| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ablfac.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
| 2 |
|
ablfac.c |
⊢ 𝐶 = { 𝑟 ∈ ( SubGrp ‘ 𝐺 ) ∣ ( 𝐺 ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } |
| 3 |
|
ablfac.1 |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
| 4 |
|
ablfac.2 |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
| 5 |
|
ablfac.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
| 6 |
|
ablfac.a |
⊢ 𝐴 = { 𝑤 ∈ ℙ ∣ 𝑤 ∥ ( ♯ ‘ 𝐵 ) } |
| 7 |
|
ablfac.s |
⊢ 𝑆 = ( 𝑝 ∈ 𝐴 ↦ { 𝑥 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) |
| 8 |
|
ablfac.w |
⊢ 𝑊 = ( 𝑔 ∈ ( SubGrp ‘ 𝐺 ) ↦ { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑔 ) } ) |
| 9 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... ( ♯ ‘ 𝐵 ) ) ∈ Fin ) |
| 10 |
|
prmnn |
⊢ ( 𝑤 ∈ ℙ → 𝑤 ∈ ℕ ) |
| 11 |
10
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ ( ♯ ‘ 𝐵 ) ) → 𝑤 ∈ ℕ ) |
| 12 |
|
prmz |
⊢ ( 𝑤 ∈ ℙ → 𝑤 ∈ ℤ ) |
| 13 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
| 14 |
1
|
grpbn0 |
⊢ ( 𝐺 ∈ Grp → 𝐵 ≠ ∅ ) |
| 15 |
3 13 14
|
3syl |
⊢ ( 𝜑 → 𝐵 ≠ ∅ ) |
| 16 |
|
hashnncl |
⊢ ( 𝐵 ∈ Fin → ( ( ♯ ‘ 𝐵 ) ∈ ℕ ↔ 𝐵 ≠ ∅ ) ) |
| 17 |
4 16
|
syl |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐵 ) ∈ ℕ ↔ 𝐵 ≠ ∅ ) ) |
| 18 |
15 17
|
mpbird |
⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℕ ) |
| 19 |
|
dvdsle |
⊢ ( ( 𝑤 ∈ ℤ ∧ ( ♯ ‘ 𝐵 ) ∈ ℕ ) → ( 𝑤 ∥ ( ♯ ‘ 𝐵 ) → 𝑤 ≤ ( ♯ ‘ 𝐵 ) ) ) |
| 20 |
12 18 19
|
syl2anr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ) → ( 𝑤 ∥ ( ♯ ‘ 𝐵 ) → 𝑤 ≤ ( ♯ ‘ 𝐵 ) ) ) |
| 21 |
20
|
3impia |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ ( ♯ ‘ 𝐵 ) ) → 𝑤 ≤ ( ♯ ‘ 𝐵 ) ) |
| 22 |
18
|
nnzd |
⊢ ( 𝜑 → ( ♯ ‘ 𝐵 ) ∈ ℤ ) |
| 23 |
22
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ ( ♯ ‘ 𝐵 ) ) → ( ♯ ‘ 𝐵 ) ∈ ℤ ) |
| 24 |
|
fznn |
⊢ ( ( ♯ ‘ 𝐵 ) ∈ ℤ → ( 𝑤 ∈ ( 1 ... ( ♯ ‘ 𝐵 ) ) ↔ ( 𝑤 ∈ ℕ ∧ 𝑤 ≤ ( ♯ ‘ 𝐵 ) ) ) ) |
| 25 |
23 24
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ ( ♯ ‘ 𝐵 ) ) → ( 𝑤 ∈ ( 1 ... ( ♯ ‘ 𝐵 ) ) ↔ ( 𝑤 ∈ ℕ ∧ 𝑤 ≤ ( ♯ ‘ 𝐵 ) ) ) ) |
| 26 |
11 21 25
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ℙ ∧ 𝑤 ∥ ( ♯ ‘ 𝐵 ) ) → 𝑤 ∈ ( 1 ... ( ♯ ‘ 𝐵 ) ) ) |
| 27 |
26
|
rabssdv |
⊢ ( 𝜑 → { 𝑤 ∈ ℙ ∣ 𝑤 ∥ ( ♯ ‘ 𝐵 ) } ⊆ ( 1 ... ( ♯ ‘ 𝐵 ) ) ) |
| 28 |
6 27
|
eqsstrid |
⊢ ( 𝜑 → 𝐴 ⊆ ( 1 ... ( ♯ ‘ 𝐵 ) ) ) |
| 29 |
9 28
|
ssfid |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
| 30 |
|
dfin5 |
⊢ ( Word 𝐶 ∩ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) = { 𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) } |
| 31 |
6
|
ssrab3 |
⊢ 𝐴 ⊆ ℙ |
| 32 |
31
|
a1i |
⊢ ( 𝜑 → 𝐴 ⊆ ℙ ) |
| 33 |
1 5 7 3 4 32
|
ablfac1b |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
| 34 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
| 35 |
34
|
rabex |
⊢ { 𝑥 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } ∈ V |
| 36 |
35 7
|
dmmpti |
⊢ dom 𝑆 = 𝐴 |
| 37 |
36
|
a1i |
⊢ ( 𝜑 → dom 𝑆 = 𝐴 ) |
| 38 |
33 37
|
dprdf2 |
⊢ ( 𝜑 → 𝑆 : 𝐴 ⟶ ( SubGrp ‘ 𝐺 ) ) |
| 39 |
38
|
ffvelcdmda |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 40 |
1 2 3 4 5 6 7 8
|
ablfaclem1 |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) = { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ) |
| 41 |
39 40
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) = { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ) |
| 42 |
|
ssrab2 |
⊢ { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ⊆ Word 𝐶 |
| 43 |
41 42
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ⊆ Word 𝐶 ) |
| 44 |
|
sseqin2 |
⊢ ( ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ⊆ Word 𝐶 ↔ ( Word 𝐶 ∩ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) = ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 45 |
43 44
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( Word 𝐶 ∩ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) = ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 46 |
30 45
|
eqtr3id |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) } = ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 47 |
46 41
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) } = { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ) |
| 48 |
|
eqid |
⊢ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) |
| 49 |
|
eqid |
⊢ { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } = { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } |
| 50 |
|
eqid |
⊢ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) = ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) |
| 51 |
50
|
subgabl |
⊢ ( ( 𝐺 ∈ Abel ∧ ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ∈ Abel ) |
| 52 |
3 39 51
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ∈ Abel ) |
| 53 |
32
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → 𝑞 ∈ ℙ ) |
| 54 |
50
|
subgbas |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 ‘ 𝑞 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 55 |
39 54
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑆 ‘ 𝑞 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 56 |
55
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ♯ ‘ ( 𝑆 ‘ 𝑞 ) ) = ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) |
| 57 |
1 5 7 3 4 32
|
ablfac1a |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ♯ ‘ ( 𝑆 ‘ 𝑞 ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) |
| 58 |
56 57
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) |
| 59 |
58
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) = ( 𝑞 pCnt ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) ) |
| 60 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ♯ ‘ 𝐵 ) ∈ ℕ ) |
| 61 |
53 60
|
pccld |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ∈ ℕ0 ) |
| 62 |
61
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ∈ ℤ ) |
| 63 |
|
pcid |
⊢ ( ( 𝑞 ∈ ℙ ∧ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ∈ ℤ ) → ( 𝑞 pCnt ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) = ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) |
| 64 |
53 62 63
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pCnt ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) = ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) |
| 65 |
59 64
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) = ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) |
| 66 |
65
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ 𝐵 ) ) ) ) |
| 67 |
58 66
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) ) ) |
| 68 |
50
|
subggrp |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ∈ Grp ) |
| 69 |
39 68
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ∈ Grp ) |
| 70 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → 𝐵 ∈ Fin ) |
| 71 |
1
|
subgss |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 ‘ 𝑞 ) ⊆ 𝐵 ) |
| 72 |
39 71
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑆 ‘ 𝑞 ) ⊆ 𝐵 ) |
| 73 |
70 72
|
ssfid |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑆 ‘ 𝑞 ) ∈ Fin ) |
| 74 |
55 73
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∈ Fin ) |
| 75 |
48
|
pgpfi2 |
⊢ ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ∈ Grp ∧ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∈ Fin ) → ( 𝑞 pGrp ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↔ ( 𝑞 ∈ ℙ ∧ ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) ) ) ) ) |
| 76 |
69 74 75
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑞 pGrp ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↔ ( 𝑞 ∈ ℙ ∧ ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) = ( 𝑞 ↑ ( 𝑞 pCnt ( ♯ ‘ ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) ) ) ) ) |
