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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 17401 | . . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌)) |
11 | 8, 10 | eleqtrd 2841 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
12 | comfval2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍)) | |
13 | 9, 2, 3, 6, 7 | homfval 17401 | . . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍)) |
14 | 12, 13 | eleqtrd 2841 | . 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍)) |
15 | 1, 2, 3, 4, 5, 6, 7, 11, 14 | comfval 17409 | 1 ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉𝑂𝑍)𝐹) = (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 〈cop 4567 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Hom chom 16973 compcco 16974 Homf chomf 17375 compfccomf 17376 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-homf 17379 df-comf 17380 |
This theorem is referenced by: (None) |
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