Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  caovdir2d Structured version   GIF version

Theorem caovdir2d 5619
 Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdir2d.1 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z)))
caovdir2d.2 (φA 𝑆)
caovdir2d.3 (φB 𝑆)
caovdir2d.4 (φ𝐶 𝑆)
caovdir2d.cl ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
caovdir2d.com ((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))
Assertion
Ref Expression
caovdir2d (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐹(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,𝑆,y,z

Proof of Theorem caovdir2d
StepHypRef Expression
1 caovdir2d.1 . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z)))
2 caovdir2d.4 . . 3 (φ𝐶 𝑆)
3 caovdir2d.2 . . 3 (φA 𝑆)
4 caovdir2d.3 . . 3 (φB 𝑆)
51, 2, 3, 4caovdid 5618 . 2 (φ → (𝐶𝐺(A𝐹B)) = ((𝐶𝐺A)𝐹(𝐶𝐺B)))
6 caovdir2d.com . . 3 ((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))
7 caovdir2d.cl . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
87, 3, 4caovcld 5596 . . 3 (φ → (A𝐹B) 𝑆)
96, 8, 2caovcomd 5599 . 2 (φ → ((A𝐹B)𝐺𝐶) = (𝐶𝐺(A𝐹B)))
106, 3, 2caovcomd 5599 . . 3 (φ → (A𝐺𝐶) = (𝐶𝐺A))
116, 4, 2caovcomd 5599 . . 3 (φ → (B𝐺𝐶) = (𝐶𝐺B))
1210, 11oveq12d 5473 . 2 (φ → ((A𝐺𝐶)𝐹(B𝐺𝐶)) = ((𝐶𝐺A)𝐹(𝐶𝐺B)))
135, 9, 123eqtr4d 2079 1 (φ → ((A𝐹B)𝐺𝐶) = ((A𝐺𝐶)𝐹(B𝐺𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  (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-bndl 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:  addcmpblnq  6351  ltanqg  6384  addcmpblnq0  6426  mulasssrg  6686  mulgt0sr  6704  mulextsr1lem  6706
 Copyright terms: Public domain W3C validator