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Theorem coaval 18113
Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o · = (compa𝐶)
homdmcoa.h 𝐻 = (Homa𝐶)
homdmcoa.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
coaval.x = (comp‘𝐶)
Assertion
Ref Expression
coaval (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)

Proof of Theorem coaval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homdmcoa.o . . 3 · = (compa𝐶)
2 eqid 2737 . . 3 (Arrow‘𝐶) = (Arrow‘𝐶)
3 coaval.x . . 3 = (comp‘𝐶)
41, 2, 3coafval 18109 . 2 · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
5 homdmcoa.h . . . . 5 𝐻 = (Homa𝐶)
62, 5homarw 18091 . . . 4 (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
86, 7sselid 3981 . . 3 (𝜑𝐺 ∈ (Arrow‘𝐶))
9 fveqeq2 6915 . . . 4 ( = 𝐹 → ((coda) = (doma𝑔) ↔ (coda𝐹) = (doma𝑔)))
102, 5homarw 18091 . . . . 5 (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
11 homdmcoa.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1211adantr 480 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
1310, 12sselid 3981 . . . 4 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
145homacd 18086 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
1512, 14syl 17 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝐹) = 𝑌)
16 simpr 484 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
1716fveq2d 6910 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝑔) = (doma𝐺))
187adantr 480 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
195homadm 18085 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (doma𝐺) = 𝑌)
2018, 19syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐺) = 𝑌)
2117, 20eqtrd 2777 . . . . 5 ((𝜑𝑔 = 𝐺) → (doma𝑔) = 𝑌)
2215, 21eqtr4d 2780 . . . 4 ((𝜑𝑔 = 𝐺) → (coda𝐹) = (doma𝑔))
239, 13, 22elrabd 3694 . . 3 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)})
24 otex 5470 . . . 4 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V
2524a1i 11 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V)
26 simprr 773 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
2726fveq2d 6910 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = (doma𝐹))
285homadm 18085 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
2912, 28syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐹) = 𝑋)
3029adantrr 717 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝐹) = 𝑋)
3127, 30eqtrd 2777 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = 𝑋)
3216fveq2d 6910 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝑔) = (coda𝐺))
335homacd 18086 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (coda𝐺) = 𝑍)
3418, 33syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝐺) = 𝑍)
3532, 34eqtrd 2777 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝑔) = 𝑍)
3635adantrr 717 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (coda𝑔) = 𝑍)
3721adantrr 717 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑔) = 𝑌)
3831, 37opeq12d 4881 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (doma𝑔)⟩ = ⟨𝑋, 𝑌⟩)
3938, 36oveq12d 7449 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)) = (⟨𝑋, 𝑌 𝑍))
40 simprl 771 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
4140fveq2d 6910 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
4226fveq2d 6910 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
4339, 41, 42oveq123d 7452 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
4431, 36, 43oteq123d 4888 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
458, 23, 25, 44ovmpodv2 7591 . 2 (𝜑 → ( · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩))
464, 45mpi 20 1 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  cop 4632  cotp 4634  cfv 6561  (class class class)co 7431  cmpo 7433  2nd c2nd 8013  compcco 17309  domacdoma 18065  codaccoda 18066  Arrowcarw 18067  Homachoma 18068  compaccoa 18099
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-doma 18069  df-coda 18070  df-homa 18071  df-arw 18072  df-coa 18101
This theorem is referenced by:  coa2  18114  coahom  18115  arwlid  18117  arwrid  18118  arwass  18119
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