Step | Hyp | Ref
| Expression |
1 | | elex 3185 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) |
2 | | symgtrf.g |
. . . . . . 7
⊢ 𝐺 = (SymGrp‘𝐷) |
3 | 2 | symggrp 17643 |
. . . . . 6
⊢ (𝐷 ∈ V → 𝐺 ∈ Grp) |
4 | | grpmnd 17252 |
. . . . . 6
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
5 | 3, 4 | syl 17 |
. . . . 5
⊢ (𝐷 ∈ V → 𝐺 ∈ Mnd) |
6 | | symgtrf.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐺) |
7 | 6 | submacs 17188 |
. . . . 5
⊢ (𝐺 ∈ Mnd →
(SubMnd‘𝐺) ∈
(ACS‘𝐵)) |
8 | | acsmre 16136 |
. . . . 5
⊢
((SubMnd‘𝐺)
∈ (ACS‘𝐵) →
(SubMnd‘𝐺) ∈
(Moore‘𝐵)) |
9 | 5, 7, 8 | 3syl 18 |
. . . 4
⊢ (𝐷 ∈ V →
(SubMnd‘𝐺) ∈
(Moore‘𝐵)) |
10 | 1, 9 | syl 17 |
. . 3
⊢ (𝐷 ∈ 𝑉 → (SubMnd‘𝐺) ∈ (Moore‘𝐵)) |
11 | | symgtrf.t |
. . . . . 6
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
12 | 11, 2, 6 | symgtrf 17712 |
. . . . 5
⊢ 𝑇 ⊆ 𝐵 |
13 | 12 | a1i 11 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵) |
14 | | 2onn 7607 |
. . . . . 6
⊢
2𝑜 ∈ ω |
15 | | nnfi 8038 |
. . . . . 6
⊢
(2𝑜 ∈ ω → 2𝑜
∈ Fin) |
16 | 14, 15 | ax-mp 5 |
. . . . 5
⊢
2𝑜 ∈ Fin |
17 | | eqid 2610 |
. . . . . . . . 9
⊢
(pmTrsp‘𝐷) =
(pmTrsp‘𝐷) |
18 | 17, 11 | pmtrfb 17708 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈
2𝑜)) |
19 | 18 | simp3bi 1071 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈
2𝑜) |
20 | | enfi 8061 |
. . . . . . 7
⊢ (dom
(𝑥 ∖ I ) ≈
2𝑜 → (dom (𝑥 ∖ I ) ∈ Fin ↔
2𝑜 ∈ Fin)) |
21 | 19, 20 | syl 17 |
. . . . . 6
⊢ (𝑥 ∈ 𝑇 → (dom (𝑥 ∖ I ) ∈ Fin ↔
2𝑜 ∈ Fin)) |
22 | 21 | adantl 481 |
. . . . 5
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔
2𝑜 ∈ Fin)) |
23 | 16, 22 | mpbiri 247 |
. . . 4
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin) |
24 | 13, 23 | ssrabdv 3644 |
. . 3
⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
25 | 2, 6 | symgfisg 17711 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈
(SubGrp‘𝐺)) |
26 | | subgsubm 17439 |
. . . 4
⊢ ({𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈
(SubGrp‘𝐺) →
{𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈
(SubMnd‘𝐺)) |
27 | 25, 26 | syl 17 |
. . 3
⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈
(SubMnd‘𝐺)) |
28 | | symggen.k |
. . . 4
⊢ 𝐾 =
(mrCls‘(SubMnd‘𝐺)) |
29 | 28 | mrcsscl 16103 |
. . 3
⊢
(((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}) |
30 | 10, 24, 27, 29 | syl3anc 1318 |
. 2
⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
31 | | vex 3176 |
. . . . . . 7
⊢ 𝑥 ∈ V |
32 | 31 | a1i 11 |
. . . . . 6
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ 𝑥 ∈
V) |
33 | | finnum 8657 |
. . . . . 6
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ dom (𝑥 ∖ I )
∈ dom card) |
34 | | domfi 8066 |
. . . . . . . 8
⊢ ((dom
(𝑥 ∖ I ) ∈ Fin
∧ dom (𝑦 ∖ I )
≼ dom (𝑥 ∖ I ))
→ dom (𝑦 ∖ I )
∈ Fin) |
35 | 2, 6 | symgbasf1o 17626 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷) |
36 | 35 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷) |
37 | | f1ofn 6051 |
. . . . . . . . . . . . . 14
⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷) |
38 | | fnnfpeq0 6349 |
. . . . . . . . . . . . . 14
⊢ (𝑦 Fn 𝐷 → (dom (𝑦 ∖ I ) = ∅ ↔ 𝑦 = ( I ↾ 𝐷))) |
39 | 36, 37, 38 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ ↔
𝑦 = ( I ↾ 𝐷))) |
40 | 2, 6 | elbasfv 15748 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) |
41 | 40 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V) |
42 | 2 | symgid 17644 |
. . . . . . . . . . . . . . . 16
⊢ (𝐷 ∈ V → ( I ↾
𝐷) =
(0g‘𝐺)) |
43 | 41, 42 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) = (0g‘𝐺)) |
44 | 41, 9 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵)) |
45 | 28 | mrccl 16094 |
. . . . . . . . . . . . . . . . 17
⊢
(((SubMnd‘𝐺)
∈ (Moore‘𝐵)
∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺)) |
46 | 44, 12, 45 | sylancl 693 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺)) |
47 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢
(0g‘𝐺) = (0g‘𝐺) |
48 | 47 | subm0cl 17175 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐾‘𝑇) ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ (𝐾‘𝑇)) |
49 | 46, 48 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) →
(0g‘𝐺)
∈ (𝐾‘𝑇)) |
50 | 43, 49 | eqeltrd 2688 |
. . . . . . . . . . . . . 14
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) ∈ (𝐾‘𝑇)) |
51 | | eleq1a 2683 |
. . . . . . . . . . . . . 14
⊢ (( I
↾ 𝐷) ∈ (𝐾‘𝑇) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇))) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (𝑦 = ( I ↾ 𝐷) → 𝑦 ∈ (𝐾‘𝑇))) |
