Proof of Theorem symgcom2
| Step | Hyp | Ref
| Expression |
| 1 | | symgcom.g |
. 2
⊢ 𝐺 = (SymGrp‘𝐴) |
| 2 | | symgcom.b |
. 2
⊢ 𝐵 = (Base‘𝐺) |
| 3 | | symgcom.x |
. 2
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| 4 | | symgcom.y |
. 2
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| 5 | 1, 2 | symgbasf 19393 |
. . . . . 6
⊢ (𝑋 ∈ 𝐵 → 𝑋:𝐴⟶𝐴) |
| 6 | 3, 5 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑋:𝐴⟶𝐴) |
| 7 | 6 | ffnd 6737 |
. . . 4
⊢ (𝜑 → 𝑋 Fn 𝐴) |
| 8 | | fnresi 6697 |
. . . . 5
⊢ ( I
↾ 𝐴) Fn 𝐴 |
| 9 | 8 | a1i 11 |
. . . 4
⊢ (𝜑 → ( I ↾ 𝐴) Fn 𝐴) |
| 10 | | difssd 4137 |
. . . 4
⊢ (𝜑 → (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ 𝐴) |
| 11 | | ssidd 4007 |
. . . . 5
⊢ (𝜑 → (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ (𝐴 ∖ dom (𝑋 ∖ I ))) |
| 12 | | nfpconfp 32642 |
. . . . . . 7
⊢ (𝑋 Fn 𝐴 → (𝐴 ∖ dom (𝑋 ∖ I )) = dom (𝑋 ∩ I )) |
| 13 | 7, 12 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∖ dom (𝑋 ∖ I )) = dom (𝑋 ∩ I )) |
| 14 | | inres 6015 |
. . . . . . . 8
⊢ (𝑋 ∩ ( I ↾ 𝐴)) = ((𝑋 ∩ I ) ↾ 𝐴) |
| 15 | | reli 5836 |
. . . . . . . . . 10
⊢ Rel
I |
| 16 | | relin2 5823 |
. . . . . . . . . 10
⊢ (Rel I
→ Rel (𝑋 ∩ I
)) |
| 17 | 15, 16 | ax-mp 5 |
. . . . . . . . 9
⊢ Rel
(𝑋 ∩ I
) |
| 18 | 13, 10 | eqsstrrd 4019 |
. . . . . . . . 9
⊢ (𝜑 → dom (𝑋 ∩ I ) ⊆ 𝐴) |
| 19 | | relssres 6040 |
. . . . . . . . 9
⊢ ((Rel
(𝑋 ∩ I ) ∧ dom
(𝑋 ∩ I ) ⊆ 𝐴) → ((𝑋 ∩ I ) ↾ 𝐴) = (𝑋 ∩ I )) |
| 20 | 17, 18, 19 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → ((𝑋 ∩ I ) ↾ 𝐴) = (𝑋 ∩ I )) |
| 21 | 14, 20 | eqtrid 2789 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∩ ( I ↾ 𝐴)) = (𝑋 ∩ I )) |
| 22 | 21 | dmeqd 5916 |
. . . . . 6
⊢ (𝜑 → dom (𝑋 ∩ ( I ↾ 𝐴)) = dom (𝑋 ∩ I )) |
| 23 | 13, 22 | eqtr4d 2780 |
. . . . 5
⊢ (𝜑 → (𝐴 ∖ dom (𝑋 ∖ I )) = dom (𝑋 ∩ ( I ↾ 𝐴))) |
| 24 | 11, 23 | sseqtrd 4020 |
. . . 4
⊢ (𝜑 → (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ dom (𝑋 ∩ ( I ↾ 𝐴))) |
| 25 | | fnreseql 7068 |
. . . . 5
⊢ ((𝑋 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴 ∧ (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ 𝐴) → ((𝑋 ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) = (( I ↾ 𝐴) ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) ↔ (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ dom (𝑋 ∩ ( I ↾ 𝐴)))) |
