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Theorem coaval 17162
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 2795 . . 3 (Arrow‘𝐶) = (Arrow‘𝐶)
3 coaval.x . . 3 = (comp‘𝐶)
41, 2, 3coafval 17158 . 2 · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
5 homdmcoa.h . . . . 5 𝐻 = (Homa𝐶)
62, 5homarw 17140 . . . 4 (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
86, 7sseldi 3891 . . 3 (𝜑𝐺 ∈ (Arrow‘𝐶))
9 fveqeq2 6552 . . . 4 ( = 𝐹 → ((coda) = (doma𝑔) ↔ (coda𝐹) = (doma𝑔)))
102, 5homarw 17140 . . . . 5 (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
11 homdmcoa.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1211adantr 481 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
1310, 12sseldi 3891 . . . 4 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
145homacd 17135 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
1512, 14syl 17 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝐹) = 𝑌)
16 simpr 485 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
1716fveq2d 6547 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝑔) = (doma𝐺))
187adantr 481 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
195homadm 17134 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (doma𝐺) = 𝑌)
2018, 19syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐺) = 𝑌)
2117, 20eqtrd 2831 . . . . 5 ((𝜑𝑔 = 𝐺) → (doma𝑔) = 𝑌)
2215, 21eqtr4d 2834 . . . 4 ((𝜑𝑔 = 𝐺) → (coda𝐹) = (doma𝑔))
239, 13, 22elrabd 3621 . . 3 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)})
24 otex 5254 . . . 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 769 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
2726fveq2d 6547 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = (doma𝐹))
285homadm 17134 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
2912, 28syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐹) = 𝑋)
3029adantrr 713 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝐹) = 𝑋)
3127, 30eqtrd 2831 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = 𝑋)
3216fveq2d 6547 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝑔) = (coda𝐺))
335homacd 17135 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (coda𝐺) = 𝑍)
3418, 33syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝐺) = 𝑍)
3532, 34eqtrd 2831 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝑔) = 𝑍)
3635adantrr 713 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (coda𝑔) = 𝑍)
3721adantrr 713 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑔) = 𝑌)
3831, 37opeq12d 4722 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (doma𝑔)⟩ = ⟨𝑋, 𝑌⟩)
3938, 36oveq12d 7039 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)) = (⟨𝑋, 𝑌 𝑍))
40 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
4140fveq2d 6547 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
4226fveq2d 6547 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
4339, 41, 42oveq123d 7042 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
4431, 36, 43oteq123d 4729 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
458, 23, 25, 44ovmpodv2 7169 . 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 396   = wceq 1522  wcel 2081  {crab 3109  Vcvv 3437  cop 4482  cotp 4484  cfv 6230  (class class class)co 7021  cmpo 7023  2nd c2nd 7549  compcco 16411  domacdoma 17114  codaccoda 17115  Arrowcarw 17116  Homachoma 17117  compaccoa 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-ot 4485  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-ov 7024  df-oprab 7025  df-mpo 7026  df-1st 7550  df-2nd 7551  df-doma 17118  df-coda 17119  df-homa 17120  df-arw 17121  df-coa 17150
This theorem is referenced by:  coa2  17163  coahom  17164  arwlid  17166  arwrid  17167  arwass  17168
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