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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 16780 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 2825 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | eqid 2825 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
10 | 1, 9, 3, 5 | idiso 16800 | . . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌)) |
11 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) |
12 | 11 | oveqd 6922 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌)) |
13 | 10, 12 | eleqtrrd 2909 | . . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 13 | invco 16783 | . . 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 2825 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
16 | 1, 15, 6, 3, 4, 5 | isohom 16788 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
17 | 16, 7 | sseldd 3828 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 15, 9, 3, 4, 8, 5, 17 | catlid 16696 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹) = 𝐹) |
19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶)) |
20 | 19 | oveqd 6922 | . . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌)) |
21 | 20 | fveq1d 6435 | . . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌))) |
22 | 1, 9, 3, 5 | idinv 16801 | . . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
23 | 21, 22 | eqtrd 2861 | . . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
24 | 23 | oveq2d 6921 | . . . 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 16788 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
26 | 1, 2, 3, 4, 5, 6 | invf 16780 | . . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
27 | 26, 7 | ffvelrnd 6609 | . . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
28 | 25, 27 | sseldd 3828 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
29 | 1, 15, 9, 3, 5, 8, 4, 28 | catrid 16697 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹)) |
30 | 24, 29 | eqtrd 2861 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹)) |
31 | 14, 18, 30 | 3brtr3d 4904 | . 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
32 | 1, 2, 3, 5, 4 | invsym 16774 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))) |
33 | 31, 32 | mpbird 249 | 1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 〈cop 4403 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Hom chom 16316 compcco 16317 Catccat 16677 Idccid 16678 Invcinv 16757 Isociso 16758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-cat 16681 df-cid 16682 df-sect 16759 df-inv 16760 df-iso 16761 |
This theorem is referenced by: invisoinvr 16803 isocoinvid 16805 |
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