Step | Hyp | Ref
| Expression |
1 | | elex 3433 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) |
2 | | symgtrf.g |
. . . . . . 7
⊢ 𝐺 = (SymGrp‘𝐷) |
3 | 2 | symggrp 18289 |
. . . . . 6
⊢ (𝐷 ∈ V → 𝐺 ∈ Grp) |
4 | | grpmnd 17898 |
. . . . . 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 17833 |
. . . . 5
⊢ (𝐺 ∈ Mnd →
(SubMnd‘𝐺) ∈
(ACS‘𝐵)) |
8 | | acsmre 16781 |
. . . . 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 18358 |
. . . . 5
⊢ 𝑇 ⊆ 𝐵 |
13 | 12 | a1i 11 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ 𝐵) |
14 | | 2onn 8067 |
. . . . . 6
⊢
2o ∈ ω |
15 | | nnfi 8506 |
. . . . . 6
⊢
(2o ∈ ω → 2o ∈
Fin) |
16 | 14, 15 | ax-mp 5 |
. . . . 5
⊢
2o ∈ Fin |
17 | | eqid 2778 |
. . . . . . . . 9
⊢
(pmTrsp‘𝐷) =
(pmTrsp‘𝐷) |
18 | 17, 11 | pmtrfb 18354 |
. . . . . . . 8
⊢ (𝑥 ∈ 𝑇 ↔ (𝐷 ∈ V ∧ 𝑥:𝐷–1-1-onto→𝐷 ∧ dom (𝑥 ∖ I ) ≈
2o)) |
19 | 18 | simp3bi 1127 |
. . . . . . 7
⊢ (𝑥 ∈ 𝑇 → dom (𝑥 ∖ I ) ≈
2o) |
20 | | enfi 8529 |
. . . . . . 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 474 |
. . . . 5
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → (dom (𝑥 ∖ I ) ∈ Fin ↔ 2o
∈ Fin)) |
23 | 16, 22 | mpbiri 250 |
. . . 4
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → dom (𝑥 ∖ I ) ∈ Fin) |
24 | 13, 23 | ssrabdv 3940 |
. . 3
⊢ (𝐷 ∈ 𝑉 → 𝑇 ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
25 | 2, 6 | symgfisg 18357 |
. . . 4
⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈
(SubGrp‘𝐺)) |
26 | | subgsubm 18085 |
. . . 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 16749 |
. . 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 1351 |
. 2
⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) ⊆ {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |
31 | | vex 3418 |
. . . . . . 7
⊢ 𝑥 ∈ V |
32 | 31 | a1i 11 |
. . . . . 6
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ 𝑥 ∈
V) |
33 | | finnum 9171 |
. . . . . 6
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ dom (𝑥 ∖ I )
∈ dom card) |
34 | | domfi 8534 |
. . . . . . . 8
⊢ ((dom
(𝑥 ∖ I ) ∈ Fin
∧ dom (𝑦 ∖ I )
≼ dom (𝑥 ∖ I ))
→ dom (𝑦 ∖ I )
∈ Fin) |
35 | 2, 6 | symgbasf1o 18272 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝐵 → 𝑦:𝐷–1-1-onto→𝐷) |
36 | 35 | adantl 474 |
. . . . . . . . . . . . . 14
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝑦:𝐷–1-1-onto→𝐷) |
37 | | f1ofn 6445 |
. . . . . . . . . . . . . 14
⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦 Fn 𝐷) |
38 | | fnnfpeq0 6763 |
. . . . . . . . . . . . . 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 16400 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) |
41 | 40 | adantl 474 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ V) |
42 | 2 | symgid 18290 |
. . . . . . . . . . . . . . . 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 16740 |
. . . . . . . . . . . . . . . . 17
⊢
(((SubMnd‘𝐺)
∈ (Moore‘𝐵)
∧ 𝑇 ⊆ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺)) |
46 | 44, 12, 45 | sylancl 577 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (𝐾‘𝑇) ∈ (SubMnd‘𝐺)) |
47 | | eqid 2778 |
. . . . . . . . . . . . . . . . 17
⊢
(0g‘𝐺) = (0g‘𝐺) |
48 | 47 | subm0cl 17820 |
. . . . . . . . . . . . . . . 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 2866 |
. . . . . . . . . . . . . 14
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → ( I ↾ 𝐷) ∈ (𝐾‘𝑇)) |
51 | | eleq1a 2861 |
. . . . . . . . . . . . . 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 232 |
. . . . . . . . . . . 12
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → (dom (𝑦 ∖ I ) = ∅ →
𝑦 ∈ (𝐾‘𝑇))) |
54 | 53 | adantr 473 |
. . . . . . . . . . 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 4196 |
. . . . . . . . . . . 12
⊢ (dom
(𝑦 ∖ I ) ≠ ∅
↔ ∃𝑢 𝑢 ∈ dom (𝑦 ∖ I )) |
56 | 41 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐷 ∈ V) |
57 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ dom (𝑦 ∖ I )) |
58 | | f1omvdmvd 18332 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦:𝐷–1-1-onto→𝐷 ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢})) |
59 | 36, 58 | sylan 572 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ (dom (𝑦 ∖ I ) ∖ {𝑢})) |
60 | 59 | eldifad 3841 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ dom (𝑦 ∖ I )) |
61 | 57, 60 | prssd 4629 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ dom (𝑦 ∖ I )) |
62 | | difss 3998 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∖ I ) ⊆ 𝑦 |
63 | | dmss 5621 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 ∖ I ) ⊆ 𝑦 → dom (𝑦 ∖ I ) ⊆ dom 𝑦) |