| 26 | 25 | biimpar 477 |
. . . 4
⊢ (((𝑋 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴 ∧ (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ 𝐴) ∧ (𝐴 ∖ dom (𝑋 ∖ I )) ⊆ dom (𝑋 ∩ ( I ↾ 𝐴))) → (𝑋 ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) = (( I ↾ 𝐴) ↾ (𝐴 ∖ dom (𝑋 ∖ I )))) |
| 27 | 7, 9, 10, 24, 26 | syl31anc 1375 |
. . 3
⊢ (𝜑 → (𝑋 ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) = (( I ↾ 𝐴) ↾ (𝐴 ∖ dom (𝑋 ∖ I )))) |
| 28 | 10 | resabs1d 6026 |
. . 3
⊢ (𝜑 → (( I ↾ 𝐴) ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) = ( I ↾ (𝐴 ∖ dom (𝑋 ∖ I )))) |
| 29 | 27, 28 | eqtrd 2777 |
. 2
⊢ (𝜑 → (𝑋 ↾ (𝐴 ∖ dom (𝑋 ∖ I ))) = ( I ↾ (𝐴 ∖ dom (𝑋 ∖ I )))) |
| 30 | 1, 2 | symgbasf 19393 |
. . . . . 6
⊢ (𝑌 ∈ 𝐵 → 𝑌:𝐴⟶𝐴) |
| 31 | 4, 30 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑌:𝐴⟶𝐴) |
| 32 | 31 | ffnd 6737 |
. . . 4
⊢ (𝜑 → 𝑌 Fn 𝐴) |
| 33 | | difss 4136 |
. . . . . 6
⊢ (𝑋 ∖ I ) ⊆ 𝑋 |
| 34 | | dmss 5913 |
. . . . . 6
⊢ ((𝑋 ∖ I ) ⊆ 𝑋 → dom (𝑋 ∖ I ) ⊆ dom 𝑋) |
| 35 | 33, 34 | ax-mp 5 |
. . . . 5
⊢ dom
(𝑋 ∖ I ) ⊆ dom
𝑋 |
| 36 | | fdm 6745 |
. . . . . 6
⊢ (𝑋:𝐴⟶𝐴 → dom 𝑋 = 𝐴) |
| 37 | 3, 5, 36 | 3syl 18 |
. . . . 5
⊢ (𝜑 → dom 𝑋 = 𝐴) |
| 38 | 35, 37 | sseqtrid 4026 |
. . . 4
⊢ (𝜑 → dom (𝑋 ∖ I ) ⊆ 𝐴) |
| 39 | | symgcom2.1 |
. . . . . . 7
⊢ (𝜑 → (dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅) |
| 40 | | reldisj 4453 |
. . . . . . . 8
⊢ (dom
(𝑋 ∖ I ) ⊆
𝐴 → ((dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅ ↔
dom (𝑋 ∖ I ) ⊆
(𝐴 ∖ dom (𝑌 ∖ I )))) |
| 41 | 38, 40 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((dom (𝑋 ∖ I ) ∩ dom (𝑌 ∖ I )) = ∅ ↔ dom (𝑋 ∖ I ) ⊆ (𝐴 ∖ dom (𝑌 ∖ I )))) |
| 42 | 39, 41 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → dom (𝑋 ∖ I ) ⊆ (𝐴 ∖ dom (𝑌 ∖ I ))) |
| 43 | | nfpconfp 32642 |
. . . . . . 7
⊢ (𝑌 Fn 𝐴 → (𝐴 ∖ dom (𝑌 ∖ I )) = dom (𝑌 ∩ I )) |
| 44 | 32, 43 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∖ dom (𝑌 ∖ I )) = dom (𝑌 ∩ I )) |
| 45 | 42, 44 | sseqtrd 4020 |
. . . . 5
⊢ (𝜑 → dom (𝑋 ∖ I ) ⊆ dom (𝑌 ∩ I )) |
| 46 | | inres 6015 |
. . . . . . 7
⊢ (𝑌 ∩ ( I ↾ 𝐴)) = ((𝑌 ∩ I ) ↾ 𝐴) |
| 47 | | relin2 5823 |