53 | 39, 52 | sylbid 229 |
. . . . . . . . . . . 12
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ →
𝑦 ∈ (𝐾‘𝑇))) |
54 | 53 | adantr 480 |
. . . . . . . . . . 11
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (dom (𝑦 ∖ I ) = ∅ → 𝑦 ∈ (𝐾‘𝑇))) |
55 | | n0 3890 |
. . . . . . . . . . . 12
⊢ (dom
(𝑦 ∖ I ) ≠ ∅
↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I )) |
56 | 41 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐷 ∈ V) |
57 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ dom (𝑦 ∖ I )) |
58 | | f1omvdmvd 17686 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢})) |
59 | 36, 58 | sylan 487 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢})) |
60 | 59 | eldifad 3552 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ dom (𝑦 ∖ I )) |
61 | | prssi 4293 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑢 ∈ dom (𝑦 ∖ I ) ∧ (𝑦‘𝑢) ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I )) |
62 | 57, 60, 61 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I )) |
63 | | difss 3699 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∖ I ) ⊆ 𝑦 |
64 | | dmss 5245 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦) |
65 | 63, 64 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ dom
(𝑦 ∖ I ) ⊆ dom
𝑦 |
66 | | f1odm 6054 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦:𝐷–1-1-onto→𝐷 → dom 𝑦 = 𝐷) |
67 | 36, 66 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → dom 𝑦 = 𝐷) |
68 | 65, 67 | syl5sseq 3616 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷) |
69 | 68 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷) |
70 | 62, 69 | sstrd 3578 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷) |
71 | 36 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦:𝐷–1-1-onto→𝐷) |
72 | 71, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 Fn 𝐷) |
73 | 68 | sselda 3568 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷) |
74 | | fnelnfp 6348 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢)) |
75 | 72, 73, 74 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢)) |
76 | 57, 75 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢) |
77 | 76 | necomd 2837 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢)) |
78 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑢 ∈ V |
79 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦‘𝑢) ∈ V |
80 | | pr2nelem 8710 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈
2𝑜) |
81 | 78, 79, 80 | mp3an12 1406 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑢 ≠ (𝑦‘𝑢) → {𝑢, (𝑦‘𝑢)} ≈
2𝑜) |
82 | 77, 81 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈
2𝑜) |
83 | 17, 11 | pmtrrn 17700 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2𝑜) →
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇) |
84 | 56, 70, 82, 83 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇) |
85 | 12, 84 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵) |
86 | | simplr 788 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵) |
87 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(+g‘𝐺) = (+g‘𝐺) |
88 | 2, 6, 87 | symgov 17633 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) |
89 | 85, 86, 88 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) |
90 | 89 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (((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))) |
91 | 41, 3 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) |
92 | 91 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp) |
93 | 6, 87 | grpcl 17253 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐺 ∈ Grp ∧
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵) |
94 | 92, 85, 86, 93 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵) |
95 | 89, 94 | eqeltrrd 2689 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵) |
96 | 2, 6, 87 | symgov 17633 |
. . . . . . . . . . . . . . . . . 18
⊢
((((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))) |
97 | 85, 95, 96 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (((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))) |
98 | | coass 5571 |
. . . . . . . . . . . . . . . . . 18
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) |
99 | 17, 11 | pmtrfinv 17704 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷)) |
100 | 84, 99 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷)) |
