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Mirrors > Home > MPE Home > Th. List > invcoisoid | Structured version Visualization version GIF version |
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 5-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)) |
invcoisoid.1 | ⊢ 1 = (Id‘𝐶) |
invcoisoid.o | ⊢ ⚬ = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) |
Ref | Expression |
---|---|
invcoisoid | ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invisoinv.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | invisoinv.i | . . . 4 ⊢ 𝐼 = (Iso‘𝐶) | |
3 | invisoinv.n | . . . 4 ⊢ 𝑁 = (Inv‘𝐶) | |
4 | invisoinv.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | invisoinv.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | invisoinv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | invisoinv.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | invisoinvr 16847 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
9 | eqid 2778 | . . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
10 | 1, 3, 4, 5, 6, 9 | isinv 16816 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))) |
11 | simpl 476 | . . . 4 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) | |
12 | 10, 11 | syl6bi 245 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹))) |
13 | 8, 12 | mpd 15 | . 2 ⊢ (𝜑 → 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) |
14 | eqid 2778 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
15 | eqid 2778 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | invcoisoid.1 | . . . 4 ⊢ 1 = (Id‘𝐶) | |
17 | 1, 14, 2, 4, 5, 6 | isohom 16832 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 17, 7 | sseldd 3822 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
19 | 1, 14, 2, 4, 6, 5 | isohom 16832 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
20 | 1, 3, 4, 5, 6, 2 | invf 16824 | . . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
21 | 20, 7 | ffvelrnd 6626 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
22 | 19, 21 | sseldd 3822 | . . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
23 | 1, 14, 15, 16, 9, 4, 5, 6, 18, 22 | issect2 16810 | . . 3 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋))) |
24 | invcoisoid.o | . . . . . . 7 ⊢ ⚬ = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ⚬ = (〈𝑋, 𝑌〉(comp‘𝐶)𝑋)) |
26 | 25 | eqcomd 2784 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = ⚬ ) |
27 | 26 | oveqd 6941 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹)) |
28 | 27 | eqeq1d 2780 | . . 3 ⊢ (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
29 | 23, 28 | bitrd 271 | . 2 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
30 | 13, 29 | mpbid 224 | 1 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 〈cop 4404 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 Hom chom 16360 compcco 16361 Catccat 16721 Idccid 16722 Sectcsect 16800 Invcinv 16801 Isociso 16802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-1st 7447 df-2nd 7448 df-cat 16725 df-cid 16726 df-sect 16803 df-inv 16804 df-iso 16805 |
This theorem is referenced by: rcaninv 16850 |
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