ ( 𝜑 → { ( 0g ‘ 𝐺 ) } = { 0 } ) |
148 |
18 143 147
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ∩ ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) = { ( 0g ‘ 𝐺 ) } ) |
149 |
63 71 85 86 87 98 130 135 148
|
dmdprdsplit2 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑇 ) |
150 |
|
eqid |
⊢ ( LSSum ‘ 𝐺 ) = ( LSSum ‘ 𝐺 ) |
151 |
63 71 85 150 149
|
dprdsplit |
⊢ ( 𝜑 → ( 𝐺 DProd 𝑇 ) = ( ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) ) |
152 |
137 140
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝐺 DProd ( 𝑇 ↾ ( 0 ..^ ( ♯ ‘ 𝑆 ) ) ) ) ( LSSum ‘ 𝐺 ) ( 𝐺 DProd ( 𝑇 ↾ { ( ♯ ‘ 𝑆 ) } ) ) ) = ( 𝑊 ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ { 𝑋 } ) ) ) |
153 |
137 132
|
eqeltrrd |
⊢ ( 𝜑 → 𝑊 ∈ ( SubGrp ‘ 𝐺 ) ) |
154 |
150
|
lsmcom |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑊 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑊 ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ { 𝑋 } ) ) = ( ( 𝐾 ‘ { 𝑋 } ) ( LSSum ‘ 𝐺 ) 𝑊 ) ) |
155 |
3 153 38 154
|
syl3anc |
⊢ ( 𝜑 → ( 𝑊 ( LSSum ‘ 𝐺 ) ( 𝐾 ‘ { 𝑋 } ) ) = ( ( 𝐾 ‘ { 𝑋 } ) ( LSSum ‘ 𝐺 ) 𝑊 ) ) |
156 |
151 152 155
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐺 DProd 𝑇 ) = ( ( 𝐾 ‘ { 𝑋 } ) ( LSSum ‘ 𝐺 ) 𝑊 ) ) |
157 |
26
|
subgss |
⊢ ( 𝑊 ∈ ( SubGrp ‘ 𝐻 ) → 𝑊 ⊆ ( Base ‘ 𝐻 ) ) |
158 |
17 157
|
syl |
⊢ ( 𝜑 → 𝑊 ⊆ ( Base ‘ 𝐻 ) ) |
159 |
158 31
|
sseqtrrd |
⊢ ( 𝜑 → 𝑊 ⊆ 𝑈 ) |
160 |
8 150 13
|
subglsm |
⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐾 ‘ { 𝑋 } ) ⊆ 𝑈 ∧ 𝑊 ⊆ 𝑈 ) → ( ( 𝐾 ‘ { 𝑋 } ) ( LSSum ‘ 𝐺 ) 𝑊 ) = ( ( 𝐾 ‘ { 𝑋 } ) ⊕ 𝑊 ) ) |
161 |
6 40 159 160
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐾 ‘ { 𝑋 } ) ( LSSum ‘ 𝐺 ) 𝑊 ) = ( ( 𝐾 ‘ { 𝑋 } ) ⊕ 𝑊 ) ) |
162 |
156 161 19
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐺 DProd 𝑇 ) = 𝑈 ) |
163 |
|
breq2 |
⊢ ( 𝑠 = 𝑇 → ( 𝐺 dom DProd 𝑠 ↔ 𝐺 dom DProd 𝑇 ) ) |
164 |
|
oveq2 |
⊢ ( 𝑠 = 𝑇 → ( 𝐺 DProd 𝑠 ) = ( 𝐺 DProd 𝑇 ) ) |
165 |
164
|
eqeq1d |
⊢ ( 𝑠 = 𝑇 → ( ( 𝐺 DProd 𝑠 ) = 𝑈 ↔ ( 𝐺 DProd 𝑇 ) = 𝑈 ) ) |
166 |
163 165
|
anbi12d |
⊢ ( 𝑠 = 𝑇 → ( ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑈 ) ↔ ( 𝐺 dom DProd 𝑇 ∧ ( 𝐺 DProd 𝑇 ) = 𝑈 ) ) ) |
167 |
166
|
rspcev |
⊢ ( ( 𝑇 ∈ Word 𝐶 ∧ ( 𝐺 dom DProd 𝑇 ∧ ( 𝐺 DProd 𝑇 ) = 𝑈 ) ) → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑈 ) ) |
168 |
58 149 162 167
|
syl12anc |
⊢ ( 𝜑 → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑈 ) ) |