| 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 |
|
ablfac2.m |
⊢ · = ( .g ‘ 𝐺 ) |
| 6 |
|
ablfac2.s |
⊢ 𝑆 = ( 𝑘 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) |
| 7 |
|
wrdf |
⊢ ( 𝑠 ∈ Word 𝐶 → 𝑠 : ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ⟶ 𝐶 ) |
| 8 |
7
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → 𝑠 : ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ⟶ 𝐶 ) |
| 9 |
8
|
fdmd |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → dom 𝑠 = ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ) |
| 10 |
|
fzofi |
⊢ ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ∈ Fin |
| 11 |
9 10
|
eqeltrdi |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → dom 𝑠 ∈ Fin ) |
| 12 |
8
|
ffdmd |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → 𝑠 : dom 𝑠 ⟶ 𝐶 ) |
| 13 |
12
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝑠 ‘ 𝑘 ) ∈ 𝐶 ) |
| 14 |
|
oveq2 |
⊢ ( 𝑟 = ( 𝑠 ‘ 𝑘 ) → ( 𝐺 ↾s 𝑟 ) = ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) |
| 15 |
14
|
eleq1d |
⊢ ( 𝑟 = ( 𝑠 ‘ 𝑘 ) → ( ( 𝐺 ↾s 𝑟 ) ∈ ( CycGrp ∩ ran pGrp ) ↔ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
| 16 |
15 2
|
elrab2 |
⊢ ( ( 𝑠 ‘ 𝑘 ) ∈ 𝐶 ↔ ( ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ ( CycGrp ∩ ran pGrp ) ) ) |
| 17 |
16
|
simplbi |
⊢ ( ( 𝑠 ‘ 𝑘 ) ∈ 𝐶 → ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 18 |
13 17
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 19 |
1
|
subgss |
⊢ ( ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑠 ‘ 𝑘 ) ⊆ 𝐵 ) |
| 20 |
18 19
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝑠 ‘ 𝑘 ) ⊆ 𝐵 ) |
| 21 |
16
|
simprbi |
⊢ ( ( 𝑠 ‘ 𝑘 ) ∈ 𝐶 → ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ ( CycGrp ∩ ran pGrp ) ) |
| 22 |
13 21
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ ( CycGrp ∩ ran pGrp ) ) |
| 23 |
22
|
elin1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ CycGrp ) |
| 24 |
|
eqid |
⊢ ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) |
| 25 |
|
eqid |
⊢ ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) = ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) |
| 26 |
24 25
|
iscyg |
⊢ ( ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ CycGrp ↔ ( ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ Grp ∧ ∃ 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) ) |
| 27 |
26
|
simprbi |
⊢ ( ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ∈ CycGrp → ∃ 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 28 |
23 27
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ∃ 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 29 |
|
eqid |
⊢ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) = ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) |
| 30 |
29
|
subgbas |
⊢ ( ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑠 ‘ 𝑘 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 31 |
18 30
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( 𝑠 ‘ 𝑘 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 32 |
31
|
rexeqdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ↔ ∃ 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) ) |
| 33 |
28 32
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 34 |
18
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℤ ) → ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
| 35 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℤ ) → 𝑛 ∈ ℤ ) |
| 36 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℤ ) → 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) |
| 37 |
5 29 25
|
subgmulg |
⊢ ( ( ( 𝑠 ‘ 𝑘 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑛 ∈ ℤ ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) → ( 𝑛 · 𝑥 ) = ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) |
| 38 |
34 35 36 37
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) ∧ 𝑛 ∈ ℤ ) → ( 𝑛 · 𝑥 ) = ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) |
| 39 |
38
|
mpteq2dva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) ) |
| 40 |
39
|
rneqd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) ) |
| 41 |
31
