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| Mirrors > Home > MPE Home > Th. List > invisoinvl | Structured version Visualization version GIF version | ||
| Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17812 and is therefore denoted by ((𝑋𝑁𝑌)‘𝐹), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.) |
| Ref | Expression |
|---|---|
| invisoinv.b | ⊢ 𝐵 = (Base‘𝐶) |
| invisoinv.i | ⊢ 𝐼 = (Iso‘𝐶) |
| invisoinv.n | ⊢ 𝑁 = (Inv‘𝐶) |
| invisoinv.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
| invisoinv.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| invisoinv.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| invisoinv.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
| Ref | Expression |
|---|---|
| invisoinvl | ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | invisoinv.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | invisoinv.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
| 3 | invisoinv.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 4 | invisoinv.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | invisoinv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | invisoinv.i | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
| 7 | invisoinv.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
| 8 | eqid 2737 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
| 10 | 1, 9, 3, 5 | idiso 17832 | . . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌)) |
| 11 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) |
| 12 | 11 | oveqd 7448 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌)) |
| 13 | 10, 12 | eleqtrrd 2844 | . . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌)) |
| 14 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 13 | invco 17815 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹)(𝑋𝑁𝑌)(((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)))) |
| 15 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 16 | 1, 15, 6, 3, 4, 5 | isohom 17820 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
| 17 | 16, 7 | sseldd 3984 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
| 18 | 1, 15, 9, 3, 4, 8, 5, 17 | catlid 17726 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹) = 𝐹) |
| 19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶)) |
| 20 | 19 | oveqd 7448 | . . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌)) |
| 21 | 20 | fveq1d 6908 | . . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌))) |
| 22 | 1, 9, 3, 5 | idinv 17833 | . . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
| 23 | 21, 22 | eqtrd 2777 | . . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
| 24 | 23 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌))) |
| 25 | 1, 15, 6, 3, 5, 4 | isohom 17820 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
| 26 | 1, 2, 3, 4, 5, 6 | invf 17812 | . . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
| 27 | 26, 7 | ffvelcdmd 7105 | . . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
| 28 | 25, 27 | sseldd 3984 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
| 29 | 1, 15, 9, 3, 5, 8, 4, 28 | catrid 17727 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹)) |
| 30 | 24, 29 | eqtrd 2777 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹)) |
| 31 | 14, 18, 30 | 3brtr3d 5174 | . 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
| 32 | 1, 2, 3, 5, 4 | invsym 17806 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))) |
| 33 | 31, 32 | mpbird 257 | 1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Hom chom 17308 compcco 17309 Catccat 17707 Idccid 17708 Invcinv 17789 Isociso 17790 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-cat 17711 df-cid 17712 df-sect 17791 df-inv 17792 df-iso 17793 |
| This theorem is referenced by: invisoinvr 17835 isocoinvid 17837 |
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