. . . . . . . . 9
⊢ (Rel I
→ Rel (𝑌 ∩ I
)) |
| 48 | 15, 47 | ax-mp 5 |
. . . . . . . 8
⊢ Rel
(𝑌 ∩ I
) |
| 49 | | difssd 4137 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∖ dom (𝑌 ∖ I )) ⊆ 𝐴) |
| 50 | 44, 49 | eqsstrrd 4019 |
. . . . . . . 8
⊢ (𝜑 → dom (𝑌 ∩ I ) ⊆ 𝐴) |
| 51 | | relssres 6040 |
. . . . . . . 8
⊢ ((Rel
(𝑌 ∩ I ) ∧ dom
(𝑌 ∩ I ) ⊆ 𝐴) → ((𝑌 ∩ I ) ↾ 𝐴) = (𝑌 ∩ I )) |
| 52 | 48, 50, 51 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → ((𝑌 ∩ I ) ↾ 𝐴) = (𝑌 ∩ I )) |
| 53 | 46, 52 | eqtrid 2789 |
. . . . . 6
⊢ (𝜑 → (𝑌 ∩ ( I ↾ 𝐴)) = (𝑌 ∩ I )) |
| 54 | 53 | dmeqd 5916 |
. . . . 5
⊢ (𝜑 → dom (𝑌 ∩ ( I ↾ 𝐴)) = dom (𝑌 ∩ I )) |
| 55 | 45, 54 | sseqtrrd 4021 |
. . . 4
⊢ (𝜑 → dom (𝑋 ∖ I ) ⊆ dom (𝑌 ∩ ( I ↾ 𝐴))) |
| 56 | | fnreseql 7068 |
. . . . 5
⊢ ((𝑌 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴 ∧ dom (𝑋 ∖ I ) ⊆ 𝐴) → ((𝑌 ↾ dom (𝑋 ∖ I )) = (( I ↾ 𝐴) ↾ dom (𝑋 ∖ I )) ↔ dom (𝑋 ∖ I ) ⊆ dom (𝑌 ∩ ( I ↾ 𝐴)))) |
| 57 | 56 | biimpar 477 |
. . . 4
⊢ (((𝑌 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴 ∧ dom (𝑋 ∖ I ) ⊆ 𝐴) ∧ dom (𝑋 ∖ I ) ⊆ dom (𝑌 ∩ ( I ↾ 𝐴))) → (𝑌 ↾ dom (𝑋 ∖ I )) = (( I ↾ 𝐴) ↾ dom (𝑋 ∖ I ))) |
| 58 | 32, 9, 38, 55, 57 | syl31anc 1375 |
. . 3
⊢ (𝜑 → (𝑌 ↾ dom (𝑋 ∖ I )) = (( I ↾ 𝐴) ↾ dom (𝑋 ∖ I ))) |
| 59 | 38 | resabs1d 6026 |
. . 3
⊢ (𝜑 → (( I ↾ 𝐴) ↾ dom (𝑋 ∖ I )) = ( I ↾ dom (𝑋 ∖ I ))) |
| 60 | 58, 59 | eqtrd 2777 |
. 2
⊢ (𝜑 → (𝑌 ↾ dom (𝑋 ∖ I )) = ( I ↾ dom (𝑋 ∖ I ))) |
| 61 | | difin2 4301 |
. . . 4
⊢ (dom
(𝑋 ∖ I ) ⊆
𝐴 → (dom (𝑋 ∖ I ) ∖ dom (𝑋 ∖ I )) = ((𝐴 ∖ dom (𝑋 ∖ I )) ∩ dom (𝑋 ∖ I ))) |
| 62 | 38, 61 | syl 17 |
. . 3
⊢ (𝜑 → (dom (𝑋 ∖ I ) ∖ dom (𝑋 ∖ I )) = ((𝐴 ∖ dom (𝑋 ∖ I )) ∩ dom (𝑋 ∖ I ))) |
| 63 | | difid 4376 |
. . 3
⊢ (dom
(𝑋 ∖ I ) ∖ dom
(𝑋 ∖ I )) =
∅ |
| 64 | 62, 63 | eqtr3di 2792 |
. 2
⊢ (𝜑 → ((𝐴 ∖ dom (𝑋 ∖ I )) ∩ dom (𝑋 ∖ I )) = ∅) |
| 65 | | undif1 4476 |
. . 3
⊢ ((𝐴 ∖ dom (𝑋 ∖ I )) ∪ dom (𝑋 ∖ I )) = (𝐴 ∪ dom (𝑋 ∖ I )) |
| 66 | | ssequn2 4189 |
. . . 4
⊢ (dom
(𝑋 ∖ I ) ⊆
𝐴 ↔ (𝐴 ∪ dom (𝑋 ∖ I )) = 𝐴) |
| 67 | 38, 66 | sylib 218 |
. . 3
⊢ (𝜑 → (𝐴 ∪ dom (𝑋 ∖ I )) = 𝐴) |
| 68 | 65, 67 | eqtrid 2789 |
. 2
⊢ (𝜑 → ((𝐴 ∖ dom (𝑋 ∖ I )) ∪ dom (𝑋 ∖ I )) = 𝐴) |
| 69 | 1, 2, 3, 4, 29, 60, 64, 68 | symgcom 33103 |
1
⊢ (𝜑 → (𝑋 ∘ 𝑌) = (𝑌 ∘ 𝑋)) |