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Mirrors > Home > MPE Home > Th. List > comfval | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffval.b | ⊢ 𝐵 = (Base‘𝐶) |
comfffval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
comfffval.x | ⊢ · = (comp‘𝐶) |
comffval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffval.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
comfval.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
comfval.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
Ref | Expression |
---|---|
comfval | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval.o | . . 3 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
3 | comfffval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | comfffval.x | . . 3 ⊢ · = (comp‘𝐶) | |
5 | comffval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | comffval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | comffval.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | comffval 16963 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
9 | oveq12 7159 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | |
10 | 9 | adantl 484 | . 2 ⊢ ((𝜑 ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) → (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
11 | comfval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
12 | comfval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
13 | ovexd 7185 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ V) | |
14 | 8, 10, 11, 12, 13 | ovmpod 7296 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4566 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 compcco 16571 compfccomf 16932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-comf 16936 |
This theorem is referenced by: comfval2 16967 comfeqval 16972 |
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