64 | 62, 63 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ dom
(𝑦 ∖ I ) ⊆ dom
𝑦 |
65 | | f1odm 6448 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦:𝐷–1-1-onto→𝐷 → dom 𝑦 = 𝐷) |
66 | 36, 65 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → dom 𝑦 = 𝐷) |
67 | 64, 66 | syl5sseq 3909 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → dom (𝑦 ∖ I ) ⊆ 𝐷) |
68 | 67 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ 𝐷) |
69 | 61, 68 | sstrd 3868 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ⊆ 𝐷) |
70 | | vex 3418 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑢 ∈ V |
71 | | fvex 6512 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦‘𝑢) ∈ V |
72 | 36 | adantr 473 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦:𝐷–1-1-onto→𝐷) |
73 | 72, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 Fn 𝐷) |
74 | 67 | sselda 3858 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ∈ 𝐷) |
75 | | fnelnfp 6762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢)) |
76 | 73, 74, 75 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑢 ∈ dom (𝑦 ∖ I ) ↔ (𝑦‘𝑢) ≠ 𝑢)) |
77 | 57, 76 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ≠ 𝑢) |
78 | 77 | necomd 3022 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑢 ≠ (𝑦‘𝑢)) |
79 | | pr2nelem 9224 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑢 ∈ V ∧ (𝑦‘𝑢) ∈ V ∧ 𝑢 ≠ (𝑦‘𝑢)) → {𝑢, (𝑦‘𝑢)} ≈ 2o) |
80 | 70, 71, 78, 79 | mp3an12i 1444 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → {𝑢, (𝑦‘𝑢)} ≈ 2o) |
81 | 17, 11 | pmtrrn 18346 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2o) →
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇) |
82 | 56, 69, 80, 81 | syl3anc 1351 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇) |
83 | 12, 82 | sseldi 3856 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵) |
84 | | simplr 756 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑦 ∈ 𝐵) |
85 | | eqid 2778 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(+g‘𝐺) = (+g‘𝐺) |
86 | 2, 6, 85 | symgov 18279 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) |
87 | 83, 84, 86 | syl2anc 576 |
. . . . . . . . . . . . . . . . . 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)) |
88 | 87 | oveq2d 6992 |
. . . . . . . . . . . . . . . . 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))) |
89 | 41, 3 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Grp) |
90 | 89 | adantr 473 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝐺 ∈ Grp) |
91 | 6, 85 | grpcl 17899 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐺 ∈ Grp ∧
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵) |
92 | 90, 83, 84, 91 | syl3anc 1351 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵) |
93 | 87, 92 | eqeltrrd 2867 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵) |
94 | 2, 6, 85 | symgov 18279 |
. . . . . . . . . . . . . . . . . 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))) |
95 | 83, 93, 94 | syl2anc 576 |
. . . . . . . . . . . . . . . . 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))) |
96 | | coass 5957 |
. . . . . . . . . . . . . . . . . 18
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) |
97 | 17, 11 | pmtrfinv 18350 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ 𝑇 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) = ( I ↾ 𝐷)) |
98 | 82, 97 | 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)) |
99 | 98 | coeq1d 5582 |
. . . . . . . . . . . . . . . . . . 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)) |
100 | | f1of 6444 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦:𝐷–1-1-onto→𝐷 → 𝑦:𝐷⟶𝐷) |
101 | | fcoi2 6382 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦:𝐷⟶𝐷 → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦) |
102 | 72, 100, 101 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (( I ↾ 𝐷) ∘ 𝑦) = 𝑦) |
103 | 99, 102 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})) ∘ 𝑦) = 𝑦) |
104 | 96, 103 | syl5eqr 2828 |
. . . . . . . . . . . . . . . . 17
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)) = 𝑦) |
105 | 88, 95, 104 | 3eqtrd 2818 |
. . . . . . . . . . . . . . . 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) |
106 | 105 | adantlr 702 |
. . . . . . . . . . . . . . 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) |
107 | 46 | ad2antrr 713 |
. . . . . . . . . . . . . . . 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)) |
108 | 28 | mrcssid 16746 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((SubMnd‘𝐺)
