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Theorem coaval 18028
Description: Value of composition for composable arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o Β· = (compaβ€˜πΆ)
homdmcoa.h 𝐻 = (Homaβ€˜πΆ)
homdmcoa.f (πœ‘ β†’ 𝐹 ∈ (π‘‹π»π‘Œ))
homdmcoa.g (πœ‘ β†’ 𝐺 ∈ (π‘Œπ»π‘))
coaval.x βˆ™ = (compβ€˜πΆ)
Assertion
Ref Expression
coaval (πœ‘ β†’ (𝐺 Β· 𝐹) = βŸ¨π‘‹, 𝑍, ((2nd β€˜πΊ)(βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍)(2nd β€˜πΉ))⟩)
Proof of Theorem coaval
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 homdmcoa.o . . 3 Β· = (compaβ€˜πΆ)
2 eqid 2726 . . 3 (Arrowβ€˜πΆ) = (Arrowβ€˜πΆ)
3 coaval.x . . 3 βˆ™ = (compβ€˜πΆ)
41, 2, 3coafval 18024 . 2 Β· = (𝑔 ∈ (Arrowβ€˜πΆ), 𝑓 ∈ {β„Ž ∈ (Arrowβ€˜πΆ) ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} ↦ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“))⟩)
5 homdmcoa.h . . . . 5 𝐻 = (Homaβ€˜πΆ)
62, 5homarw 18006 . . . 4 (π‘Œπ»π‘) βŠ† (Arrowβ€˜πΆ)
7 homdmcoa.g . . . 4 (πœ‘ β†’ 𝐺 ∈ (π‘Œπ»π‘))
86, 7sselid 3975 . . 3 (πœ‘ β†’ 𝐺 ∈ (Arrowβ€˜πΆ))
9 fveqeq2 6893 . . . 4 (β„Ž = 𝐹 β†’ ((codaβ€˜β„Ž) = (domaβ€˜π‘”) ↔ (codaβ€˜πΉ) = (domaβ€˜π‘”)))
102, 5homarw 18006 . . . . 5 (π‘‹π»π‘Œ) βŠ† (Arrowβ€˜πΆ)
11 homdmcoa.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ (π‘‹π»π‘Œ))
1211adantr 480 . . . . 5 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝐹 ∈ (π‘‹π»π‘Œ))
1310, 12sselid 3975 . . . 4 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝐹 ∈ (Arrowβ€˜πΆ))
145homacd 18001 . . . . . 6 (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (codaβ€˜πΉ) = π‘Œ)
1512, 14syl 17 . . . . 5 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (codaβ€˜πΉ) = π‘Œ)
16 simpr 484 . . . . . . 7 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝑔 = 𝐺)
1716fveq2d 6888 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (domaβ€˜π‘”) = (domaβ€˜πΊ))
187adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝐺 ∈ (π‘Œπ»π‘))
195homadm 18000 . . . . . . 7 (𝐺 ∈ (π‘Œπ»π‘) β†’ (domaβ€˜πΊ) = π‘Œ)
2018, 19syl 17 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (domaβ€˜πΊ) = π‘Œ)
2117, 20eqtrd 2766 . . . . 5 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (domaβ€˜π‘”) = π‘Œ)
2215, 21eqtr4d 2769 . . . 4 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (codaβ€˜πΉ) = (domaβ€˜π‘”))
239, 13, 22elrabd 3680 . . 3 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ 𝐹 ∈ {β„Ž ∈ (Arrowβ€˜πΆ) ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)})
24 otex 5458 . . . 4 ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“))⟩ ∈ V
2524a1i 11 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“))⟩ ∈ V)
26 simprr 770 . . . . . 6 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑓 = 𝐹)
2726fveq2d 6888 . . . . 5 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (domaβ€˜π‘“) = (domaβ€˜πΉ))
285homadm 18000 . . . . . . 7 (𝐹 ∈ (π‘‹π»π‘Œ) β†’ (domaβ€˜πΉ) = 𝑋)
2912, 28syl 17 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (domaβ€˜πΉ) = 𝑋)
3029adantrr 714 . . . . 5 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (domaβ€˜πΉ) = 𝑋)
3127, 30eqtrd 2766 . . . 4 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (domaβ€˜π‘“) = 𝑋)
3216fveq2d 6888 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (codaβ€˜π‘”) = (codaβ€˜πΊ))
335homacd 18001 . . . . . . 7 (𝐺 ∈ (π‘Œπ»π‘) β†’ (codaβ€˜πΊ) = 𝑍)
3418, 33syl 17 . . . . . 6 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (codaβ€˜πΊ) = 𝑍)
3532, 34eqtrd 2766 . . . . 5 ((πœ‘ ∧ 𝑔 = 𝐺) β†’ (codaβ€˜π‘”) = 𝑍)
3635adantrr 714 . . . 4 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (codaβ€˜π‘”) = 𝑍)
3721adantrr 714 . . . . . . 7 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (domaβ€˜π‘”) = π‘Œ)
3831, 37opeq12d 4876 . . . . . 6 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ ⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ = βŸ¨π‘‹, π‘ŒβŸ©)
3938, 36oveq12d 7422 . . . . 5 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”)) = (βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍))
40 simprl 768 . . . . . 6 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑔 = 𝐺)
4140fveq2d 6888 . . . . 5 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (2nd β€˜π‘”) = (2nd β€˜πΊ))
4226fveq2d 6888 . . . . 5 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (2nd β€˜π‘“) = (2nd β€˜πΉ))
4339, 41, 42oveq123d 7425 . . . 4 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“)) = ((2nd β€˜πΊ)(βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍)(2nd β€˜πΉ)))
4431, 36, 43oteq123d 4883 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“))⟩ = βŸ¨π‘‹, 𝑍, ((2nd β€˜πΊ)(βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍)(2nd β€˜πΉ))⟩)
458, 23, 25, 44ovmpodv2 7561 . 2 (πœ‘ β†’ ( Β· = (𝑔 ∈ (Arrowβ€˜πΆ), 𝑓 ∈ {β„Ž ∈ (Arrowβ€˜πΆ) ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} ↦ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩ βˆ™ (codaβ€˜π‘”))(2nd β€˜π‘“))⟩) β†’ (𝐺 Β· 𝐹) = βŸ¨π‘‹, 𝑍, ((2nd β€˜πΊ)(βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍)(2nd β€˜πΉ))⟩))
464, 45mpi 20 1 (πœ‘ β†’ (𝐺 Β· 𝐹) = βŸ¨π‘‹, 𝑍, ((2nd β€˜πΊ)(βŸ¨π‘‹, π‘ŒβŸ© βˆ™ 𝑍)(2nd β€˜πΉ))⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468  βŸ¨cop 4629  βŸ¨cotp 4631  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  2nd c2nd 7970  compcco 17216  domacdoma 17980  codaccoda 17981  Arrowcarw 17982  Homachoma 17983  compaccoa 18014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-ot 4632  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-doma 17984  df-coda 17985  df-homa 17986  df-arw 17987  df-coa 18016
This theorem is referenced by:  coa2  18029  coahom  18030  arwlid  18032  arwrid  18033  arwass  18034
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