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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 17784 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 2726 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | eqid 2726 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
10 | 1, 9, 3, 5 | idiso 17804 | . . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌)) |
11 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) |
12 | 11 | oveqd 7441 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌)) |
13 | 10, 12 | eleqtrrd 2829 | . . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 13 | invco 17787 | . . 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 2726 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
16 | 1, 15, 6, 3, 4, 5 | isohom 17792 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
17 | 16, 7 | sseldd 3980 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 15, 9, 3, 4, 8, 5, 17 | catlid 17696 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹) = 𝐹) |
19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶)) |
20 | 19 | oveqd 7441 | . . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌)) |
21 | 20 | fveq1d 6903 | . . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌))) |
22 | 1, 9, 3, 5 | idinv 17805 | . . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
23 | 21, 22 | eqtrd 2766 | . . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
24 | 23 | oveq2d 7440 | . . . 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 17792 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
26 | 1, 2, 3, 4, 5, 6 | invf 17784 | . . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
27 | 26, 7 | ffvelcdmd 7099 | . . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
28 | 25, 27 | sseldd 3980 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
29 | 1, 15, 9, 3, 5, 8, 4, 28 | catrid 17697 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹)) |
30 | 24, 29 | eqtrd 2766 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹)) |
31 | 14, 18, 30 | 3brtr3d 5184 | . 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
32 | 1, 2, 3, 5, 4 | invsym 17778 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))) |
33 | 31, 32 | mpbird 256 | 1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 〈cop 4639 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 Hom chom 17277 compcco 17278 Catccat 17677 Idccid 17678 Invcinv 17761 Isociso 17762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-cat 17681 df-cid 17682 df-sect 17763 df-inv 17764 df-iso 17765 |
This theorem is referenced by: invisoinvr 17807 isocoinvid 17809 |
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