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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 16251 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-2017.) |
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 2610 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | eqid 2610 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
10 | 1, 9, 3, 5 | idiso 16271 | . . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌)) |
11 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) |
12 | 11 | oveqd 6566 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌)) |
13 | 10, 12 | eleqtrrd 2691 | . . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 13 | invco 16254 | . . 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 2610 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
16 | 1, 15, 6, 3, 4, 5 | isohom 16259 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
17 | 16, 7 | sseldd 3569 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 15, 9, 3, 4, 8, 5, 17 | catlid 16167 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹) = 𝐹) |
19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶)) |
20 | 19 | oveqd 6566 | . . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌)) |
21 | 20 | fveq1d 6105 | . . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌))) |
22 | 1, 9, 3, 5 | idinv 16272 | . . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
23 | 21, 22 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
24 | 23 | oveq2d 6565 | . . . 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 16259 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
26 | 1, 2, 3, 4, 5, 6 | invf 16251 | . . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
27 | 26, 7 | ffvelrnd 6268 | . . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
28 | 25, 27 | sseldd 3569 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
29 | 1, 15, 9, 3, 5, 8, 4, 28 | catrid 16168 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹)) |
30 | 24, 29 | eqtrd 2644 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹)) |
31 | 14, 18, 30 | 3brtr3d 4614 | . 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
32 | 1, 2, 3, 5, 4 | invsym 16245 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))) |
33 | 31, 32 | mpbird 246 | 1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 Catccat 16148 Idccid 16149 Invcinv 16228 Isociso 16229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-cat 16152 df-cid 16153 df-sect 16230 df-inv 16231 df-iso 16232 |
This theorem is referenced by: invisoinvr 16274 isocoinvid 16276 |
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