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Theorem coaval 16487
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 2609 . . 3 (Arrow‘𝐶) = (Arrow‘𝐶)
3 coaval.x . . 3 = (comp‘𝐶)
41, 2, 3coafval 16483 . 2 · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
5 homdmcoa.h . . . . 5 𝐻 = (Homa𝐶)
62, 5homarw 16465 . . . 4 (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
86, 7sseldi 3565 . . 3 (𝜑𝐺 ∈ (Arrow‘𝐶))
92, 5homarw 16465 . . . . 5 (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
10 homdmcoa.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1110adantr 479 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
129, 11sseldi 3565 . . . 4 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
135homacd 16460 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
1411, 13syl 17 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝐹) = 𝑌)
15 simpr 475 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
1615fveq2d 6092 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝑔) = (doma𝐺))
177adantr 479 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
185homadm 16459 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (doma𝐺) = 𝑌)
1917, 18syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐺) = 𝑌)
2016, 19eqtrd 2643 . . . . 5 ((𝜑𝑔 = 𝐺) → (doma𝑔) = 𝑌)
2114, 20eqtr4d 2646 . . . 4 ((𝜑𝑔 = 𝐺) → (coda𝐹) = (doma𝑔))
22 fveq2 6088 . . . . . 6 ( = 𝐹 → (coda) = (coda𝐹))
2322eqeq1d 2611 . . . . 5 ( = 𝐹 → ((coda) = (doma𝑔) ↔ (coda𝐹) = (doma𝑔)))
2423elrab 3330 . . . 4 (𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ (coda𝐹) = (doma𝑔)))
2512, 21, 24sylanbrc 694 . . 3 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)})
26 otex 4854 . . . 4 ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V
2726a1i 11 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ ∈ V)
28 simprr 791 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
2928fveq2d 6092 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = (doma𝐹))
305homadm 16459 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
3111, 30syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐹) = 𝑋)
3231adantrr 748 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝐹) = 𝑋)
3329, 32eqtrd 2643 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = 𝑋)
3415fveq2d 6092 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝑔) = (coda𝐺))
355homacd 16460 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (coda𝐺) = 𝑍)
3617, 35syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝐺) = 𝑍)
3734, 36eqtrd 2643 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝑔) = 𝑍)
3837adantrr 748 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (coda𝑔) = 𝑍)
3920adantrr 748 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑔) = 𝑌)
4033, 39opeq12d 4342 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (doma𝑔)⟩ = ⟨𝑋, 𝑌⟩)
4140, 38oveq12d 6545 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)) = (⟨𝑋, 𝑌 𝑍))
42 simprl 789 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
4342fveq2d 6092 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
4428fveq2d 6092 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
4541, 43, 44oveq123d 6548 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
4633, 38, 45oteq123d 4349 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
478, 25, 27, 46ovmpt2dv2 6670 . 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))⟩))
484, 47mpi 20 1 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  cop 4130  cotp 4132  cfv 5790  (class class class)co 6527  cmpt2 6529  2nd c2nd 7035  compcco 15726  domacdoma 16439  codaccoda 16440  Arrowcarw 16441  Homachoma 16442  compaccoa 16473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-ot 4133  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-doma 16443  df-coda 16444  df-homa 16445  df-arw 16446  df-coa 16475
This theorem is referenced by:  coa2  16488  coahom  16489  arwlid  16491  arwrid  16492  arwass  16493
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