![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isoco | Structured version Visualization version GIF version |
Description: The composition of two isomorphisms is an isomorphism. Proposition 3.14(2) of [Adamek] p. 29. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
isoco.b | ⊢ 𝐵 = (Base‘𝐶) |
isoco.o | ⊢ · = (comp‘𝐶) |
isoco.n | ⊢ 𝐼 = (Iso‘𝐶) |
isoco.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
isoco.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
isoco.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
isoco.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
isoco.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) |
isoco.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍)) |
Ref | Expression |
---|---|
isoco | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐼𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isoco.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2798 | . 2 ⊢ (Inv‘𝐶) = (Inv‘𝐶) | |
3 | isoco.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | isoco.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
5 | isoco.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
6 | isoco.n | . 2 ⊢ 𝐼 = (Iso‘𝐶) | |
7 | isoco.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | isoco.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌)) | |
9 | isoco.o | . . 3 ⊢ · = (comp‘𝐶) | |
10 | isoco.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐼𝑍)) | |
11 | 1, 2, 3, 4, 7, 6, 8, 9, 5, 10 | invco 17033 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)(𝑋(Inv‘𝐶)𝑍)(((𝑋(Inv‘𝐶)𝑌)‘𝐹)(〈𝑍, 𝑌〉 · 𝑋)((𝑌(Inv‘𝐶)𝑍)‘𝐺))) |
12 | 1, 2, 3, 4, 5, 6, 11 | inviso1 17028 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐼𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 〈cop 4531 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 compcco 16569 Catccat 16927 Invcinv 17007 Isociso 17008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-cat 16931 df-cid 16932 df-sect 17009 df-inv 17010 df-iso 17011 |
This theorem is referenced by: cictr 17067 |
Copyright terms: Public domain | W3C validator |