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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 16182 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
9 | oveq12 6558 | . . 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 | ovex 6577 | . . 3 ⊢ (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ V | |
14 | 13 | a1i 11 | . 2 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) ∈ V) |
15 | 8, 10, 11, 12, 14 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 Hom chom 15779 compcco 15780 compfccomf 16151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-comf 16155 |
This theorem is referenced by: comfval2 16186 comfeqval 16191 |
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