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Mirrors > Home > MPE Home > Th. List > isocoinvid | Structured version Visualization version GIF version |
Description: The inverse of an isomorphism composed with the isomorphism is the identity. (Contributed by AV, 10-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) |
isocoinvid.o | ⊢ ⚬ = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) |
Ref | Expression |
---|---|
isocoinvid | ⊢ (𝜑 → (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌)) |
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 | invisoinvl 16809 | . . 3 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹) |
9 | eqid 2825 | . . . . 5 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
10 | 1, 3, 4, 6, 5, 9 | isinv 16779 | . . . 4 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 ↔ (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)))) |
11 | simpl 476 | . . . 4 ⊢ ((((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ∧ 𝐹(𝑋(Sect‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) | |
12 | 10, 11 | syl6bi 245 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹)) |
13 | 8, 12 | mpd 15 | . 2 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹) |
14 | eqid 2825 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
15 | eqid 2825 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | invcoisoid.1 | . . . 4 ⊢ 1 = (Id‘𝐶) | |
17 | 1, 14, 2, 4, 6, 5 | isohom 16795 | . . . . 5 ⊢ (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋)) |
18 | 1, 3, 4, 5, 6, 2 | invf 16787 | . . . . . 6 ⊢ (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋)) |
19 | 18, 7 | ffvelrnd 6614 | . . . . 5 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋)) |
20 | 17, 19 | sseldd 3828 | . . . 4 ⊢ (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋)) |
21 | 1, 14, 2, 4, 5, 6 | isohom 16795 | . . . . 5 ⊢ (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌)) |
22 | 21, 7 | sseldd 3828 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
23 | 1, 14, 15, 16, 9, 4, 6, 5, 20, 22 | issect2 16773 | . . 3 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))) |
24 | isocoinvid.o | . . . . . . 7 ⊢ ⚬ = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) | |
25 | 24 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ⚬ = (〈𝑌, 𝑋〉(comp‘𝐶)𝑌)) |
26 | 25 | eqcomd 2831 | . . . . 5 ⊢ (𝜑 → (〈𝑌, 𝑋〉(comp‘𝐶)𝑌) = ⚬ ) |
27 | 26 | oveqd 6927 | . . . 4 ⊢ (𝜑 → (𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹))) |
28 | 27 | eqeq1d 2827 | . . 3 ⊢ (𝜑 → ((𝐹(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌) ↔ (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))) |
29 | 23, 28 | bitrd 271 | . 2 ⊢ (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌(Sect‘𝐶)𝑋)𝐹 ↔ (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌))) |
30 | 13, 29 | mpbid 224 | 1 ⊢ (𝜑 → (𝐹 ⚬ ((𝑋𝑁𝑌)‘𝐹)) = ( 1 ‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 〈cop 4405 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 Hom chom 16323 compcco 16324 Catccat 16684 Idccid 16685 Sectcsect 16763 Invcinv 16764 Isociso 16765 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-cat 16688 df-cid 16689 df-sect 16766 df-inv 16767 df-iso 16768 |
This theorem is referenced by: (None) |
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