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) → ( 𝑠 ‘ 𝑘 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) |
| 42 |
40 41
|
eqeq12d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) ∧ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) ) |
| 43 |
42
|
rexbidva |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ( ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ↔ ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) 𝑥 ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑠 ‘ 𝑘 ) ) ) ) ) |
| 44 |
33 43
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ) |
| 45 |
|
ssrexv |
⊢ ( ( 𝑠 ‘ 𝑘 ) ⊆ 𝐵 → ( ∃ 𝑥 ∈ ( 𝑠 ‘ 𝑘 ) ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) → ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ) ) |
| 46 |
20 44 45
|
sylc |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ 𝑘 ∈ dom 𝑠 ) → ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ) |
| 47 |
46
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ∀ 𝑘 ∈ dom 𝑠 ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ) |
| 48 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑤 ‘ 𝑘 ) → ( 𝑛 · 𝑥 ) = ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) |
| 49 |
48
|
mpteq2dv |
⊢ ( 𝑥 = ( 𝑤 ‘ 𝑘 ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) |
| 50 |
49
|
rneqd |
⊢ ( 𝑥 = ( 𝑤 ‘ 𝑘 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) |
| 51 |
50
|
eqeq1d |
⊢ ( 𝑥 = ( 𝑤 ‘ 𝑘 ) → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) |
| 52 |
51
|
ac6sfi |
⊢ ( ( dom 𝑠 ∈ Fin ∧ ∀ 𝑘 ∈ dom 𝑠 ∃ 𝑥 ∈ 𝐵 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · 𝑥 ) ) = ( 𝑠 ‘ 𝑘 ) ) → ∃ 𝑤 ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) |
| 53 |
11 47 52
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ∃ 𝑤 ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) |
| 54 |
|
simprl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑤 : dom 𝑠 ⟶ 𝐵 ) |
| 55 |
9
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → dom 𝑠 = ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ) |
| 56 |
55
|
feq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝑤 : dom 𝑠 ⟶ 𝐵 ↔ 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ⟶ 𝐵 ) ) |
| 57 |
54 56
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ⟶ 𝐵 ) |
| 58 |
|
iswrdi |
⊢ ( 𝑤 : ( 0 ..^ ( ♯ ‘ 𝑠 ) ) ⟶ 𝐵 → 𝑤 ∈ Word 𝐵 ) |
| 59 |
57 58
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑤 ∈ Word 𝐵 ) |
| 60 |
54
|
fdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → dom 𝑤 = dom 𝑠 ) |
| 61 |
60
|
eleq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝑗 ∈ dom 𝑤 ↔ 𝑗 ∈ dom 𝑠 ) ) |
| 62 |
61
|
biimpa |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ dom 𝑤 ) → 𝑗 ∈ dom 𝑠 ) |
| 63 |
|
simprr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) |
| 64 |
|
simpl |
⊢ ( ( 𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ ) → 𝑘 = 𝑗 ) |
| 65 |
64
|
fveq2d |
⊢ ( ( 𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ ) → ( 𝑤 ‘ 𝑘 ) = ( 𝑤 ‘ 𝑗 ) ) |
| 66 |
65
|
oveq2d |
⊢ ( ( 𝑘 = 𝑗 ∧ 𝑛 ∈ ℤ ) → ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) = ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) |
| 67 |
66
|
mpteq2dva |
⊢ ( 𝑘 = 𝑗 → ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 68 |
67
|
rneqd |
⊢ ( 𝑘 = 𝑗 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 69 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑠 ‘ 𝑘 ) = ( 𝑠 ‘ 𝑗 ) ) |
| 70 |
68 69
|
eqeq12d |
⊢ ( 𝑘 = 𝑗 → ( ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ↔ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) = ( 𝑠 ‘ 𝑗 ) ) ) |
| 71 |
70
|
rspccva |
⊢ ( ( ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ∧ 𝑗 ∈ dom 𝑠 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) = ( 𝑠 ‘ 𝑗 ) ) |
| 72 |
63 71
|
sylan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ dom 𝑠 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) = ( 𝑠 ‘ 𝑗 ) ) |
| 73 |
12
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑠 : dom 𝑠 ⟶ 𝐶 ) |
| 74 |
73
|
ffvelrnda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ dom 𝑠 ) → ( 𝑠 ‘ 𝑗 ) ∈ 𝐶 ) |
| 75 |
72 74
|
eqeltrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ dom 𝑠 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ∈ 𝐶 ) |