| 77 |
53 67 76
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → 𝑞 pGrp ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) |
| 78 |
48 49 52 77 74
|
pgpfac |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ∃ 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) |
| 79 |
|
ssrab2 |
⊢ { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) |
| 80 |
|
sswrd |
⊢ ( { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) → Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 81 |
79 80
|
ax-mp |
⊢ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) |
| 82 |
81
|
sseli |
⊢ ( 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } → 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 83 |
39
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 84 |
83
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 85 |
50
|
subgdmdprd |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ↔ ( 𝐺 dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 86 |
83 85
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ↔ ( 𝐺 dom DProd 𝑠 ∧ ran 𝑠 ⊆ 𝒫 ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 87 |
86
|
simprbda |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → 𝐺 dom DProd 𝑠 ) |
| 88 |
86
|
simplbda |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ran 𝑠 ⊆ 𝒫 ( 𝑆 ‘ 𝑞 ) ) |
| 89 |
50 84 87 88
|
subgdprd |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( 𝐺 DProd 𝑠 ) ) |
| 90 |
55
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( 𝑆 ‘ 𝑞 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 91 |
90
|
eqcomd |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) = ( 𝑆 ‘ 𝑞 ) ) |
| 92 |
89 91
|
eqeq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ↔ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) |
| 93 |
92
|
biimpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) → ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) |
| 94 |
93 87
|
jctild |
⊢ ( ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) → ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 95 |
94
|
expimpd |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 96 |
82 95
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 97 |
|
oveq2 |
⊢ ( 𝑟 = 𝑦 → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) = ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ) |
| 98 |
97
|
eleq1d |
⊢ ( 𝑟 = 𝑦 → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) ↔ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
| 99 |
98
|
cbvrabv |
⊢ { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } = { 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) } |
| 100 |
50
|
subsubg |
⊢ ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ↔ ( 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑦 ⊆ ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 101 |
39 100
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ↔ ( 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑦 ⊆ ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 102 |
101
|
simprbda |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 103 |
102
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ) |
| 104 |
39
|
3ad2ant1 |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 105 |
101
|
simplbda |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → 𝑦 ⊆ ( 𝑆 ‘ 𝑞 ) ) |
| 106 |
105
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → 𝑦 ⊆ ( 𝑆 ‘ 𝑞 ) ) |
| 107 |
|
ressabs |
⊢ ( ( ( 𝑆 ‘ 𝑞 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑦 ⊆ ( 𝑆 ‘ 𝑞 ) ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) = ( 𝐺 ↾s 𝑦 ) ) |
| 108 |
104 106 107
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) = ( 𝐺 ↾s 𝑦 ) ) |
| 109 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) |
| 110 |
108 109
