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Theorem caovdilemd 5631
Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
Hypotheses
Ref Expression
caovdilemd.com ((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))
caovdilemd.distr ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z)))
caovdilemd.ass ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z)))
caovdilemd.cl ((φ (x 𝑆 y 𝑆)) → (x𝐺y) 𝑆)
caovdilemd.a (φA 𝑆)
caovdilemd.b (φB 𝑆)
caovdilemd.c (φ𝐶 𝑆)
caovdilemd.d (φ𝐷 𝑆)
caovdilemd.h (φ𝐻 𝑆)
Assertion
Ref Expression
caovdilemd (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,𝐶,y,z   x,𝐷,y,z   φ,x,y,z   x,𝐹,y,z   x,𝐺,y,z   x,𝐻,y,z   x,𝑆,y,z

Proof of Theorem caovdilemd
StepHypRef Expression
1 caovdilemd.distr . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z)))
2 caovdilemd.cl . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐺y) 𝑆)
3 caovdilemd.a . . . 4 (φA 𝑆)
4 caovdilemd.c . . . 4 (φ𝐶 𝑆)
52, 3, 4caovcld 5593 . . 3 (φ → (A𝐺𝐶) 𝑆)
6 caovdilemd.b . . . 4 (φB 𝑆)
7 caovdilemd.d . . . 4 (φ𝐷 𝑆)
82, 6, 7caovcld 5593 . . 3 (φ → (B𝐺𝐷) 𝑆)
9 caovdilemd.h . . 3 (φ𝐻 𝑆)
101, 5, 8, 9caovdird 5618 . 2 (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = (((A𝐺𝐶)𝐺𝐻)𝐹((B𝐺𝐷)𝐺𝐻)))
11 caovdilemd.ass . . . 4 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z)))
1211, 3, 4, 9caovassd 5599 . . 3 (φ → ((A𝐺𝐶)𝐺𝐻) = (A𝐺(𝐶𝐺𝐻)))
1311, 6, 7, 9caovassd 5599 . . 3 (φ → ((B𝐺𝐷)𝐺𝐻) = (B𝐺(𝐷𝐺𝐻)))
1412, 13oveq12d 5470 . 2 (φ → (((A𝐺𝐶)𝐺𝐻)𝐹((B𝐺𝐷)𝐺𝐻)) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))
1510, 14eqtrd 2069 1 (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  (class class class)co 5452
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 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-iota 4809  df-fv 4852  df-ov 5455
This theorem is referenced by:  caovlem2d  5632  addassnqg  6359  addassnq0  6437  axmulass  6709
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