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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 17064 | . . 3 ⊢ (𝜑 → 𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)) |
9 | eqid 2824 | . . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
10 | 1, 3, 4, 5, 6, 9 | isinv 17033 | . . . 4 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ∧ ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹))) |
11 | simpl 485 | . . . 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 255 | . . 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 2824 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
15 | eqid 2824 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | invcoisoid.1 | . . . 4 ⊢ 1 = (Id‘𝐶) | |
17 | 1, 14, 2, 4, 5, 6 | isohom 17049 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 17, 7 | sseldd 3971 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
19 | 1, 14, 2, 4, 6, 5 | isohom 17049 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
20 | 1, 3, 4, 5, 6, 2 | invf 17041 | . . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
21 | 20, 7 | ffvelrnd 6855 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
22 | 19, 21 | sseldd 3971 | . . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
23 | 1, 14, 15, 16, 9, 4, 5, 6, 18, 22 | issect2 17027 | . . 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 2830 | . . . . 5 ⊢ (𝜑 → (〈𝑋, 𝑌〉(comp‘𝐶)𝑋) = ⚬ ) |
27 | 26 | oveqd 7176 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹)) |
28 | 27 | eqeq1d 2826 | . . 3 ⊢ (𝜑 → ((((𝑋𝑁𝑌)‘𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐹) = ( 1 ‘𝑋) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
29 | 23, 28 | bitrd 281 | . 2 ⊢ (𝜑 → (𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹) ↔ (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋))) |
30 | 13, 29 | mpbid 234 | 1 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹) ⚬ 𝐹) = ( 1 ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 〈cop 4576 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 Hom chom 16579 compcco 16580 Catccat 16938 Idccid 16939 Sectcsect 17017 Invcinv 17018 Isociso 17019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-1st 7692 df-2nd 7693 df-cat 16942 df-cid 16943 df-sect 17020 df-inv 17021 df-iso 17022 |
This theorem is referenced by: rcaninv 17067 |
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