Theorem evlf2val 18270

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 18269	. 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 782	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴)
15	14	fveq1d 6883	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌))
16		simprr 784	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾)
17	16	fveq2d 6885	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑋(2nd ‘𝐹)𝑌)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
18	15, 17	oveq12d 7428	. 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 7445	. 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 7562	1 ⊢ (𝜑 → (𝐴𝐿𝐾) = ((𝐴‘𝑌)(⟨((1st ‘𝐹)‘𝑋), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐺)‘𝑌))((𝑋(2nd ‘𝐹)𝑌)‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Basecbs 17264  Hom chom 17316  compcco 17317  Catccat 17715   Func cfunc 17906   Nat cnat 17996   eval_F cevlf 18260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-evlf 18264

This theorem is referenced by:  evlfcllem  18272  evlfcl  18273  uncf2  18288  yonedalem3b  18330