Theorem coa2 17991

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 17990	. . 3 ⊢ (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩)
7	6	fveq2d 6836	. 2 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘⟨𝑋, 𝑍, ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹))⟩))
8		ovex 7389	. . 3 ⊢ ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)) ∈ V
9		ot3rdg 7947	. . 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 2785	1 ⊢ (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd ‘𝐺)(⟨𝑋, 𝑌⟩ ∙ 𝑍)(2nd ‘𝐹)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438  ⟨cop 4584  ⟨cotp 4586  ‘cfv 6490  (class class class)co 7356  2^nd c2nd 7930  compcco 17187  Hom_achoma 17945  comp_accoa 17976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-ot 4587  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-doma 17946  df-coda 17947  df-homa 17948  df-arw 17949  df-coa 17978

This theorem is referenced by:  arwass  17996