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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 17741 | . . 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 6910 | . . . . 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 8027 | . . . . . . 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 2777 | . . . 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 6910 | . . . 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 2795 | . . 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 5724 | . 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 8104 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 2nd c2nd 8013 Basecbs 17247 Hom chom 17308 compcco 17309 compfccomf 17710 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-comf 17714 |
| This theorem is referenced by: comfval 17743 comffval2 17745 comffn 17748 |
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