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → ( 𝐺 ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) |
| 111 |
|
oveq2 |
⊢ ( 𝑟 = 𝑦 → ( 𝐺 ↾s 𝑟 ) = ( 𝐺 ↾s 𝑦 ) ) |
| 112 |
111
|
eleq1d |
⊢ ( 𝑟 = 𝑦 → ( ( 𝐺 ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) ↔ ( 𝐺 ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
| 113 |
112 2
|
elrab2 |
⊢ ( 𝑦 ∈ 𝐶 ↔ ( 𝑦 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
| 114 |
103 110 113
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) ) → 𝑦 ∈ 𝐶 ) |
| 115 |
114
|
rabssdv |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑦 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑦 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ 𝐶 ) |
| 116 |
99 115
|
eqsstrid |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ 𝐶 ) |
| 117 |
|
sswrd |
⊢ ( { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ 𝐶 → Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ Word 𝐶 ) |
| 118 |
116 117
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ⊆ Word 𝐶 ) |
| 119 |
118
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ) → 𝑠 ∈ Word 𝐶 ) |
| 120 |
96 119
|
jctild |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) ∧ 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ) → ( ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 𝑠 ∈ Word 𝐶 ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) ) |
| 121 |
120
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ( 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) ) → ( 𝑠 ∈ Word 𝐶 ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) ) |
| 122 |
121
|
reximdv2 |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ( ∃ 𝑠 ∈ Word { 𝑟 ∈ ( SubGrp ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ∣ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) } ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) dom DProd 𝑠 ∧ ( ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) DProd 𝑠 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑆 ‘ 𝑞 ) ) ) ) → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 123 |
78 122
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) |
| 124 |
|
rabn0 |
⊢ ( { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ≠ ∅ ↔ ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) ) |
| 125 |
123 124
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = ( 𝑆 ‘ 𝑞 ) ) } ≠ ∅ ) |
| 126 |
47 125
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → { 𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) } ≠ ∅ ) |
| 127 |
|
rabn0 |
⊢ ( { 𝑦 ∈ Word 𝐶 ∣ 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) } ≠ ∅ ↔ ∃ 𝑦 ∈ Word 𝐶 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 128 |
126 127
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝐴 ) → ∃ 𝑦 ∈ Word 𝐶 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 129 |
128
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑞 ∈ 𝐴 ∃ 𝑦 ∈ Word 𝐶 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 130 |
|
eleq1 |
⊢ ( 𝑦 = ( 𝑓 ‘ 𝑞 ) → ( 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ↔ ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 131 |
130
|
ac6sfi |
⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑞 ∈ 𝐴 ∃ 𝑦 ∈ Word 𝐶 𝑦 ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 132 |
29 129 131
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) |
| 133 |
|
sneq |
⊢ ( 𝑞 = 𝑦 → { 𝑞 } = { 𝑦 } ) |
| 134 |
|
fveq2 |
⊢ ( 𝑞 = 𝑦 → ( 𝑓 ‘ 𝑞 ) = ( 𝑓 ‘ 𝑦 ) ) |
| 135 |
134
|
dmeqd |
⊢ ( 𝑞 = 𝑦 → dom ( 𝑓 ‘ 𝑞 ) = dom ( 𝑓 ‘ 𝑦 ) ) |
| 136 |
133 135
|
xpeq12d |
⊢ ( 𝑞 = 𝑦 → ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) = ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ) |
| 137 |
136
|
cbviunv |
⊢ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) = ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) |
| 138 |
|
snfi |
⊢ { 𝑦 } ∈ Fin |
