Step |
Hyp |
Ref |
Expression |
1 |
|
pi1xfr.p |
⊢ 𝑃 = ( 𝐽 π1 ( 𝐹 ‘ 0 ) ) |
2 |
|
pi1xfr.q |
⊢ 𝑄 = ( 𝐽 π1 ( 𝐹 ‘ 1 ) ) |
3 |
|
pi1xfr.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
pi1xfr.g |
⊢ 𝐺 = ran ( 𝑔 ∈ ∪ 𝐵 ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
5 |
|
pi1xfr.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
6 |
|
pi1xfr.f |
⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) ) |
7 |
|
pi1xfr.i |
⊢ 𝐼 = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ ( 1 − 𝑥 ) ) ) |
8 |
|
pi1xfrcnv.h |
⊢ 𝐻 = ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
9 |
1 2 3 4 5 6 7 8
|
pi1xfrcnvlem |
⊢ ( 𝜑 → ◡ 𝐺 ⊆ 𝐻 ) |
10 |
|
fvex |
⊢ ( ≃ph ‘ 𝐽 ) ∈ V |
11 |
|
ecexg |
⊢ ( ( ≃ph ‘ 𝐽 ) ∈ V → [ ℎ ] ( ≃ph ‘ 𝐽 ) ∈ V ) |
12 |
10 11
|
mp1i |
⊢ ( ( 𝜑 ∧ ℎ ∈ ∪ ( Base ‘ 𝑄 ) ) → [ ℎ ] ( ≃ph ‘ 𝐽 ) ∈ V ) |
13 |
|
ecexg |
⊢ ( ( ≃ph ‘ 𝐽 ) ∈ V → [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) ∈ V ) |
14 |
10 13
|
mp1i |
⊢ ( ( 𝜑 ∧ ℎ ∈ ∪ ( Base ‘ 𝑄 ) ) → [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) ∈ V ) |
15 |
8 12 14
|
fliftrel |
⊢ ( 𝜑 → 𝐻 ⊆ ( V × V ) ) |
16 |
|
df-rel |
⊢ ( Rel 𝐻 ↔ 𝐻 ⊆ ( V × V ) ) |
17 |
15 16
|
sylibr |
⊢ ( 𝜑 → Rel 𝐻 ) |
18 |
|
dfrel2 |
⊢ ( Rel 𝐻 ↔ ◡ ◡ 𝐻 = 𝐻 ) |
19 |
17 18
|
sylib |
⊢ ( 𝜑 → ◡ ◡ 𝐻 = 𝐻 ) |
20 |
|
0elunit |
⊢ 0 ∈ ( 0 [,] 1 ) |
21 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 1 − 𝑥 ) = ( 1 − 0 ) ) |
22 |
|
1m0e1 |
⊢ ( 1 − 0 ) = 1 |
23 |
21 22
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 1 − 𝑥 ) = 1 ) |
24 |
23
|
fveq2d |
⊢ ( 𝑥 = 0 → ( 𝐹 ‘ ( 1 − 𝑥 ) ) = ( 𝐹 ‘ 1 ) ) |
25 |
|
fvex |
⊢ ( 𝐹 ‘ 1 ) ∈ V |
26 |
24 7 25
|
fvmpt |
⊢ ( 0 ∈ ( 0 [,] 1 ) → ( 𝐼 ‘ 0 ) = ( 𝐹 ‘ 1 ) ) |
27 |
20 26
|
ax-mp |
⊢ ( 𝐼 ‘ 0 ) = ( 𝐹 ‘ 1 ) |
28 |
27
|
oveq2i |
⊢ ( 𝐽 π1 ( 𝐼 ‘ 0 ) ) = ( 𝐽 π1 ( 𝐹 ‘ 1 ) ) |
29 |
2 28
|
eqtr4i |
⊢ 𝑄 = ( 𝐽 π1 ( 𝐼 ‘ 0 ) ) |
30 |
|
1elunit |
⊢ 1 ∈ ( 0 [,] 1 ) |
31 |
|
oveq2 |
⊢ ( 𝑥 = 1 → ( 1 − 𝑥 ) = ( 1 − 1 ) ) |
32 |
31
|
fveq2d |
⊢ ( 𝑥 = 1 → ( 𝐹 ‘ ( 1 − 𝑥 ) ) = ( 𝐹 ‘ ( 1 − 1 ) ) ) |
33 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
34 |
33
|
fveq2i |
⊢ ( 𝐹 ‘ ( 1 − 1 ) ) = ( 𝐹 ‘ 0 ) |
35 |
32 34
|
eqtrdi |
⊢ ( 𝑥 = 1 → ( 𝐹 ‘ ( 1 − 𝑥 ) ) = ( 𝐹 ‘ 0 ) ) |
36 |
|
fvex |
⊢ ( 𝐹 ‘ 0 ) ∈ V |
37 |
35 7 36
|
fvmpt |
⊢ ( 1 ∈ ( 0 [,] 1 ) → ( 𝐼 ‘ 1 ) = ( 𝐹 ‘ 0 ) ) |
38 |
30 37
|
ax-mp |
⊢ ( 𝐼 ‘ 1 ) = ( 𝐹 ‘ 0 ) |
39 |
38
|
oveq2i |
⊢ ( 𝐽 π1 ( 𝐼 ‘ 1 ) ) = ( 𝐽 π1 ( 𝐹 ‘ 0 ) ) |
40 |
1 39
|
eqtr4i |
⊢ 𝑃 = ( 𝐽 π1 ( 𝐼 ‘ 1 ) ) |
41 |
|
eqid |
⊢ ( Base ‘ 𝑄 ) = ( Base ‘ 𝑄 ) |
42 |
|
eqid |
⊢ ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
43 |
7
|
pcorevcl |
⊢ ( 𝐹 ∈ ( II Cn 𝐽 ) → ( 𝐼 ∈ ( II Cn 𝐽 ) ∧ ( 𝐼 ‘ 0 ) = ( 𝐹 ‘ 1 ) ∧ ( 𝐼 ‘ 1 ) = ( 𝐹 ‘ 0 ) ) ) |
