Theorem coa2 18087

Description: The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homdmcoa.o	⊢ · = (compa‘𝐶)
homdmcoa.h	⊢ 𝐻 = (Homa‘𝐶)
homdmcoa.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
coaval.x	⊢ ∙ = (comp‘𝐶)

Assertion

Ref	Expression
coa2	⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))

Proof of Theorem coa2

Step	Hyp	Ref	Expression
1		homdmcoa.o	. . . 4 ⊢ · = (compa‘𝐶)
2		homdmcoa.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3		homdmcoa.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		homdmcoa.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
5		coaval.x	. . . 4 ⊢ ∙ = (comp‘𝐶)
6	1, 2, 3, 4, 5	coaval 18086	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)
7	6	fveq2d 6885	. 2 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩))
8		ovex 7443	. . 3 ⊢ ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)) ∈ V
9		ot3rdg 8009	. . 3 ⊢ (((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)) ∈ V → (2nd ‘⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))
10	8, 9	ax-mp 5	. 2 ⊢ (2nd ‘⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))
11	7, 10	eqtrdi 2787	1 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  ⟨cop 4612  ⟨cotp 4614  ‘cfv 6536  (class class class)co 7410  2^nd c2nd 7992  compcco 17288  Hom_achoma 18041  comp_accoa 18072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  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 7993  df-2nd 7994  df-doma 18042  df-coda 18043  df-homa 18044  df-arw 18045  df-coa 18074

This theorem is referenced by:  arwass  18092