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Mirrors > Home > MPE Home > Th. List > comffval | 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) |
Ref | Expression |
---|---|
comffval | ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | comfffval.o | . . . 4 ⊢ 𝑂 = (compf‘𝐶) | |
2 | comfffval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | comfffval.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | comfffval.x | . . . 4 ⊢ · = (comp‘𝐶) | |
5 | 1, 2, 3, 4 | comfffval 16181 | . . 3 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓))) |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)))) |
7 | simprl 790 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑥 = 〈𝑋, 𝑌〉) | |
8 | 7 | fveq2d 6107 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = (2nd ‘〈𝑋, 𝑌〉)) |
9 | comffval.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | comffval.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
11 | op2ndg 7072 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
12 | 9, 10, 11 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
14 | 8, 13 | eqtrd 2644 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌) |
15 | simprr 792 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
16 | 14, 15 | oveq12d 6567 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍)) |
17 | 7 | fveq2d 6107 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘〈𝑋, 𝑌〉)) |
18 | df-ov 6552 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
19 | 17, 18 | syl6eqr 2662 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌)) |
20 | 7, 15 | oveq12d 6567 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (〈𝑋, 𝑌〉 · 𝑍)) |
21 | 20 | oveqd 6566 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) |
22 | 16, 19, 21 | mpt2eq123dv 6615 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑧), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑧)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓))) |
23 | opelxpi 5072 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
24 | 9, 10, 23 | syl2anc 691 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
25 | comffval.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
26 | ovex 6577 | . . . 4 ⊢ (𝑌𝐻𝑍) ∈ V | |
27 | ovex 6577 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
28 | 26, 27 | mpt2ex 7136 | . . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V |
29 | 28 | a1i 11 | . 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V) |
30 | 6, 22, 24, 25, 29 | ovmpt2d 6686 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (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 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 2nd c2nd 7058 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: comfval 16183 comffval2 16185 comffn 16188 |
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