44 |
6 43
|
syl |
⊢ ( 𝜑 → ( 𝐼 ∈ ( II Cn 𝐽 ) ∧ ( 𝐼 ‘ 0 ) = ( 𝐹 ‘ 1 ) ∧ ( 𝐼 ‘ 1 ) = ( 𝐹 ‘ 0 ) ) ) |
45 |
44
|
simp1d |
⊢ ( 𝜑 → 𝐼 ∈ ( II Cn 𝐽 ) ) |
46 |
|
oveq2 |
⊢ ( 𝑧 = 𝑦 → ( 1 − 𝑧 ) = ( 1 − 𝑦 ) ) |
47 |
46
|
fveq2d |
⊢ ( 𝑧 = 𝑦 → ( 𝐼 ‘ ( 1 − 𝑧 ) ) = ( 𝐼 ‘ ( 1 − 𝑦 ) ) ) |
48 |
47
|
cbvmptv |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) = ( 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑦 ) ) ) |
49 |
|
eqid |
⊢ ran ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ran ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
50 |
29 40 41 42 5 45 48 49
|
pi1xfrcnvlem |
⊢ ( 𝜑 → ◡ ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ⊆ ran ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
51 |
|
iitopon |
⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) |
52 |
|
cnf2 |
⊢ ( ( II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) ∧ 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ∈ ( II Cn 𝐽 ) ) → 𝐹 : ( 0 [,] 1 ) ⟶ 𝑋 ) |
53 |
51 5 6 52
|
mp3an2i |
⊢ ( 𝜑 → 𝐹 : ( 0 [,] 1 ) ⟶ 𝑋 ) |
54 |
53
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ 𝑧 ) ) ) |
55 |
|
iirev |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → ( 1 − 𝑧 ) ∈ ( 0 [,] 1 ) ) |
56 |
|
oveq2 |
⊢ ( 𝑥 = ( 1 − 𝑧 ) → ( 1 − 𝑥 ) = ( 1 − ( 1 − 𝑧 ) ) ) |
57 |
56
|
fveq2d |
⊢ ( 𝑥 = ( 1 − 𝑧 ) → ( 𝐹 ‘ ( 1 − 𝑥 ) ) = ( 𝐹 ‘ ( 1 − ( 1 − 𝑧 ) ) ) ) |
58 |
|
fvex |
⊢ ( 𝐹 ‘ ( 1 − ( 1 − 𝑧 ) ) ) ∈ V |
59 |
57 7 58
|
fvmpt |
⊢ ( ( 1 − 𝑧 ) ∈ ( 0 [,] 1 ) → ( 𝐼 ‘ ( 1 − 𝑧 ) ) = ( 𝐹 ‘ ( 1 − ( 1 − 𝑧 ) ) ) ) |
60 |
55 59
|
syl |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → ( 𝐼 ‘ ( 1 − 𝑧 ) ) = ( 𝐹 ‘ ( 1 − ( 1 − 𝑧 ) ) ) ) |
61 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
62 |
|
unitssre |
⊢ ( 0 [,] 1 ) ⊆ ℝ |
63 |
62
|
sseli |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → 𝑧 ∈ ℝ ) |
64 |
63
|
recnd |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → 𝑧 ∈ ℂ ) |
65 |
|
nncan |
⊢ ( ( 1 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( 1 − ( 1 − 𝑧 ) ) = 𝑧 ) |
66 |
61 64 65
|
sylancr |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → ( 1 − ( 1 − 𝑧 ) ) = 𝑧 ) |
67 |
66
|
fveq2d |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → ( 𝐹 ‘ ( 1 − ( 1 − 𝑧 ) ) ) = ( 𝐹 ‘ 𝑧 ) ) |
68 |
60 67
|
eqtrd |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) → ( 𝐼 ‘ ( 1 − 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) |
69 |
68
|
mpteq2ia |
⊢ ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ 𝑧 ) ) |
70 |
54 69
|
eqtr4di |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) |
71 |
70
|
oveq1d |
⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) = ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ) |
72 |
71
|
eceq1d |
⊢ ( 𝜑 → [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) = [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) ) |
73 |
72
|
opeq2d |
⊢ ( 𝜑 → 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 = 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
74 |
73
|
mpteq2dv |
⊢ ( 𝜑 → ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
75 |
74
|
rneqd |
⊢ ( 𝜑 → ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐹 ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
76 |
8 75
|
syl5eq |
⊢ ( 𝜑 → 𝐻 = ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
77 |
76
|
cnveqd |
⊢ ( 𝜑 → ◡ 𝐻 = ◡ ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
78 |
3
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝑃 ) ) |
79 |
78
|
unieqd |
⊢ ( 𝜑 → ∪ 𝐵 = ∪ ( Base ‘ 𝑃 ) ) |
80 |
70
|
oveq2d |
⊢ ( 𝜑 → ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) = ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) |
81 |
80
|
oveq2d |
⊢ ( 𝜑 → ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) = ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ) |
82 |
81
|
eceq1d |
⊢ ( 𝜑 → [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) ] ( ≃ph ‘ 𝐽 ) = [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) ) |
83 |
82
|
opeq2d |
⊢ ( 𝜑 → 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 = 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) |
84 |
79 83
|
mpteq12dv |
⊢ ( 𝜑 → ( 𝑔 ∈ ∪ 𝐵 ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
85 |
84
|
rneqd |
⊢ ( 𝜑 → ran ( 𝑔 ∈ ∪ 𝐵 ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) 𝐹 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) = ran ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
86 |
4 85
|
syl5eq |
⊢ ( 𝜑 → 𝐺 = ran ( 𝑔 ∈ ∪ ( Base ‘ 𝑃 ) ↦ 〈 [ 𝑔 ] ( ≃ph ‘ 𝐽 ) , [ ( 𝐼 ( *𝑝 ‘ 𝐽 ) ( 𝑔 ( *𝑝 ‘ 𝐽 ) ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
87 |
50 77 86
|
3sstr4d |
⊢ ( 𝜑 → ◡ 𝐻 ⊆ 𝐺 ) |
88 |
|
cnvss |
⊢ ( ◡ 𝐻 ⊆ 𝐺 → ◡ ◡ 𝐻 ⊆ ◡ 𝐺 ) |
89 |
87 88
|
syl |
⊢ ( 𝜑 → ◡ ◡ 𝐻 ⊆ ◡ 𝐺 ) |
90 |
19 89
|
eqsstrrd |
⊢ ( 𝜑 → 𝐻 ⊆ ◡ 𝐺 ) |
91 |
9 90
|
eqssd |
⊢ ( 𝜑 → ◡ 𝐺 = 𝐻 ) |
92 |
91 76
|
eqtrd |
⊢ ( 𝜑 → ◡ 𝐺 = ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ) |
93 |
29 40 41 42 5 45 48
|
pi1xfr |
⊢ ( 𝜑 → ran ( ℎ ∈ ∪ ( Base ‘ 𝑄 ) ↦ 〈 [ ℎ ] ( ≃ph ‘ 𝐽 ) , [ ( ( 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 𝐼 ‘ ( 1 − 𝑧 ) ) ) ( *𝑝 ‘ 𝐽 ) ( ℎ ( *𝑝 ‘ 𝐽 ) 𝐼 ) ) ] ( ≃ph ‘ 𝐽 ) 〉 ) ∈ ( 𝑄 GrpHom 𝑃 ) ) |
94 |
92 93
|
eqeltrd |
⊢ ( 𝜑 → ◡ 𝐺 ∈ ( 𝑄 GrpHom 𝑃 ) ) |
95 |
91 94
|
jca |
⊢ ( 𝜑 → ( ◡ 𝐺 = 𝐻 ∧ ◡ 𝐺 ∈ ( 𝑄 GrpHom 𝑃 ) ) ) |