| 139 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → 𝑓 : 𝐴 ⟶ Word 𝐶 ) |
| 140 |
139
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑓 ‘ 𝑦 ) ∈ Word 𝐶 ) |
| 141 |
|
wrdf |
⊢ ( ( 𝑓 ‘ 𝑦 ) ∈ Word 𝐶 → ( 𝑓 ‘ 𝑦 ) : ( 0 ..^ ( ♯ ‘ ( 𝑓 ‘ 𝑦 ) ) ) ⟶ 𝐶 ) |
| 142 |
|
fdm |
⊢ ( ( 𝑓 ‘ 𝑦 ) : ( 0 ..^ ( ♯ ‘ ( 𝑓 ‘ 𝑦 ) ) ) ⟶ 𝐶 → dom ( 𝑓 ‘ 𝑦 ) = ( 0 ..^ ( ♯ ‘ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 143 |
140 141 142
|
3syl |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ 𝑦 ∈ 𝐴 ) → dom ( 𝑓 ‘ 𝑦 ) = ( 0 ..^ ( ♯ ‘ ( 𝑓 ‘ 𝑦 ) ) ) ) |
| 144 |
|
fzofi |
⊢ ( 0 ..^ ( ♯ ‘ ( 𝑓 ‘ 𝑦 ) ) ) ∈ Fin |
| 145 |
143 144
|
eqeltrdi |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ 𝑦 ∈ 𝐴 ) → dom ( 𝑓 ‘ 𝑦 ) ∈ Fin ) |
| 146 |
|
xpfi |
⊢ ( ( { 𝑦 } ∈ Fin ∧ dom ( 𝑓 ‘ 𝑦 ) ∈ Fin ) → ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) |
| 147 |
138 145 146
|
sylancr |
⊢ ( ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) ∧ 𝑦 ∈ 𝐴 ) → ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) |
| 148 |
147
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ∀ 𝑦 ∈ 𝐴 ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) |
| 149 |
|
iunfi |
⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑦 ∈ 𝐴 ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) → ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) |
| 150 |
29 148 149
|
syl2an2r |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ∪ 𝑦 ∈ 𝐴 ( { 𝑦 } × dom ( 𝑓 ‘ 𝑦 ) ) ∈ Fin ) |
| 151 |
137 150
|
eqeltrid |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ∈ Fin ) |
| 152 |
|
hashcl |
⊢ ( ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ∈ Fin → ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ∈ ℕ0 ) |
| 153 |
|
hashfzo0 |
⊢ ( ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ∈ ℕ0 → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ) = ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) |
| 154 |
151 152 153
|
3syl |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ♯ ‘ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ) = ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) |
| 155 |
|
fzofi |
⊢ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ∈ Fin |
| 156 |
|
hashen |
⊢ ( ( ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ∈ Fin ∧ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ∈ Fin ) → ( ( ♯ ‘ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ) = ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ↔ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ≈ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) |
| 157 |
155 151 156
|
sylancr |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ( ♯ ‘ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ) = ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ↔ ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ≈ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) |
| 158 |
154 157
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ≈ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) |
| 159 |
|
bren |
⊢ ( ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) ≈ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ↔ ∃ ℎ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) |
| 160 |
158 159
|
sylib |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ∃ ℎ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) |
| 161 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → 𝐺 ∈ Abel ) |
| 162 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → 𝐵 ∈ Fin ) |
| 163 |
|
breq1 |
⊢ ( 𝑤 = 𝑎 → ( 𝑤 ∥ ( ♯ ‘ 𝐵 ) ↔ 𝑎 ∥ ( ♯ ‘ 𝐵 ) ) ) |
| 164 |
163
|
cbvrabv |
⊢ { 𝑤 ∈ ℙ ∣ 𝑤 ∥ ( ♯ ‘ 𝐵 ) } = { 𝑎 ∈ ℙ ∣ 𝑎 ∥ ( ♯ ‘ 𝐵 ) } |
| 165 |
6 164
|
eqtri |
⊢ 𝐴 = { 𝑎 ∈ ℙ ∣ 𝑎 ∥ ( ♯ ‘ 𝐵 ) } |
| 166 |
|
fveq2 |
⊢ ( 𝑥 = 𝑐 → ( 𝑂 ‘ 𝑥 ) = ( 𝑂 ‘ 𝑐 ) ) |
| 167 |
166
|
breq1d |