∈ (Moore‘𝐵)
∧ 𝑇 ⊆ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇)) |
109 | 44, 12, 108 | sylancl 577 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) → 𝑇 ⊆ (𝐾‘𝑇)) |
110 | 109 | adantr 473 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → 𝑇 ⊆ (𝐾‘𝑇)) |
111 | 110, 82 | sseldd 3859 |
. . . . . . . . . . . . . . . . 17
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∈ (𝐾‘𝑇)) |
112 | 111 | adantlr 702 |
. . . . . . . . . . . . . . . 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)) |
113 | 87 | difeq1d 3988 |
. . . . . . . . . . . . . . . . . . . 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 )) |
114 | 113 | dmeqd 5624 |
. . . . . . . . . . . . . . . . . . 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 )) |
115 | | simpll 754 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ∈ Fin) |
116 | | mvdco 18334 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ (dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) |
117 | 17 | pmtrmvd 18345 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐷 ∈ V ∧ {𝑢, (𝑦‘𝑢)} ⊆ 𝐷 ∧ {𝑢, (𝑦‘𝑢)} ≈ 2o) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)}) |
118 | 56, 69, 80, 117 | syl3anc 1351 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) = {𝑢, (𝑦‘𝑢)}) |
119 | 118, 61 | eqsstrd 3895 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ⊆ dom (𝑦 ∖ I )) |
120 | | ssidd 3880 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom (𝑦 ∖ I ) ⊆ dom (𝑦 ∖ I )) |
121 | 119, 120 | unssd 4050 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (dom
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∖ I ) ∪ dom (𝑦 ∖ I )) ⊆ dom (𝑦 ∖ I )) |
122 | 116, 121 | syl5ss 3869 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊆ dom (𝑦 ∖ I )) |
123 | | fvco2 6586 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 Fn 𝐷 ∧ 𝑢 ∈ 𝐷) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢))) |
124 | 73, 74, 123 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . . 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))) |
125 | | prcom 4542 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ {𝑢, (𝑦‘𝑢)} = {(𝑦‘𝑢), 𝑢} |
126 | 125 | fveq2i 6502 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) = ((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢}) |
127 | 126 | fveq1i 6500 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) |
128 | 68, 60 | sseldd 3859 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (𝑦‘𝑢) ∈ 𝐷) |
129 | 17 | pmtrprfv 18342 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐷 ∈ V ∧ ((𝑦‘𝑢) ∈ 𝐷 ∧ 𝑢 ∈ 𝐷 ∧ (𝑦‘𝑢) ≠ 𝑢)) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢) |
130 | 56, 128, 74, 77, 129 | syl13anc 1352 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{(𝑦‘𝑢), 𝑢})‘(𝑦‘𝑢)) = 𝑢) |
131 | 127, 130 | syl5eq 2826 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})‘(𝑦‘𝑢)) = 𝑢) |
132 | 124, 131 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦)‘𝑢) = 𝑢) |
133 | 2, 6 | symgbasf1o 18272 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∈ 𝐵 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷) |
134 | | f1ofn 6445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦):𝐷–1-1-onto→𝐷 → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷) |
135 | 93, 133, 134 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) Fn 𝐷) |
136 | | fnelnfp 6762 |
. . . . . . . . . . . . . . . . . . . . . . . 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)) |
137 | 136 | necon2bbid 3010 |
. . . . . . . . . . . . . . . . . . . . . . 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 ))) |
138 | 135, 74, 137 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . . . . 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 ))) |
139 | 132, 138 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → ¬ 𝑢 ∈ dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I )) |
140 | 122, 57, 139 | ssnelpssd 3979 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ⊊ dom (𝑦 ∖ I )) |
141 | | php3 8499 |
. . . . . . . . . . . . . . . . . . . 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 )) |
142 | 115, 140,
141 | syl2anc 576 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)}) ∘ 𝑦) ∖ I ) ≺ dom (𝑦 ∖ I )) |
143 | 114, 142 | eqbrtrd 4951 |
. . . . . . . . . . . . . . . . . 18
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ dom (𝑦 ∖ I )) → dom
((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I ) ≺ dom (𝑦 ∖ I )) |
144 | 143 | adantlr 702 |
. . . . . . . . . . . . . . . . 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 )) |
