Theorem evlf2val 18176

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 18175	. 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 771	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴)
15	14	fveq1d 6836	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌))
16		simprr 773	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾)
17	16	fveq2d 6838	. . 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 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ⟨cop 4574  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621   Func cfunc 17812   Nat cnat 17902   eval_F cevlf 18166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-evlf 18170

This theorem is referenced by:  evlfcllem  18178  evlfcl  18179  uncf2  18194  yonedalem3b  18236