| Step |
Hyp |
Ref |
Expression |
| 1 |
|
symgtrf.t |
⊢ 𝑇 = ran ( pmTrsp ‘ 𝐷 ) |
| 2 |
|
symgtrf.g |
⊢ 𝐺 = ( SymGrp ‘ 𝐷 ) |
| 3 |
|
symgtrf.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
| 4 |
|
symggen.k |
⊢ 𝐾 = ( mrCls ‘ ( SubMnd ‘ 𝐺 ) ) |
| 5 |
|
elex |
⊢ ( 𝐷 ∈ 𝑉 → 𝐷 ∈ V ) |
| 6 |
2
|
symggrp |
⊢ ( 𝐷 ∈ V → 𝐺 ∈ Grp ) |
| 7 |
6
|
grpmndd |
⊢ ( 𝐷 ∈ V → 𝐺 ∈ Mnd ) |
| 8 |
3
|
submacs |
⊢ ( 𝐺 ∈ Mnd → ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |
| 9 |
|
acsmre |
⊢ ( ( SubMnd ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 10 |
7 8 9
|
3syl |
⊢ ( 𝐷 ∈ V → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 11 |
5 10
|
syl |
⊢ ( 𝐷 ∈ 𝑉 → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 12 |
1 2 3
|
symgtrf |
⊢ 𝑇 ⊆ 𝐵 |
| 13 |
12
|
a1i |
⊢ ( 𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵 ) |
| 14 |
|
2onn |
⊢ 2o ∈ ω |
| 15 |
|
nnfi |
⊢ ( 2o ∈ ω → 2o ∈ Fin ) |
| 16 |
14 15
|
ax-mp |
⊢ 2o ∈ Fin |
| 17 |
|
eqid |
⊢ ( pmTrsp ‘ 𝐷 ) = ( pmTrsp ‘ 𝐷 ) |
| 18 |
17 1
|
pmtrfb |
⊢ ( 𝑥 ∈ 𝑇 ↔ ( 𝐷 ∈ V ∧ 𝑥 : 𝐷 –1-1-onto→ 𝐷 ∧ dom ( 𝑥 ∖ I ) ≈ 2o ) ) |
| 19 |
18
|
simp3bi |
⊢ ( 𝑥 ∈ 𝑇 → dom ( 𝑥 ∖ I ) ≈ 2o ) |
| 20 |
|
enfi |
⊢ ( dom ( 𝑥 ∖ I ) ≈ 2o → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 21 |
19 20
|
syl |
⊢ ( 𝑥 ∈ 𝑇 → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 22 |
21
|
adantl |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) → ( dom ( 𝑥 ∖ I ) ∈ Fin ↔ 2o ∈ Fin ) ) |
| 23 |
16 22
|
mpbiri |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇 ) → dom ( 𝑥 ∖ I ) ∈ Fin ) |
| 24 |
13 23
|
ssrabdv |
⊢ ( 𝐷 ∈ 𝑉 → 𝑇 ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 25 |
2 3
|
symgfisg |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubGrp ‘ 𝐺 ) ) |
| 26 |
|
subgsubm |
⊢ ( { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubGrp ‘ 𝐺 ) → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) |
| 27 |
25 26
|
syl |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) |
| 28 |
4
|
mrcsscl |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∧ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ∈ ( SubMnd ‘ 𝐺 ) ) → ( 𝐾 ‘ 𝑇 ) ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 29 |
11 24 27 28
|
syl3anc |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝐾 ‘ 𝑇 ) ⊆ { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |
| 30 |
|
vex |
⊢ 𝑥 ∈ V |
| 31 |
30
|
a1i |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → 𝑥 ∈ V ) |
| 32 |
|
finnum |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → dom ( 𝑥 ∖ I ) ∈ dom card ) |
| 33 |
|
domfi |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ) → dom ( 𝑦 ∖ I ) ∈ Fin ) |
| 34 |
2 3
|
symgbasf1o |
⊢ ( 𝑦 ∈ 𝐵 → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 35 |
34
|
adantl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 36 |
|
f1ofn |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → 𝑦 Fn 𝐷 ) |
| 37 |
|
fnnfpeq0 |
⊢ ( 𝑦 Fn 𝐷 → ( dom ( 𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷 ) ) ) |
| 38 |
35 36 37
|
3syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( dom ( 𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷 ) ) ) |
| 39 |
2 3
|
elbasfv |
⊢ ( 𝑦 ∈ 𝐵 → 𝐷 ∈ V ) |
| 40 |
39
|
adantl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝐷 ∈ V ) |
| 41 |
2
|
symgid |
⊢ ( 𝐷 ∈ V → ( I ↾ 𝐷 ) = ( 0g ‘ 𝐺 ) ) |
| 42 |
40 41
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( I ↾ 𝐷 ) = ( 0g ‘ 𝐺 ) ) |
| 43 |
40 10
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
| 44 |
4
|
mrccl |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ 𝐵 ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 45 |
43 12 44
|
sylancl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 46 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
| 47 |
46
|
subm0cl |
⊢ ( ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) → ( 0g ‘ 𝐺 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 48 |
45 47
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 0g ‘ 𝐺 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 49 |
42 48
|
eqeltrd |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( I ↾ 𝐷 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 50 |
|
eleq1a |
⊢ ( ( I ↾ 𝐷 ) ∈ ( 𝐾 ‘ 𝑇 ) → ( 𝑦 = ( I ↾ 𝐷 ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 51 |
49 50
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 = ( I ↾ 𝐷 ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 52 |
38 51
|
sylbid |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → ( dom ( 𝑦 ∖ I ) = ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 53 |
52
