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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 17743 | . . 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 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑥 = 〈𝑋, 𝑌〉) | |
8 | 7 | fveq2d 6911 | . . . . 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 8026 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
12 | 9, 10, 11 | syl2anc 584 | . . . . . 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 2775 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌) |
15 | simprr 773 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
16 | 14, 15 | oveq12d 7449 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍)) |
17 | 7 | fveq2d 6911 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘〈𝑋, 𝑌〉)) |
18 | df-ov 7434 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
19 | 17, 18 | eqtr4di 2793 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌)) |
20 | 7, 15 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (〈𝑋, 𝑌〉 · 𝑍)) |
21 | 20 | oveqd 7448 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) |
22 | 16, 19, 21 | mpoeq123dv 7508 | . 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 | 9, 10 | opelxpd 5728 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
24 | comffval.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | ovex 7464 | . . . 4 ⊢ (𝑌𝐻𝑍) ∈ V | |
26 | ovex 7464 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
27 | 25, 26 | mpoex 8103 | . . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V |
28 | 27 | a1i 11 | . 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) ∈ V) |
29 | 6, 22, 23, 24, 28 | ovmpod 7585 | 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 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 2nd c2nd 8012 Basecbs 17245 Hom chom 17309 compcco 17310 compfccomf 17712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-comf 17716 |
This theorem is referenced by: comfval 17745 comffval2 17747 comffn 17750 |
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