Step | Hyp | Ref
| Expression |
1 | | symgext.s |
. . 3
⊢ 𝑆 =
(Base‘(SymGrp‘(𝑁 ∖ {𝐾}))) |
2 | | symgext.e |
. . 3
⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥))) |
3 | 1, 2 | symgextf 18187 |
. 2
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁⟶𝑁) |
4 | | difsnid 4559 |
. . . . . . . 8
⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁) |
5 | 4 | eqcomd 2831 |
. . . . . . 7
⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾})) |
6 | 5 | eleq2d 2892 |
. . . . . 6
⊢ (𝐾 ∈ 𝑁 → (𝑦 ∈ 𝑁 ↔ 𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}))) |
7 | 5 | eleq2d 2892 |
. . . . . 6
⊢ (𝐾 ∈ 𝑁 → (𝑧 ∈ 𝑁 ↔ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}))) |
8 | 6, 7 | anbi12d 626 |
. . . . 5
⊢ (𝐾 ∈ 𝑁 → ((𝑦 ∈ 𝑁 ∧ 𝑧 ∈ 𝑁) ↔ (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})))) |
9 | 8 | adantr 474 |
. . . 4
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑦 ∈ 𝑁 ∧ 𝑧 ∈ 𝑁) ↔ (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})))) |
10 | | elun 3980 |
. . . . . 6
⊢ (𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↔ (𝑦 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑦 ∈ {𝐾})) |
11 | | elun 3980 |
. . . . . 6
⊢ (𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ↔ (𝑧 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑧 ∈ {𝐾})) |
12 | 1, 2 | symgextfv 18188 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑦 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑦) = (𝑍‘𝑦))) |
13 | 12 | com12 32 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ (𝑁 ∖ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑦) = (𝑍‘𝑦))) |
14 | 13 | adantr 474 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑦) = (𝑍‘𝑦))) |
15 | 14 | imp 397 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆)) → (𝐸‘𝑦) = (𝑍‘𝑦)) |
16 | 1, 2 | symgextfv 18188 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝑧 ∈ (𝑁 ∖ {𝐾}) → (𝐸‘𝑧) = (𝑍‘𝑧))) |
17 | 16 | com12 32 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ (𝑁 ∖ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑧) = (𝑍‘𝑧))) |
18 | 17 | adantl 475 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → (𝐸‘𝑧) = (𝑍‘𝑧))) |
19 | 18 | imp 397 |
. . . . . . . . . 10
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆)) → (𝐸‘𝑧) = (𝑍‘𝑧)) |
20 | 15, 19 | eqeq12d 2840 |
. . . . . . . . 9
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆)) → ((𝐸‘𝑦) = (𝐸‘𝑧) ↔ (𝑍‘𝑦) = (𝑍‘𝑧))) |
21 | | eqid 2825 |
. . . . . . . . . . . . 13
⊢
(SymGrp‘(𝑁
∖ {𝐾})) =
(SymGrp‘(𝑁 ∖
{𝐾})) |
22 | 21, 1 | symgbasf1o 18153 |
. . . . . . . . . . . 12
⊢ (𝑍 ∈ 𝑆 → 𝑍:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾})) |
23 | | f1of1 6377 |
. . . . . . . . . . . 12
⊢ (𝑍:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾})) |
24 | | dff13 6767 |
. . . . . . . . . . . . 13
⊢ (𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾}) ↔ (𝑍:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗))) |
25 | | fveqeq2 6442 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑦 → ((𝑍‘𝑖) = (𝑍‘𝑗) ↔ (𝑍‘𝑦) = (𝑍‘𝑗))) |
26 | | equequ1 2131 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑦 → (𝑖 = 𝑗 ↔ 𝑦 = 𝑗)) |
27 | 25, 26 | imbi12d 336 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑦 → (((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗) ↔ ((𝑍‘𝑦) = (𝑍‘𝑗) → 𝑦 = 𝑗))) |
28 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑧 → (𝑍‘𝑗) = (𝑍‘𝑧)) |
29 | 28 | eqeq2d 2835 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑧 → ((𝑍‘𝑦) = (𝑍‘𝑗) ↔ (𝑍‘𝑦) = (𝑍‘𝑧))) |
30 | | equequ2 2132 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑧 → (𝑦 = 𝑗 ↔ 𝑦 = 𝑧)) |
31 | 29, 30 | imbi12d 336 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑧 → (((𝑍‘𝑦) = (𝑍‘𝑗) → 𝑦 = 𝑗) ↔ ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧))) |
32 | 27, 31 | rspc2va 3540 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗)) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)) |
33 | 32 | expcom 404 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑖 ∈
(𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧))) |
34 | 33 | a1d 25 |
. . . . . . . . . . . . . 14
⊢
(∀𝑖 ∈
(𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗) → (𝐾 ∈ 𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)))) |
35 | 34 | adantl 475 |
. . . . . . . . . . . . 13
⊢ ((𝑍:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})∀𝑗 ∈ (𝑁 ∖ {𝐾})((𝑍‘𝑖) = (𝑍‘𝑗) → 𝑖 = 𝑗)) → (𝐾 ∈ 𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)))) |
36 | 24, 35 | sylbi 209 |
. . . . . . . . . . . 12
⊢ (𝑍:(𝑁 ∖ {𝐾})–1-1→(𝑁 ∖ {𝐾}) → (𝐾 ∈ 𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)))) |
37 | 22, 23, 36 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝑍 ∈ 𝑆 → (𝐾 ∈ 𝑁 → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)))) |
38 | 37 | impcom 398 |
. . . . . . . . . 10
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧))) |
39 | 38 | impcom 398 |
. . . . . . . . 9
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆)) → ((𝑍‘𝑦) = (𝑍‘𝑧) → 𝑦 = 𝑧)) |
40 | 20, 39 | sylbid 232 |
. . . . . . . 8
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) ∧ (𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆)) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧)) |
41 | 40 | ex 403 |
. . . . . . 7
⊢ ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
42 | 1, 2 | symgextf1lem 18190 |
. . . . . . . . 9
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑧 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑦 ∈ {𝐾}) → (𝐸‘𝑧) ≠ (𝐸‘𝑦))) |
43 | | eqneqall 3010 |
. . . . . . . . . . 11
⊢ ((𝐸‘𝑧) = (𝐸‘𝑦) → ((𝐸‘𝑧) ≠ (𝐸‘𝑦) → 𝑦 = 𝑧)) |
44 | 43 | eqcoms 2833 |
. . . . . . . . . 10
⊢ ((𝐸‘𝑦) = (𝐸‘𝑧) → ((𝐸‘𝑧) ≠ (𝐸‘𝑦) → 𝑦 = 𝑧)) |
45 | 44 | com12 32 |
. . . . . . . . 9
⊢ ((𝐸‘𝑧) ≠ (𝐸‘𝑦) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧)) |
46 | 42, 45 | syl6com 37 |
. . . . . . . 8
⊢ ((𝑧 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑦 ∈ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
47 | 46 | ancoms 452 |
. . . . . . 7
⊢ ((𝑦 ∈ {𝐾} ∧ 𝑧 ∈ (𝑁 ∖ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
48 | 1, 2 | symgextf1lem 18190 |
. . . . . . . 8
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ {𝐾}) → (𝐸‘𝑦) ≠ (𝐸‘𝑧))) |
49 | | eqneqall 3010 |
. . . . . . . . 9
⊢ ((𝐸‘𝑦) = (𝐸‘𝑧) → ((𝐸‘𝑦) ≠ (𝐸‘𝑧) → 𝑦 = 𝑧)) |
50 | 49 | com12 32 |
. . . . . . . 8
⊢ ((𝐸‘𝑦) ≠ (𝐸‘𝑧) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧)) |
51 | 48, 50 | syl6com 37 |
. . . . . . 7
⊢ ((𝑦 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑧 ∈ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
52 | | elsni 4414 |
. . . . . . . 8
⊢ (𝑦 ∈ {𝐾} → 𝑦 = 𝐾) |
53 | | elsni 4414 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝐾} → 𝑧 = 𝐾) |
54 | | eqtr3 2848 |
. . . . . . . . 9
⊢ ((𝑦 = 𝐾 ∧ 𝑧 = 𝐾) → 𝑦 = 𝑧) |
55 | 54 | 2a1d 26 |
. . . . . . . 8
⊢ ((𝑦 = 𝐾 ∧ 𝑧 = 𝐾) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
56 | 52, 53, 55 | syl2an 591 |
. . . . . . 7
⊢ ((𝑦 ∈ {𝐾} ∧ 𝑧 ∈ {𝐾}) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
57 | 41, 47, 51, 56 | ccase 1066 |
. . . . . 6
⊢ (((𝑦 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑦 ∈ {𝐾}) ∧ (𝑧 ∈ (𝑁 ∖ {𝐾}) ∨ 𝑧 ∈ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
58 | 10, 11, 57 | syl2anb 593 |
. . . . 5
⊢ ((𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})) → ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
59 | 58 | com12 32 |
. . . 4
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑦 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∧ 𝑧 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
60 | 9, 59 | sylbid 232 |
. . 3
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ((𝑦 ∈ 𝑁 ∧ 𝑧 ∈ 𝑁) → ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
61 | 60 | ralrimivv 3179 |
. 2
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧)) |
62 | | dff13 6767 |
. 2
⊢ (𝐸:𝑁–1-1→𝑁 ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝐸‘𝑦) = (𝐸‘𝑧) → 𝑦 = 𝑧))) |
63 | 3, 61, 62 | sylanbrc 580 |
1
⊢ ((𝐾 ∈ 𝑁 ∧ 𝑍 ∈ 𝑆) → 𝐸:𝑁–1-1→𝑁) |