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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 16968 | . . 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 769 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑥 = 〈𝑋, 𝑌〉) | |
8 | 7 | fveq2d 6674 | . . . . 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 7702 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) | |
12 | 9, 10, 11 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
13 | 12 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘〈𝑋, 𝑌〉) = 𝑌) |
14 | 8, 13 | eqtrd 2856 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (2nd ‘𝑥) = 𝑌) |
15 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍) | |
16 | 14, 15 | oveq12d 7174 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → ((2nd ‘𝑥)𝐻𝑧) = (𝑌𝐻𝑍)) |
17 | 7 | fveq2d 6674 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝐻‘〈𝑋, 𝑌〉)) |
18 | df-ov 7159 | . . . 4 ⊢ (𝑋𝐻𝑌) = (𝐻‘〈𝑋, 𝑌〉) | |
19 | 17, 18 | syl6eqr 2874 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝐻‘𝑥) = (𝑋𝐻𝑌)) |
20 | 7, 15 | oveq12d 7174 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑥 · 𝑧) = (〈𝑋, 𝑌〉 · 𝑍)) |
21 | 20 | oveqd 7173 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 〈𝑋, 𝑌〉 ∧ 𝑧 = 𝑍)) → (𝑔(𝑥 · 𝑧)𝑓) = (𝑔(〈𝑋, 𝑌〉 · 𝑍)𝑓)) |
22 | 16, 19, 21 | mpoeq123dv 7229 | . 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 5593 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
24 | comffval.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
25 | ovex 7189 | . . . 4 ⊢ (𝑌𝐻𝑍) ∈ V | |
26 | ovex 7189 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
27 | 25, 26 | mpoex 7777 | . . 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 7302 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 × cxp 5553 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 2nd c2nd 7688 Basecbs 16483 Hom chom 16576 compcco 16577 compfccomf 16938 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-comf 16942 |
This theorem is referenced by: comfval 16970 comffval2 16972 comffn 16975 |
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