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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 2825 | . 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 16790 | . 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 16785 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ (𝑋𝐼𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 〈cop 4405 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 compcco 16324 Catccat 16684 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: cictr 16824 |
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