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Theorem caovdirg 5620
Description: Convert an operation reverse distributive law to class notation. (Contributed by Mario Carneiro, 19-Oct-2014.)
Hypothesis
Ref Expression
caovdirg.1 ((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))
Assertion
Ref Expression
caovdirg ((φ (A 𝑆 B 𝑆 𝐶 𝐾)) → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   φ,x,y,z   x,𝐹,y,z   x,𝐺,y,z   x,𝐻,y,z   x,𝐾,y,z   x,𝑆,y,z

Proof of Theorem caovdirg
StepHypRef Expression
1 caovdirg.1 . . 3 ((φ (x 𝑆 y 𝑆 z 𝐾)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))
21ralrimivvva 2396 . 2 (φx 𝑆 y 𝑆 z 𝐾 ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)))
3 oveq1 5462 . . . . 5 (x = A → (x𝐹y) = (A𝐹y))
43oveq1d 5470 . . . 4 (x = A → ((x𝐹y)𝐺z) = ((A𝐹y)𝐺z))
5 oveq1 5462 . . . . 5 (x = A → (x𝐺z) = (A𝐺z))
65oveq1d 5470 . . . 4 (x = A → ((x𝐺z)𝐻(y𝐺z)) = ((A𝐺z)𝐻(y𝐺z)))
74, 6eqeq12d 2051 . . 3 (x = A → (((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)) ↔ ((A𝐹y)𝐺z) = ((A𝐺z)𝐻(y𝐺z))))
8 oveq2 5463 . . . . 5 (y = B → (A𝐹y) = (A𝐹B))
98oveq1d 5470 . . . 4 (y = B → ((A𝐹y)𝐺z) = ((A𝐹B)𝐺z))
10 oveq1 5462 . . . . 5 (y = B → (y𝐺z) = (B𝐺z))
1110oveq2d 5471 . . . 4 (y = B → ((A𝐺z)𝐻(y𝐺z)) = ((A𝐺z)𝐻(B𝐺z)))
129, 11eqeq12d 2051 . . 3 (y = B → (((A𝐹y)𝐺z) = ((A𝐺z)𝐻(y𝐺z)) ↔ ((A𝐹B)𝐺z) = ((A𝐺z)𝐻(B𝐺z))))
13 oveq2 5463 . . . 4 (z = 𝐶 → ((A𝐹B)𝐺z) = ((A𝐹B)𝐺𝐶))
14 oveq2 5463 . . . . 5 (z = 𝐶 → (A𝐺z) = (A𝐺𝐶))
15 oveq2 5463 . . . . 5 (z = 𝐶 → (B𝐺z) = (B𝐺𝐶))
1614, 15oveq12d 5473 . . . 4 (z = 𝐶 → ((A𝐺z)𝐻(B𝐺z)) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
1713, 16eqeq12d 2051 . . 3 (z = 𝐶 → (((A𝐹B)𝐺z) = ((A𝐺z)𝐻(B𝐺z)) ↔ ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶))))
187, 12, 17rspc3v 2659 . 2 ((A 𝑆 B 𝑆 𝐶 𝐾) → (x 𝑆 y 𝑆 z 𝐾 ((x𝐹y)𝐺z) = ((x𝐺z)𝐻(y𝐺z)) → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶))))
192, 18mpan9 265 1 ((φ (A 𝑆 B 𝑆 𝐶 𝐾)) → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐻(B𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wral 2300  (class class class)co 5455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-ov 5458
This theorem is referenced by:  caovdird  5621  caovlem2d  5635
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