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( dom ( 𝑦 ∖ I ) = ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 54 |
|
n0 |
⊢ ( dom ( 𝑦 ∖ I ) ≠ ∅ ↔ ∃ 𝑢 𝑢 ∈ dom ( 𝑦 ∖ I ) ) |
| 55 |
40
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝐷 ∈ V ) |
| 56 |
|
simpr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ∈ dom ( 𝑦 ∖ I ) ) |
| 57 |
|
f1omvdmvd |
⊢ ( ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ ( dom ( 𝑦 ∖ I ) ∖ { 𝑢 } ) ) |
| 58 |
35 57
|
sylan |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ ( dom ( 𝑦 ∖ I ) ∖ { 𝑢 } ) ) |
| 59 |
58
|
eldifad |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ dom ( 𝑦 ∖ I ) ) |
| 60 |
56 59
|
prssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ dom ( 𝑦 ∖ I ) ) |
| 61 |
|
difss |
⊢ ( 𝑦 ∖ I ) ⊆ 𝑦 |
| 62 |
|
dmss |
⊢ ( ( 𝑦 ∖ I ) ⊆ 𝑦 → dom ( 𝑦 ∖ I ) ⊆ dom 𝑦 ) |
| 63 |
61 62
|
ax-mp |
⊢ dom ( 𝑦 ∖ I ) ⊆ dom 𝑦 |
| 64 |
|
f1odm |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → dom 𝑦 = 𝐷 ) |
| 65 |
35 64
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → dom 𝑦 = 𝐷 ) |
| 66 |
63 65
|
sseqtrid |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → dom ( 𝑦 ∖ I ) ⊆ 𝐷 ) |
| 67 |
66
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ⊆ 𝐷 ) |
| 68 |
60 67
|
sstrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ) |
| 69 |
|
vex |
⊢ 𝑢 ∈ V |
| 70 |
|
fvex |
⊢ ( 𝑦 ‘ 𝑢 ) ∈ V |
| 71 |
35
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 : 𝐷 –1-1-onto→ 𝐷 ) |
| 72 |
71 36
|
syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 Fn 𝐷 ) |
| 73 |
66
|
sselda |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ∈ 𝐷 ) |
| 74 |
|
fnelnfp |
⊢ ( ( 𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) ↔ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 75 |
72 73 74
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) ↔ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 76 |
56 75
|
mpbid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) |
| 77 |
76
|
necomd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑢 ≠ ( 𝑦 ‘ 𝑢 ) ) |
| 78 |
|
pr2nelem |
⊢ ( ( 𝑢 ∈ V ∧ ( 𝑦 ‘ 𝑢 ) ∈ V ∧ 𝑢 ≠ ( 𝑦 ‘ 𝑢 ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) |
| 79 |
69 70 77 78
|
mp3an12i |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) |
| 80 |
17 1
|
pmtrrn |
⊢ ( ( 𝐷 ∈ V ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 ) |
| 81 |
55 68 79 80
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 ) |
| 82 |
12 81
|
sselid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ) |
| 83 |
|
simplr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 ∈ 𝐵 ) |
| 84 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
| 85 |
2 3 84
|
symgov |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 86 |
82 83 85
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 87 |
86
|
oveq2d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 88 |
40 6
|
syl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝐺 ∈ Grp ) |
| 89 |
88
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝐺 ∈ Grp ) |
| 90 |
3 84
|
grpcl |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 91 |
89 82 83 90
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 92 |
86 91
|
eqeltrrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 ) |
| 93 |
2 3 84
|
symgov |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝐵 ∧ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 94 |
82 92 93
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) ) |
| 95 |
|
coass |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) |
| 96 |
17 1
|
pmtrfinv |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ 𝑇 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) = ( I ↾ 𝐷 ) ) |
| 97 |
81 96
|
syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) = ( I ↾ 𝐷 ) ) |
| 98 |
97
|
coeq1d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = ( ( I ↾ 𝐷 ) ∘ 𝑦 ) ) |
| 99 |
|
f1of |
⊢ ( 𝑦 : 𝐷 –1-1-onto→ 𝐷 → 𝑦 : 𝐷 ⟶ 𝐷 ) |
| 100 |
|
fcoi2 |
⊢ ( 𝑦 : 𝐷 ⟶ 𝐷 → ( ( I ↾ 𝐷 ) ∘ 𝑦 ) = 𝑦 ) |
| 101 |
71 99 100
|
3syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( I ↾ 𝐷 ) ∘ 𝑦 ) = 𝑦 ) |
| 102 |
98 101
|
eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ) ∘ 𝑦 ) = 𝑦 ) |
| 103 |
95 102
|
eqtr3id |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ) = 𝑦 ) |
| 104 |
87 94 103
|
3eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = 𝑦 ) |
| 105 |
104
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) = 𝑦 ) |
| 106 |
45
|
ad2antrr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ) |
| 107 |
4
|
mrcssid |
