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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 17673 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
9 | oveq12 7424 | . . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑓 = 𝐹) → (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) | |
10 | 9 | adantl 481 | . 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 7450 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ V) | |
14 | 8, 10, 11, 12, 13 | ovmpod 7568 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 〈cop 4631 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 Hom chom 17238 compcco 17239 compfccomf 17641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7988 df-2nd 7989 df-comf 17645 |
This theorem is referenced by: comfval2 17677 comfeqval 17682 |
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