Step |
Hyp |
Ref |
Expression |
1 |
|
psgnfval.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
2 |
|
psgnfval.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
3 |
|
psgnfval.f |
⊢ 𝐹 = { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } |
4 |
|
psgnfval.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
5 |
|
psgnfval.n |
⊢ 𝑁 = ( pmSgn ‘ 𝐷 ) |
6 |
|
fveq2 |
⊢ ( 𝑑 = 𝐷 → ( SymGrp ‘ 𝑑 ) = ( SymGrp ‘ 𝐷 ) ) |
7 |
6 1
|
eqtr4di |
⊢ ( 𝑑 = 𝐷 → ( SymGrp ‘ 𝑑 ) = 𝐺 ) |
8 |
7
|
fveq2d |
⊢ ( 𝑑 = 𝐷 → ( Base ‘ ( SymGrp ‘ 𝑑 ) ) = ( Base ‘ 𝐺 ) ) |
9 |
8 2
|
eqtr4di |
⊢ ( 𝑑 = 𝐷 → ( Base ‘ ( SymGrp ‘ 𝑑 ) ) = 𝐵 ) |
10 |
|
rabeq |
⊢ ( ( Base ‘ ( SymGrp ‘ 𝑑 ) ) = 𝐵 → { 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑑 ) ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ) |
11 |
9 10
|
syl |
⊢ ( 𝑑 = 𝐷 → { 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑑 ) ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ) |
12 |
11 3
|
eqtr4di |
⊢ ( 𝑑 = 𝐷 → { 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑑 ) ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = 𝐹 ) |
13 |
|
fveq2 |
⊢ ( 𝑑 = 𝐷 → ( pmTrsp ‘ 𝑑 ) = ( pmTrsp ‘ 𝐷 ) ) |
14 |
13
|
rneqd |
⊢ ( 𝑑 = 𝐷 → ran ( pmTrsp ‘ 𝑑 ) = ran ( pmTrsp ‘ 𝐷 ) ) |
15 |
14 4
|
eqtr4di |
⊢ ( 𝑑 = 𝐷 → ran ( pmTrsp ‘ 𝑑 ) = 𝑇 ) |
16 |
|
wrdeq |
⊢ ( ran ( pmTrsp ‘ 𝑑 ) = 𝑇 → Word ran ( pmTrsp ‘ 𝑑 ) = Word 𝑇 ) |
17 |
15 16
|
syl |
⊢ ( 𝑑 = 𝐷 → Word ran ( pmTrsp ‘ 𝑑 ) = Word 𝑇 ) |
18 |
7
|
oveq1d |
⊢ ( 𝑑 = 𝐷 → ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) = ( 𝐺 Σg 𝑤 ) ) |
19 |
18
|
eqeq2d |
⊢ ( 𝑑 = 𝐷 → ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ↔ 𝑥 = ( 𝐺 Σg 𝑤 ) ) ) |
20 |
19
|
anbi1d |
⊢ ( 𝑑 = 𝐷 → ( ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
21 |
17 20
|
rexeqbidv |
⊢ ( 𝑑 = 𝐷 → ( ∃ 𝑤 ∈ Word ran ( pmTrsp ‘ 𝑑 ) ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ↔ ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
22 |
21
|
iotabidv |
⊢ ( 𝑑 = 𝐷 → ( ℩ 𝑠 ∃ 𝑤 ∈ Word ran ( pmTrsp ‘ 𝑑 ) ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) = ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
23 |
12 22
|
mpteq12dv |
⊢ ( 𝑑 = 𝐷 → ( 𝑥 ∈ { 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑑 ) ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word ran ( pmTrsp ‘ 𝑑 ) ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) = ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
24 |
|
df-psgn |
⊢ pmSgn = ( 𝑑 ∈ V ↦ ( 𝑥 ∈ { 𝑝 ∈ ( Base ‘ ( SymGrp ‘ 𝑑 ) ) ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word ran ( pmTrsp ‘ 𝑑 ) ( 𝑥 = ( ( SymGrp ‘ 𝑑 ) Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
25 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
26 |
3 25
|
rabex2 |
⊢ 𝐹 ∈ V |
27 |
26
|
mptex |
⊢ ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ∈ V |
28 |
23 24 27
|
fvmpt |
⊢ ( 𝐷 ∈ V → ( pmSgn ‘ 𝐷 ) = ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
29 |
|
fvprc |
⊢ ( ¬ 𝐷 ∈ V → ( pmSgn ‘ 𝐷 ) = ∅ ) |
30 |
|
fvprc |
⊢ ( ¬ 𝐷 ∈ V → ( SymGrp ‘ 𝐷 ) = ∅ ) |
31 |
1 30
|
syl5eq |
⊢ ( ¬ 𝐷 ∈ V → 𝐺 = ∅ ) |
32 |
31
|
fveq2d |
⊢ ( ¬ 𝐷 ∈ V → ( Base ‘ 𝐺 ) = ( Base ‘ ∅ ) ) |
33 |
|
base0 |
⊢ ∅ = ( Base ‘ ∅ ) |
34 |
32 33
|
eqtr4di |
⊢ ( ¬ 𝐷 ∈ V → ( Base ‘ 𝐺 ) = ∅ ) |
35 |
2 34
|
syl5eq |
⊢ ( ¬ 𝐷 ∈ V → 𝐵 = ∅ ) |
36 |
|
rabeq |
⊢ ( 𝐵 = ∅ → { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ ∅ ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ) |
37 |
35 36
|
syl |
⊢ ( ¬ 𝐷 ∈ V → { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = { 𝑝 ∈ ∅ ∣ dom ( 𝑝 ∖ I ) ∈ Fin } ) |
38 |
|
rab0 |
⊢ { 𝑝 ∈ ∅ ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = ∅ |
39 |
37 38
|
eqtrdi |
⊢ ( ¬ 𝐷 ∈ V → { 𝑝 ∈ 𝐵 ∣ dom ( 𝑝 ∖ I ) ∈ Fin } = ∅ ) |
40 |
3 39
|
syl5eq |
⊢ ( ¬ 𝐷 ∈ V → 𝐹 = ∅ ) |
41 |
40
|
mpteq1d |
⊢ ( ¬ 𝐷 ∈ V → ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) = ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
42 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) = ∅ |
43 |
41 42
|
eqtrdi |
⊢ ( ¬ 𝐷 ∈ V → ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) = ∅ ) |
44 |
29 43
|
eqtr4d |
⊢ ( ¬ 𝐷 ∈ V → ( pmSgn ‘ 𝐷 ) = ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) ) |
45 |
28 44
|
pm2.61i |
⊢ ( pmSgn ‘ 𝐷 ) = ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |
46 |
5 45
|
eqtri |
⊢ 𝑁 = ( 𝑥 ∈ 𝐹 ↦ ( ℩ 𝑠 ∃ 𝑤 ∈ Word 𝑇 ( 𝑥 = ( 𝐺 Σg 𝑤 ) ∧ 𝑠 = ( - 1 ↑ ( ♯ ‘ 𝑤 ) ) ) ) ) |