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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 2823 | . 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 16964 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌)) |
11 | 8, 10 | eleqtrd 2917 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
12 | comfval2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
13 | 9, 2, 3, 6, 7 | homfval 16964 | . . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍)) |
14 | 12, 13 | eleqtrd 2917 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍)) |
15 | 1, 2, 3, 4, 5, 6, 7, 11, 14 | comfval 16972 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 〈cop 4575 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Hom chom 16578 compcco 16579 Homf chomf 16939 compfccomf 16940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-homf 16943 df-comf 16944 |
This theorem is referenced by: (None) |
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