Theorem evlf2val 18180

Description: Value of the evaluation natural transformation at an object. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

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))

Assertion

Ref	Expression
evlf2val	⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))

Proof of Theorem evlf2val

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

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 18179	. 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 777	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴)
15	14	fveq1d 6833	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌))
16		simprr 779	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾)
17	16	fveq2d 6835	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑋(2nd ‘𝐹)𝑌)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
18	15, 17	oveq12d 7378	. 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 7395	. 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 7512	1 ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  ⟨cop 4564  ‘cfv 6489  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17174  Hom chom 17226  compcco 17227  Catccat 17625   Func cfunc 17816   Nat cnat 17906   eval_F cevlf 18170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-evlf 18174

This theorem is referenced by:  evlfcllem  18182  evlfcl  18183  uncf2  18198  yonedalem3b  18240