145 | 92 | adantlr 702 |
. . . . . . . . . . . . . . . . 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) |
146 | | ovex 7008 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ V |
147 | | difeq1 3982 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∖ I ) = ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I )) |
148 | 147 | dmeqd 5624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → dom (𝑧 ∖ I ) = dom ((((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∖ I )) |
149 | 148 | breq1d 4939 |
. . . . . . . . . . . . . . . . . . . 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 ))) |
150 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∈ 𝐵 ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ 𝐵)) |
151 | | eleq1 2853 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) → (𝑧 ∈ (𝐾‘𝑇) ↔ (((pmTrsp‘𝐷)‘{𝑢, (𝑦‘𝑢)})(+g‘𝐺)𝑦) ∈ (𝐾‘𝑇))) |
152 | 150, 151 | imbi12d 337 |
. . . . . . . . . . . . . . . . . . . 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)))) |
153 | 149, 152 | imbi12d 337 |
. . . . . . . . . . . . . . . . . . 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))))) |
154 | 146, 153 | spcv 3524 |
. . . . . . . . . . . . . . . . . 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)))) |
155 | 154 | ad2antlr 714 |
. . . . . . . . . . . . . . . . 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)))) |
156 | 144, 145,
155 | mp2d 49 |
. . . . . . . . . . . . . . . 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)) |
157 | 85 | submcl 17821 |
. . . . . . . . . . . . . . . 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)) |
158 | 107, 112,
156, 157 | syl3anc 1351 |
. . . . . . . . . . . . . . 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)) |
159 | 106, 158 | eqeltrrd 2867 |
. . . . . . . . . . . . . 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)) |
160 | 159 | ex 405 |
. . . . . . . . . . . . 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))) |
161 | 160 | exlimdv 1892 |
. . . . . . . . . . . 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))) |
162 | 55, 161 | syl5bi 234 |
. . . . . . . . . . 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))) |
163 | 54, 162 | pm2.61dne 3054 |
. . . . . . . . . 10
⊢ (((dom
(𝑦 ∖ I ) ∈ Fin
∧ 𝑦 ∈ 𝐵) ∧ ∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) → 𝑦 ∈ (𝐾‘𝑇)) |
164 | 163 | exp31 412 |
. . . . . . . . 9
⊢ (dom
(𝑦 ∖ I ) ∈ Fin
→ (𝑦 ∈ 𝐵 → (∀𝑧(dom (𝑧 ∖ I ) ≺ dom (𝑦 ∖ I ) → (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → 𝑦 ∈ (𝐾‘𝑇)))) |
165 | 164 | com23 86 |
. . . . . . . 8
⊢ (dom
(𝑦 ∖ I ) ∈ Fin
→ (∀𝑧(dom
(𝑧 ∖ I ) ≺ dom
(𝑦 ∖ I ) →
(𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇))) → (𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)))) |
166 | 34, 165 | 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)))) |
167 | 166 | 3impia 1097 |
. . . . . 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))) |
168 | | eleq1w 2848 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵)) |
169 | | eleq1w 2848 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑧 ∈ (𝐾‘𝑇))) |
170 | 168, 169 | imbi12d 337 |
. . . . . 6
⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑧 ∈ 𝐵 → 𝑧 ∈ (𝐾‘𝑇)))) |
171 | | eleq1w 2848 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) |
172 | | eleq1w 2848 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∈ (𝐾‘𝑇) ↔ 𝑥 ∈ (𝐾‘𝑇))) |
173 | 171, 172 | imbi12d 337 |
. . . . . 6
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ (𝐾‘𝑇)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇)))) |
174 | | difeq1 3982 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → (𝑦 ∖ I ) = (𝑧 ∖ I )) |
175 | 174 | dmeqd 5624 |
. . . . . 6
⊢ (𝑦 = 𝑧 → dom (𝑦 ∖ I ) = dom (𝑧 ∖ I )) |
176 | | difeq1 3982 |
. . . . . . 7
⊢ (𝑦 = 𝑥 → (𝑦 ∖ I ) = (𝑥 ∖ I )) |
177 | 176 | dmeqd 5624 |
. . . . . 6
⊢ (𝑦 = 𝑥 → dom (𝑦 ∖ I ) = dom (𝑥 ∖ I )) |
178 | 32, 33, 167, 170, 173, 175, 177 | indcardi 9261 |
. . . . 5
⊢ (dom
(𝑥 ∖ I ) ∈ Fin
→ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐾‘𝑇))) |
179 | 178 | impcom 399 |
. . . 4
⊢ ((𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇)) |
180 | 179 | 3adant1 1110 |
. . 3
⊢ ((𝐷 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵 ∧ dom (𝑥 ∖ I ) ∈ Fin) → 𝑥 ∈ (𝐾‘𝑇)) |
181 | 180 | rabssdv 3941 |
. 2
⊢ (𝐷 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ⊆ (𝐾‘𝑇)) |
182 | 30, 181 | eqssd 3875 |
1
⊢ (𝐷 ∈ 𝑉 → (𝐾‘𝑇) = {𝑥 ∈ 𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin}) |