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Mirrors > Home > MPE Home > Th. List > comfval2 | Structured version Visualization version GIF version |
Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffval2.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffval2.b | ⊢ 𝐵 = (Base‘𝐶) |
comfffval2.h | ⊢ 𝐻 = (Homf ‘𝐶) |
comfffval2.x | ⊢ · = (comp‘𝐶) |
comffval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffval2.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
comfval2.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
comfval2.g | ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) |
Ref | Expression |
---|---|
comfval2 | ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval2.o | . 2 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffval2.b | . 2 ⊢ 𝐵 = (Base‘𝐶) | |
3 | eqid 2738 | . 2 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | comfffval2.x | . 2 ⊢ · = (comp‘𝐶) | |
5 | comffval2.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | comffval2.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | comffval2.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
8 | comfval2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
9 | comfffval2.h | . . . 4 ⊢ 𝐻 = (Homf ‘𝐶) | |
10 | 9, 2, 3, 5, 6 | homfval 17066 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌)) |
11 | 8, 10 | eleqtrd 2835 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
12 | comfval2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
13 | 9, 2, 3, 6, 7 | homfval 17066 | . . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍)) |
14 | 12, 13 | eleqtrd 2835 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍)) |
15 | 1, 2, 3, 4, 5, 6, 7, 11, 14 | comfval 17074 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 〈cop 4522 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Hom chom 16679 compcco 16680 Homf chomf 17040 compfccomf 17041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-homf 17044 df-comf 17045 |
This theorem is referenced by: (None) |
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