101 | 100 | coeq1d 5205 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (( I ↾ 𝐷) ∘ 𝑦)) |
102 | | f1of 6050 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷) |
103 | | fcoi2 5992 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦) |
104 | 71, 102, 103 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦) |
105 | 101, 104 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = 𝑦) |
106 | 98, 105 | syl5eqr 2658 |
. . . . . . . . . . . . . . . . 17
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = 𝑦) |
107 | 90, 97, 106 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . 16
⊢ (((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) |
108 | 107 | adantlr 747 |
. . . . . . . . . . . . . . 15
⊢ ((((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) |
109 | 46 | ad2antrr 758 |
. . . . . . . . . . . . . . . 16
⊢ ((((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)) |
110 | 28 | mrcssid 16100 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((SubMnd‘𝐺)
∈ (Moore‘𝐵)
∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇)) |
111 | 44, 12, 110 | sylancl 693 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇)) |
112 | 111 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇)) |
113 | 112, 84 | sseldd 3569 |
. . . . . . . . . . . . . . . . 17
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇)) |
114 | 113 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((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)) |
115 | 89 | difeq1d 3689 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((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 )) |
116 | 115 | dmeqd 5248 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((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 )) |
117 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin) |
118 | | mvdco 17688 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) |
119 | 17 | pmtrmvd 17699 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2𝑜) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)}) |
120 | 56, 70, 82, 119 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)}) |
121 | 120, 62 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I )) |
122 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ dom
(𝑦 ∖ I ) ⊆ dom
(𝑦 ∖ I
) |
123 | 122 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I )) |
124 | 121, 123 | unssd 3751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) ⊆ dom (𝑦 ∖ I )) |
125 | 118, 124 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I )) |
126 | | fvco2 6183 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢))) |
127 | 72, 73, 126 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢))) |
128 | | prcom 4211 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢} |
129 | 128 | fveq2i 6106 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢}) |
130 | 129 | fveq1i 6104 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) |
131 | 69, 60 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷) |
132 | 17 | pmtrprfv 17696 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐷 ∈ V ∧ ((𝑦‘𝑢) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ (𝑦‘𝑢) ≠ 𝑢)) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢) |
133 | 56, 131, 73, 76, 132 | syl13anc 1320 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢) |
134 | 130, 133 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢) |
135 | 127, 134 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢) |
136 | 2, 6 | symgbasf1o 17626 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷) |
137 | | f1ofn 6051 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷) |
138 | 95, 136, 137 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷) |
139 | | fnelnfp 6348 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((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)) |
140 | 139 | necon2bbid 2825 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((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 ))) |
141 | 138, 73, 140 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((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 ))) |
142 | 135, 141 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I )) |
143 | 125, 57, 142 | ssnelpssd 3681 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I )) |
144 | | php3 8031 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((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 )) |