⊢ ( 𝑥 = 𝑐 → ( ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) ↔ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) ) ) |
| 168 |
167
|
cbvrabv |
⊢ { 𝑥 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } = { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } |
| 169 |
|
id |
⊢ ( 𝑝 = 𝑏 → 𝑝 = 𝑏 ) |
| 170 |
|
oveq1 |
⊢ ( 𝑝 = 𝑏 → ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) = ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) |
| 171 |
169 170
|
oveq12d |
⊢ ( 𝑝 = 𝑏 → ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) = ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) ) |
| 172 |
171
|
breq2d |
⊢ ( 𝑝 = 𝑏 → ( ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) ↔ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) ) ) |
| 173 |
172
|
rabbidv |
⊢ ( 𝑝 = 𝑏 → { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } = { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) |
| 174 |
168 173
|
eqtrid |
⊢ ( 𝑝 = 𝑏 → { 𝑥 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } = { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) |
| 175 |
174
|
cbvmptv |
⊢ ( 𝑝 ∈ 𝐴 ↦ { 𝑥 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑥 ) ∥ ( 𝑝 ↑ ( 𝑝 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) = ( 𝑏 ∈ 𝐴 ↦ { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) |
| 176 |
7 175
|
eqtri |
⊢ 𝑆 = ( 𝑏 ∈ 𝐴 ↦ { 𝑐 ∈ 𝐵 ∣ ( 𝑂 ‘ 𝑐 ) ∥ ( 𝑏 ↑ ( 𝑏 pCnt ( ♯ ‘ 𝐵 ) ) ) } ) |
| 177 |
|
breq2 |
⊢ ( 𝑠 = 𝑡 → ( 𝐺 dom DProd 𝑠 ↔ 𝐺 dom DProd 𝑡 ) ) |
| 178 |
|
oveq2 |
⊢ ( 𝑠 = 𝑡 → ( 𝐺 DProd 𝑠 ) = ( 𝐺 DProd 𝑡 ) ) |
| 179 |
178
|
eqeq1d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝐺 DProd 𝑠 ) = 𝑔 ↔ ( 𝐺 DProd 𝑡 ) = 𝑔 ) ) |
| 180 |
177 179
|
anbi12d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑔 ) ↔ ( 𝐺 dom DProd 𝑡 ∧ ( 𝐺 DProd 𝑡 ) = 𝑔 ) ) ) |
| 181 |
180
|
cbvrabv |
⊢ { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑔 ) } = { 𝑡 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑡 ∧ ( 𝐺 DProd 𝑡 ) = 𝑔 ) } |
| 182 |
181
|
mpteq2i |
⊢ ( 𝑔 ∈ ( SubGrp ‘ 𝐺 ) ↦ { 𝑠 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝑔 ) } ) = ( 𝑔 ∈ ( SubGrp ‘ 𝐺 ) ↦ { 𝑡 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑡 ∧ ( 𝐺 DProd 𝑡 ) = 𝑔 ) } ) |
| 183 |
8 182
|
eqtri |
⊢ 𝑊 = ( 𝑔 ∈ ( SubGrp ‘ 𝐺 ) ↦ { 𝑡 ∈ Word 𝐶 ∣ ( 𝐺 dom DProd 𝑡 ∧ ( 𝐺 DProd 𝑡 ) = 𝑔 ) } ) |
| 184 |
|
simprll |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → 𝑓 : 𝐴 ⟶ Word 𝐶 ) |
| 185 |
|
simprlr |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) |
| 186 |
|
2fveq3 |
⊢ ( 𝑞 = 𝑦 → ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) = ( 𝑊 ‘ ( 𝑆 ‘ 𝑦 ) ) ) |
| 187 |
134 186
|
eleq12d |
⊢ ( 𝑞 = 𝑦 → ( ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ↔ ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑦 ) ) ) ) |
| 188 |
187
|
cbvralvw |
⊢ ( ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑦 ) ) ) |
| 189 |
185 188
|
sylib |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝑓 ‘ 𝑦 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑦 ) ) ) |
| 190 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) |
| 191 |
1 2 161 162 5 165 176 183 184 189 137 190
|
ablfaclem2 |
⊢ ( ( 𝜑 ∧ ( ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ∧ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) → ( 𝑊 ‘ 𝐵 ) ≠ ∅ ) |
| 192 |
191
|
expr |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) → ( 𝑊 ‘ 𝐵 ) ≠ ∅ ) ) |
| 193 |
192
|
exlimdv |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( ∃ ℎ ℎ : ( 0 ..^ ( ♯ ‘ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) ) ) –1-1-onto→ ∪ 𝑞 ∈ 𝐴 ( { 𝑞 } × dom ( 𝑓 ‘ 𝑞 ) ) → ( 𝑊 ‘ 𝐵 ) ≠ ∅ ) ) |
| 194 |
160 193
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑓 : 𝐴 ⟶ Word 𝐶 ∧ ∀ 𝑞 ∈ 𝐴 ( 𝑓 ‘ 𝑞 ) ∈ ( 𝑊 ‘ ( 𝑆 ‘ 𝑞 ) ) ) ) → ( 𝑊 ‘ 𝐵 ) ≠ ∅ ) |
| 195 |
132 194
|
exlimddv |
⊢ ( 𝜑 → ( 𝑊 ‘ 𝐵 ) ≠ ∅ ) |