Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 17032 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 2821 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | eqid 2821 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
10 | 1, 9, 3, 5 | idiso 17052 | . . . . 5 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌)) |
11 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝐼 = (Iso‘𝐶)) |
12 | 11 | oveqd 7167 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌)) |
13 | 10, 12 | eleqtrrd 2916 | . . . 4 ⊢ (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 5, 13 | invco 17035 | . . 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 2821 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
16 | 1, 15, 6, 3, 4, 5 | isohom 17040 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
17 | 16, 7 | sseldd 3967 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 15, 9, 3, 4, 8, 5, 17 | catlid 16948 | . . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐹) = 𝐹) |
19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (Inv‘𝐶)) |
20 | 19 | oveqd 7167 | . . . . . . 7 ⊢ (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌)) |
21 | 20 | fveq1d 6666 | . . . . . 6 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌))) |
22 | 1, 9, 3, 5 | idinv 17053 | . . . . . 6 ⊢ (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
23 | 21, 22 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
24 | 23 | oveq2d 7166 | . . . 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 17040 | . . . . . 6 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
26 | 1, 2, 3, 4, 5, 6 | invf 17032 | . . . . . . 7 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
27 | 26, 7 | ffvelrnd 6846 | . . . . . 6 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
28 | 25, 27 | sseldd 3967 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
29 | 1, 15, 9, 3, 5, 8, 4, 28 | catrid 16949 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹)) |
30 | 24, 29 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑌, 𝑌〉(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹)) |
31 | 14, 18, 30 | 3brtr3d 5089 | . 2 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
32 | 1, 2, 3, 5, 4 | invsym 17026 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))) |
33 | 31, 32 | mpbird 259 | 1 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Idccid 16930 Invcinv 17009 Isociso 17010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-cat 16933 df-cid 16934 df-sect 17011 df-inv 17012 df-iso 17013 |
This theorem is referenced by: invisoinvr 17055 isocoinvid 17057 |
Copyright terms: Public domain | W3C validator |