⊢ ( ( ( SubMnd ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝑇 ⊆ 𝐵 ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 108 |
43 12 107
|
sylancl |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 109 |
108
|
adantr |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑇 ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 110 |
109 81
|
sseldd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 111 |
110
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 112 |
86
|
difeq1d |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) = ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 113 |
112
|
dmeqd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) = dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 114 |
|
simpll |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ∈ Fin ) |
| 115 |
|
mvdco |
⊢ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊆ ( dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ∪ dom ( 𝑦 ∖ I ) ) |
| 116 |
17
|
pmtrmvd |
⊢ ( ( 𝐷 ∈ V ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ⊆ 𝐷 ∧ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ≈ 2o ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) = { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) |
| 117 |
55 68 79 116
|
syl3anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) = { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) |
| 118 |
117 60
|
eqsstrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 119 |
|
ssidd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( 𝑦 ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 120 |
118 119
|
unssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( dom ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∖ I ) ∪ dom ( 𝑦 ∖ I ) ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 121 |
115 120
|
sstrid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊆ dom ( 𝑦 ∖ I ) ) |
| 122 |
|
fvco2 |
⊢ ( ( 𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) ) |
| 123 |
72 73 122
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) ) |
| 124 |
|
prcom |
⊢ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } = { ( 𝑦 ‘ 𝑢 ) , 𝑢 } |
| 125 |
124
|
fveq2i |
⊢ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) = ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) |
| 126 |
125
|
fveq1i |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) |
| 127 |
67 59
|
sseldd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( 𝑦 ‘ 𝑢 ) ∈ 𝐷 ) |
| 128 |
17
|
pmtrprfv |
⊢ ( ( 𝐷 ∈ V ∧ ( ( 𝑦 ‘ 𝑢 ) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ ( 𝑦 ‘ 𝑢 ) ≠ 𝑢 ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 129 |
55 127 73 76 128
|
syl13anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { ( 𝑦 ‘ 𝑢 ) , 𝑢 } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 130 |
126 129
|
syl5eq |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ‘ ( 𝑦 ‘ 𝑢 ) ) = 𝑢 ) |
| 131 |
123 130
|
eqtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ) |
| 132 |
2 3
|
symgbasf1o |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) : 𝐷 –1-1-onto→ 𝐷 ) |
| 133 |
|
f1ofn |
⊢ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) : 𝐷 –1-1-onto→ 𝐷 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ) |
| 134 |
92 132 133
|
3syl |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ) |
| 135 |
|
fnelnfp |
⊢ ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ↔ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) ≠ 𝑢 ) ) |
| 136 |
135
|
necon2bbid |
⊢ ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) Fn 𝐷 ∧ 𝑢 ∈ 𝐷 ) → ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ↔ ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) ) |
| 137 |
134 73 136
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ‘ 𝑢 ) = 𝑢 ↔ ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) ) |
| 138 |
131 137
|
mpbid |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ¬ 𝑢 ∈ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ) |
| 139 |
121 56 138
|
ssnelpssd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊊ dom ( 𝑦 ∖ I ) ) |
| 140 |
|
php3 |
⊢ ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ⊊ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 141 |
114 139 140
|
syl2anc |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∘ 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 142 |
113 141
|
eqbrtrd |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 143 |
142
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) |
| 144 |
91
|
adantlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) |
| 145 |
|
ovex |