| 76 |
62 75
|
syldan |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) ∧ 𝑗 ∈ dom 𝑤 ) → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ∈ 𝐶 ) |
| 77 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑤 ‘ 𝑘 ) = ( 𝑤 ‘ 𝑗 ) ) |
| 78 |
77
|
oveq2d |
⊢ ( 𝑘 = 𝑗 → ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) = ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) |
| 79 |
78
|
mpteq2dv |
⊢ ( 𝑘 = 𝑗 → ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 80 |
79
|
rneqd |
⊢ ( 𝑘 = 𝑗 → ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 81 |
80
|
cbvmptv |
⊢ ( 𝑘 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) = ( 𝑗 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 82 |
6 81
|
eqtri |
⊢ 𝑆 = ( 𝑗 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑗 ) ) ) ) |
| 83 |
76 82
|
fmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑆 : dom 𝑤 ⟶ 𝐶 ) |
| 84 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → 𝐺 dom DProd 𝑠 ) |
| 85 |
84
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝐺 dom DProd 𝑠 ) |
| 86 |
60
|
raleqdv |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( ∀ 𝑘 ∈ dom 𝑤 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ↔ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) |
| 87 |
63 86
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ dom 𝑤 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) |
| 88 |
|
mpteq12 |
⊢ ( ( dom 𝑤 = dom 𝑠 ∧ ∀ 𝑘 ∈ dom 𝑤 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) → ( 𝑘 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) = ( 𝑘 ∈ dom 𝑠 ↦ ( 𝑠 ‘ 𝑘 ) ) ) |
| 89 |
60 87 88
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝑘 ∈ dom 𝑤 ↦ ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) ) = ( 𝑘 ∈ dom 𝑠 ↦ ( 𝑠 ‘ 𝑘 ) ) ) |
| 90 |
6 89
|
eqtrid |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑆 = ( 𝑘 ∈ dom 𝑠 ↦ ( 𝑠 ‘ 𝑘 ) ) ) |
| 91 |
|
dprdf |
⊢ ( 𝐺 dom DProd 𝑠 → 𝑠 : dom 𝑠 ⟶ ( SubGrp ‘ 𝐺 ) ) |
| 92 |
85 91
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑠 : dom 𝑠 ⟶ ( SubGrp ‘ 𝐺 ) ) |
| 93 |
92
|
feqmptd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑠 = ( 𝑘 ∈ dom 𝑠 ↦ ( 𝑠 ‘ 𝑘 ) ) ) |
| 94 |
90 93
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝑆 = 𝑠 ) |
| 95 |
85 94
|
breqtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → 𝐺 dom DProd 𝑆 ) |
| 96 |
94
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝐺 DProd 𝑆 ) = ( 𝐺 DProd 𝑠 ) ) |
| 97 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝐺 DProd 𝑠 ) = 𝐵 ) |
| 98 |
96 97
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝐺 DProd 𝑆 ) = 𝐵 ) |
| 99 |
83 95 98
|
3jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) |
| 100 |
59 99
|
jca |
⊢ ( ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) ∧ ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) ) → ( 𝑤 ∈ Word 𝐵 ∧ ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) ) |
| 101 |
100
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ( ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) → ( 𝑤 ∈ Word 𝐵 ∧ ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) ) ) |
| 102 |
101
|
eximdv |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ( ∃ 𝑤 ( 𝑤 : dom 𝑠 ⟶ 𝐵 ∧ ∀ 𝑘 ∈ dom 𝑠 ran ( 𝑛 ∈ ℤ ↦ ( 𝑛 · ( 𝑤 ‘ 𝑘 ) ) ) = ( 𝑠 ‘ 𝑘 ) ) → ∃ 𝑤 ( 𝑤 ∈ Word 𝐵 ∧ ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) ) ) |
| 103 |
53 102
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ∃ 𝑤 ( 𝑤 ∈ Word 𝐵 ∧ ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) ) |
| 104 |
|
df-rex |
⊢ ( ∃ 𝑤 ∈ Word 𝐵 ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ↔ ∃ 𝑤 ( 𝑤 ∈ Word 𝐵 ∧ ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) ) |
| 105 |
103 104
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ Word 𝐶 ) ∧ ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) → ∃ 𝑤 ∈ Word 𝐵 ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) |
| 106 |
1 2 3 4
|
ablfac |
⊢ ( 𝜑 → ∃ 𝑠 ∈ Word 𝐶 ( 𝐺 dom DProd 𝑠 ∧ ( 𝐺 DProd 𝑠 ) = 𝐵 ) ) |
| 107 |
105 106
|
r19.29a |
⊢ ( 𝜑 → ∃ 𝑤 ∈ Word 𝐵 ( 𝑆 : dom 𝑤 ⟶ 𝐶 ∧ 𝐺 dom DProd 𝑆 ∧ ( 𝐺 DProd 𝑆 ) = 𝐵 ) ) |