Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > evlf2val | Structured version Visualization version GIF version |
Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
evlfval.e | ⊢ 𝐸 = (𝐶 evalF 𝐷) |
evlfval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
evlfval.d | ⊢ (𝜑 → 𝐷 ∈ Cat) |
evlfval.b | ⊢ 𝐵 = (Base‘𝐶) |
evlfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
evlfval.o | ⊢ · = (comp‘𝐷) |
evlfval.n | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
evlf2.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
evlf2.g | ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
evlf2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
evlf2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
evlf2.l | ⊢ 𝐿 = (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑌〉) |
evlf2val.a | ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) |
evlf2val.k | ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
evlf2val | ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlfval.e | . . 3 ⊢ 𝐸 = (𝐶 evalF 𝐷) | |
2 | evlfval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
3 | evlfval.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
4 | evlfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
5 | evlfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
6 | evlfval.o | . . 3 ⊢ · = (comp‘𝐷) | |
7 | evlfval.n | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
8 | evlf2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
9 | evlf2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) | |
10 | evlf2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
11 | evlf2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | evlf2.l | . . 3 ⊢ 𝐿 = (〈𝐹, 𝑋〉(2nd ‘𝐸)〈𝐺, 𝑌〉) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | evlf2 17467 | . 2 ⊢ (𝜑 → 𝐿 = (𝑎 ∈ (𝐹𝑁𝐺), 𝑔 ∈ (𝑋𝐻𝑌) ↦ ((𝑎‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔)))) |
14 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴) | |
15 | 14 | fveq1d 6671 | . . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌)) |
16 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾) | |
17 | 16 | fveq2d 6673 | . . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑋(2nd ‘𝐹)𝑌)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) |
18 | 15, 17 | oveq12d 7173 | . 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑎‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝑔)) = ((𝐴‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾))) |
19 | evlf2val.a | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺)) | |
20 | evlf2val.k | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌)) | |
21 | ovexd 7190 | . 2 ⊢ (𝜑 → ((𝐴‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)) ∈ V) | |
22 | 13, 18, 19, 20, 21 | ovmpod 7301 | 1 ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(〈((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)〉 · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 Basecbs 16482 Hom chom 16575 compcco 16576 Catccat 16934 Func cfunc 17123 Nat cnat 17210 evalF cevlf 17458 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-evlf 17462 |
This theorem is referenced by: evlfcllem 17470 evlfcl 17471 uncf2 17486 yonedalem3b 17528 |
Copyright terms: Public domain | W3C validator |