⊢ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ V |
| 146 |
|
difeq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∖ I ) = ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ) |
| 147 |
146
|
dmeqd |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → dom ( 𝑧 ∖ I ) = dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ) |
| 148 |
147
|
breq1d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) ↔ dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) ) ) |
| 149 |
|
eleq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∈ 𝐵 ↔ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 ) ) |
| 150 |
|
eleq1 |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ↔ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 151 |
149 150
|
imbi12d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 152 |
148 151
|
imbi12d |
⊢ ( 𝑧 = ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) → ( ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ↔ ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ) |
| 153 |
145 152
|
spcv |
⊢ ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 154 |
153
|
ad2antlr |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( dom ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝐵 → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 155 |
143 144 154
|
mp2d |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 156 |
84
|
submcl |
⊢ ( ( ( 𝐾 ‘ 𝑇 ) ∈ ( SubMnd ‘ 𝐺 ) ∧ ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ∈ ( 𝐾 ‘ 𝑇 ) ∧ ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ∈ ( 𝐾 ‘ 𝑇 ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 157 |
106 111 155 156
|
syl3anc |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) ( ( ( pmTrsp ‘ 𝐷 ) ‘ { 𝑢 , ( 𝑦 ‘ 𝑢 ) } ) ( +g ‘ 𝐺 ) 𝑦 ) ) ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 158 |
105 157
|
eqeltrrd |
⊢ ( ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) ∧ 𝑢 ∈ dom ( 𝑦 ∖ I ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 159 |
158
|
ex |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( 𝑢 ∈ dom ( 𝑦 ∖ I ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 160 |
159
|
exlimdv |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( ∃ 𝑢 𝑢 ∈ dom ( 𝑦 ∖ I ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 161 |
54 160
|
syl5bi |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( dom ( 𝑦 ∖ I ) ≠ ∅ → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 162 |
53 161
|
pm2.61dne |
⊢ ( ( ( dom ( 𝑦 ∖ I ) ∈ Fin ∧ 𝑦 ∈ 𝐵 ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 163 |
162
|
exp31 |
⊢ ( dom ( 𝑦 ∖ I ) ∈ Fin → ( 𝑦 ∈ 𝐵 → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 164 |
163
|
com23 |
⊢ ( dom ( 𝑦 ∖ I ) ∈ Fin → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 165 |
33 164
|
syl |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ) → ( ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 166 |
165
|
3impia |
⊢ ( ( dom ( 𝑥 ∖ I ) ∈ Fin ∧ dom ( 𝑦 ∖ I ) ≼ dom ( 𝑥 ∖ I ) ∧ ∀ 𝑧 ( dom ( 𝑧 ∖ I ) ≺ dom ( 𝑦 ∖ I ) → ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 167 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) ) |
| 168 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ↔ 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 169 |
167 168
|
imbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( 𝑧 ∈ 𝐵 → 𝑧 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 170 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) ) |
| 171 |
|
eleq1w |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ↔ 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 172 |
170 171
|
imbi12d |
⊢ ( 𝑦 = 𝑥 → ( ( 𝑦 ∈ 𝐵 → 𝑦 ∈ ( 𝐾 ‘ 𝑇 ) ) ↔ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) ) |
| 173 |
|
difeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∖ I ) = ( 𝑧 ∖ I ) ) |
| 174 |
173
|
dmeqd |
⊢ ( 𝑦 = 𝑧 → dom ( 𝑦 ∖ I ) = dom ( 𝑧 ∖ I ) ) |
| 175 |
|
difeq1 |
⊢ ( 𝑦 = 𝑥 → ( 𝑦 ∖ I ) = ( 𝑥 ∖ I ) ) |
| 176 |
175
|
dmeqd |
⊢ ( 𝑦 = 𝑥 → dom ( 𝑦 ∖ I ) = dom ( 𝑥 ∖ I ) ) |
| 177 |
31 32 166 169 172 174 176
|
indcardi |
⊢ ( dom ( 𝑥 ∖ I ) ∈ Fin → ( 𝑥 ∈ 𝐵 → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) ) |
| 178 |
177
|
impcom |
⊢ ( ( 𝑥 ∈ 𝐵 ∧ dom ( 𝑥 ∖ I ) ∈ Fin ) → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 179 |
178
|
3adant1 |
⊢ ( ( 𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom ( 𝑥 ∖ I ) ∈ Fin ) → 𝑥 ∈ ( 𝐾 ‘ 𝑇 ) ) |
| 180 |
179
|
rabssdv |
⊢ ( 𝐷 ∈ 𝑉 → { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ⊆ ( 𝐾 ‘ 𝑇 ) ) |
| 181 |
29 180
|
eqssd |
⊢ ( 𝐷 ∈ 𝑉 → ( 𝐾 ‘ 𝑇 ) = { 𝑥 ∈ 𝐵 ∣ dom ( 𝑥 ∖ I ) ∈ Fin } ) |