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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 17745 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
9 | eqid 2731 | . . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
10 | 1, 3, 4, 5, 6, 9 | isinv 17714 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))) |
11 | simpl 482 | . . . 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 253 | . . 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 2731 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
15 | eqid 2731 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | invcoisoid.1 | . . . 4 ⊢ 1 = (Id‘𝐶) | |
17 | 1, 14, 2, 4, 5, 6 | isohom 17730 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 17, 7 | sseldd 3983 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
19 | 1, 14, 2, 4, 6, 5 | isohom 17730 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
20 | 1, 3, 4, 5, 6, 2 | invf 17722 | . . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
21 | 20, 7 | ffvelcdmd 7087 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
22 | 19, 21 | sseldd 3983 | . . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
23 | 1, 14, 15, 16, 9, 4, 5, 6, 18, 22 | issect2 17708 | . . 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 2737 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = ⚬ ) |
27 | 26 | oveqd 7429 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹)) |
28 | 27 | eqeq1d 2733 | . . 3 ⊢ (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
29 | 23, 28 | bitrd 279 | . 2 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
30 | 13, 29 | mpbid 231 | 1 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 〈cop 4634 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 Hom chom 17215 compcco 17216 Catccat 17615 Idccid 17616 Sectcsect 17698 Invcinv 17699 Isociso 17700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-cat 17619 df-cid 17620 df-sect 17701 df-inv 17702 df-iso 17703 |
This theorem is referenced by: rcaninv 17748 |
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