Theorem evlf2val 18289

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 18288	. 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑎 = 𝐴)
15	14	fveq1d 6922	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → (𝑎‘𝑌) = (𝐴‘𝑌))
16		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → 𝑔 = 𝐾)
17	16	fveq2d 6924	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑔 = 𝐾)) → ((𝑋(2nd ‘𝐹)𝑌)‘𝑔) = ((𝑋(2nd ‘𝐹)𝑌)‘𝐾))
18	15, 17	oveq12d 7466	. 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 7483	. 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 7602	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 1537   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  Basecbs 17258  Hom chom 17322  compcco 17323  Catccat 17722   Func cfunc 17918   Nat cnat 18009   eval_F cevlf 18279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-evlf 18283

This theorem is referenced by:  evlfcllem  18291  evlfcl  18292  uncf2  18307  yonedalem3b  18349