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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 2823 | . 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 17043 | . 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 17038 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐼𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 compcco 16579 Catccat 16937 Invcinv 17017 Isociso 17018 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-cat 16941 df-cid 16942 df-sect 17019 df-inv 17020 df-iso 17021 |
This theorem is referenced by: cictr 17077 |
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