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Theorem coaval 16925
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 2771 . . 3 (Arrow‘𝐶) = (Arrow‘𝐶)
3 coaval.x . . 3 = (comp‘𝐶)
41, 2, 3coafval 16921 . 2 · = (𝑔 ∈ (Arrow‘𝐶), 𝑓 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
5 homdmcoa.h . . . . 5 𝐻 = (Homa𝐶)
62, 5homarw 16903 . . . 4 (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
86, 7sseldi 3750 . . 3 (𝜑𝐺 ∈ (Arrow‘𝐶))
92, 5homarw 16903 . . . . 5 (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
10 homdmcoa.f . . . . . 6 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
1110adantr 466 . . . . 5 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (𝑋𝐻𝑌))
129, 11sseldi 3750 . . . 4 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ (Arrow‘𝐶))
135homacd 16898 . . . . . 6 (𝐹 ∈ (𝑋𝐻𝑌) → (coda𝐹) = 𝑌)
1411, 13syl 17 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝐹) = 𝑌)
15 simpr 471 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝑔 = 𝐺)
1615fveq2d 6336 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝑔) = (doma𝐺))
177adantr 466 . . . . . . 7 ((𝜑𝑔 = 𝐺) → 𝐺 ∈ (𝑌𝐻𝑍))
185homadm 16897 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (doma𝐺) = 𝑌)
1917, 18syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐺) = 𝑌)
2016, 19eqtrd 2805 . . . . 5 ((𝜑𝑔 = 𝐺) → (doma𝑔) = 𝑌)
2114, 20eqtr4d 2808 . . . 4 ((𝜑𝑔 = 𝐺) → (coda𝐹) = (doma𝑔))
22 fveq2 6332 . . . . . 6 ( = 𝐹 → (coda) = (coda𝐹))
2322eqeq1d 2773 . . . . 5 ( = 𝐹 → ((coda) = (doma𝑔) ↔ (coda𝐹) = (doma𝑔)))
2423elrab 3515 . . . 4 (𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)} ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ (coda𝐹) = (doma𝑔)))
2512, 21, 24sylanbrc 572 . . 3 ((𝜑𝑔 = 𝐺) → 𝐹 ∈ { ∈ (Arrow‘𝐶) ∣ (coda) = (doma𝑔)})
26 otex 5061 . . . 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 756 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
2928fveq2d 6336 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = (doma𝐹))
305homadm 16897 . . . . . . 7 (𝐹 ∈ (𝑋𝐻𝑌) → (doma𝐹) = 𝑋)
3111, 30syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (doma𝐹) = 𝑋)
3231adantrr 696 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝐹) = 𝑋)
3329, 32eqtrd 2805 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑓) = 𝑋)
3415fveq2d 6336 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝑔) = (coda𝐺))
355homacd 16898 . . . . . . 7 (𝐺 ∈ (𝑌𝐻𝑍) → (coda𝐺) = 𝑍)
3617, 35syl 17 . . . . . 6 ((𝜑𝑔 = 𝐺) → (coda𝐺) = 𝑍)
3734, 36eqtrd 2805 . . . . 5 ((𝜑𝑔 = 𝐺) → (coda𝑔) = 𝑍)
3837adantrr 696 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (coda𝑔) = 𝑍)
3920adantrr 696 . . . . . . 7 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (doma𝑔) = 𝑌)
4033, 39opeq12d 4547 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (doma𝑔)⟩ = ⟨𝑋, 𝑌⟩)
4140, 38oveq12d 6811 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)) = (⟨𝑋, 𝑌 𝑍))
42 simprl 754 . . . . . 6 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
4342fveq2d 6336 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑔) = (2nd𝐺))
4428fveq2d 6336 . . . . 5 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (2nd𝑓) = (2nd𝐹))
4541, 43, 44oveq123d 6814 . . . 4 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
4633, 38, 45oteq123d 4554 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
478, 25, 27, 46ovmpt2dv2 6941 . 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 1631  wcel 2145  {crab 3065  Vcvv 3351  cop 4322  cotp 4324  cfv 6031  (class class class)co 6793  cmpt2 6795  2nd c2nd 7314  compcco 16161  domacdoma 16877  codaccoda 16878  Arrowcarw 16879  Homachoma 16880  compaccoa 16911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-doma 16881  df-coda 16882  df-homa 16883  df-arw 16884  df-coa 16913
This theorem is referenced by:  coa2  16926  coahom  16927  arwlid  16929  arwrid  16930  arwass  16931
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