145 | 117, 143,
144 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I )) |
146 | 116, 145 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I )) |
147 | 146 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ ((((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 )) |
148 | 94 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ ((((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) |
149 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ V |
150 | | difeq1 3683 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I )) |
151 | 150 | dmeqd 5248 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → dom (𝑧 ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I )) |
152 | 151 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (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 ))) |
153 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵)) |
154 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∈ (𝐾‘𝑇) ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ (𝐾‘𝑇))) |
155 | 153, 154 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (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)))) |
156 | 152, 155 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (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))))) |
157 | 149, 156 | spcv 3272 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀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)))) |
158 | 157 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢ ((((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)))) |
159 | 147, 148,
158 | mp2d 47 |
. . . . . . . . . . . . . . . 16
⊢ ((((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)) |
160 | 87 | submcl 17176 |
. . . . . . . . . . . . . . . 16
⊢ (((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)) |
161 | 109, 114,
159, 160 | syl3anc 1318 |
. . . . . . . . . . . . . . 15
⊢ ((((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)) |
162 | 108, 161 | eqeltrrd 2689 |
. . . . . . . . . . . . . 14
⊢ ((((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)) |
163 | 162 | ex 449 |
. . . . . . . . . . . . 13
⊢ (((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))) |
164 | 163 | exlimdv 1848 |
. . . . . . . . . . . 12
⊢ (((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))) |
165 | 55, 164 | syl5bi 231 |
. . . . . . . . . . 11
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → (dom (𝑦 ∖ I ) ≠ ∅ → 𝑦 ∈ (𝐾‘𝑇))) |
166 | 54, 165 | pm2.61dne 2868 |
. . . . . . . . . 10
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → 𝑦 ∈ (𝐾‘𝑇)) |
167 | 166 | exp31 628 |
. . . . . . . . 9
⊢ (dom
(𝑦 ∖ I ) ∈ Fin
→ (𝑦 ∈ 𝐵 → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → 𝑦 ∈ (𝐾‘𝑇)))) |
168 | 167 | com23 84 |
. . . . . . . 8
⊢ (dom
(𝑦 ∖ I ) ∈ Fin
→ (∀𝑧(dom
(𝑧 ∖ I ) ≺ dom
(𝑦 ∖ I ) →
(𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)))) |
169 | 34, 168 | syl 17 |
. . . . . . 7
⊢ ((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)))) |
170 | 169 | 3impia 1253 |
. . . . . 6
⊢ ((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))) |
171 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)) |
172 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇))) |
173 | 171, 172 | imbi12d 333 |
. . . . . 6
⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) |
174 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
175 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇))) |
176 | 174, 175 | imbi12d 333 |
. . . . . 6
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))) |
177 | | difeq1 3683 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I )) |
178 | 177 | dmeqd 5248 |
. . . . . 6
⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I )) |
179 | | difeq1 3683 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I )) |
180 | 179 | dmeqd 5248 |
. . . . . 6
⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I )) |
181 | 32, 33, 170, 173, 176, 178, 180 | indcardi 8747 |
. . . . 5
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))) |
182 | 181 | impcom 445 |
. . . 4
⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇)) |
183 | 182 | 3adant1 1072 |
. . 3
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇)) |
184 | 183 | rabssdv 3645 |
. 2
⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇)) |
185 | 30, 184 | eqssd 3585 |
1
⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |