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Theorem caovlem2d 5635
 Description: Rearrangement of expression involving multiplication (𝐺) and addition (𝐹). (Contributed by Jim Kingdon, 3-Jan-2020.)
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)
caovdl2.6 (φ𝑅 𝑆)
caovdl2.com ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
caovdl2.ass ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
caovdl2.cl ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
Assertion
Ref Expression
caovlem2d (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = ((A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(B𝐺((𝐶𝐺𝑅)𝐹(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,𝑅,y,z   x,𝑆,y,z

Proof of Theorem caovlem2d
Dummy variables 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caovdilemd.cl . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐺y) 𝑆)
2 caovdilemd.a . . . 4 (φA 𝑆)
3 caovdilemd.c . . . . 5 (φ𝐶 𝑆)
4 caovdilemd.h . . . . 5 (φ𝐻 𝑆)
51, 3, 4caovcld 5596 . . . 4 (φ → (𝐶𝐺𝐻) 𝑆)
61, 2, 5caovcld 5596 . . 3 (φ → (A𝐺(𝐶𝐺𝐻)) 𝑆)
7 caovdilemd.b . . . 4 (φB 𝑆)
8 caovdilemd.d . . . . 5 (φ𝐷 𝑆)
91, 8, 4caovcld 5596 . . . 4 (φ → (𝐷𝐺𝐻) 𝑆)
101, 7, 9caovcld 5596 . . 3 (φ → (B𝐺(𝐷𝐺𝐻)) 𝑆)
11 caovdl2.6 . . . . 5 (φ𝑅 𝑆)
121, 8, 11caovcld 5596 . . . 4 (φ → (𝐷𝐺𝑅) 𝑆)
131, 2, 12caovcld 5596 . . 3 (φ → (A𝐺(𝐷𝐺𝑅)) 𝑆)
14 caovdl2.com . . 3 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) = (y𝐹x))
15 caovdl2.ass . . 3 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z)))
161, 3, 11caovcld 5596 . . . 4 (φ → (𝐶𝐺𝑅) 𝑆)
171, 7, 16caovcld 5596 . . 3 (φ → (B𝐺(𝐶𝐺𝑅)) 𝑆)
18 caovdl2.cl . . 3 ((φ (x 𝑆 y 𝑆)) → (x𝐹y) 𝑆)
196, 10, 13, 14, 15, 17, 18caov42d 5629 . 2 (φ → (((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻)))𝐹((A𝐺(𝐷𝐺𝑅))𝐹(B𝐺(𝐶𝐺𝑅)))) = (((A𝐺(𝐶𝐺𝐻))𝐹(A𝐺(𝐷𝐺𝑅)))𝐹((B𝐺(𝐶𝐺𝑅))𝐹(B𝐺(𝐷𝐺𝐻)))))
20 caovdilemd.com . . . 4 ((φ (x 𝑆 y 𝑆)) → (x𝐺y) = (y𝐺x))
21 caovdilemd.distr . . . 4 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐹y)𝐺z) = ((x𝐺z)𝐹(y𝐺z)))
22 caovdilemd.ass . . . 4 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((x𝐺y)𝐺z) = (x𝐺(y𝐺z)))
2320, 21, 22, 1, 2, 7, 3, 8, 4caovdilemd 5634 . . 3 (φ → (((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻) = ((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻))))
2420, 21, 22, 1, 2, 7, 8, 3, 11caovdilemd 5634 . . 3 (φ → (((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅) = ((A𝐺(𝐷𝐺𝑅))𝐹(B𝐺(𝐶𝐺𝑅))))
2523, 24oveq12d 5473 . 2 (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = (((A𝐺(𝐶𝐺𝐻))𝐹(B𝐺(𝐷𝐺𝐻)))𝐹((A𝐺(𝐷𝐺𝑅))𝐹(B𝐺(𝐶𝐺𝑅)))))
26 simpr1 909 . . . . . . 7 ((φ (x 𝑆 y 𝑆 z 𝑆)) → x 𝑆)
2718caovclg 5595 . . . . . . . . 9 ((φ (𝑟 𝑆 𝑠 𝑆)) → (𝑟𝐹𝑠) 𝑆)
2827caovclg 5595 . . . . . . . 8 ((φ (y 𝑆 z 𝑆)) → (y𝐹z) 𝑆)
29283adantr1 1062 . . . . . . 7 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (y𝐹z) 𝑆)
3026, 29jca 290 . . . . . 6 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x 𝑆 (y𝐹z) 𝑆))
3120caovcomg 5598 . . . . . . 7 ((φ (𝑟 𝑆 𝑠 𝑆)) → (𝑟𝐺𝑠) = (𝑠𝐺𝑟))
3231caovcomg 5598 . . . . . 6 ((φ (x 𝑆 (y𝐹z) 𝑆)) → (x𝐺(y𝐹z)) = ((y𝐹z)𝐺x))
3330, 32syldan 266 . . . . 5 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((y𝐹z)𝐺x))
34 3anrot 889 . . . . . 6 ((x 𝑆 y 𝑆 z 𝑆) ↔ (y 𝑆 z 𝑆 x 𝑆))
3521caovdirg 5620 . . . . . . 7 ((φ (𝑟 𝑆 𝑠 𝑆 𝑡 𝑆)) → ((𝑟𝐹𝑠)𝐺𝑡) = ((𝑟𝐺𝑡)𝐹(𝑠𝐺𝑡)))
3635caovdirg 5620 . . . . . 6 ((φ (y 𝑆 z 𝑆 x 𝑆)) → ((y𝐹z)𝐺x) = ((y𝐺x)𝐹(z𝐺x)))
3734, 36sylan2b 271 . . . . 5 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((y𝐹z)𝐺x) = ((y𝐺x)𝐹(z𝐺x)))
3820eqcomd 2042 . . . . . . 7 ((φ (x 𝑆 y 𝑆)) → (y𝐺x) = (x𝐺y))
39383adantr3 1064 . . . . . 6 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (y𝐺x) = (x𝐺y))
4031caovcomg 5598 . . . . . . . 8 ((φ (z 𝑆 x 𝑆)) → (z𝐺x) = (x𝐺z))
4140ancom2s 500 . . . . . . 7 ((φ (x 𝑆 z 𝑆)) → (z𝐺x) = (x𝐺z))
42413adantr2 1063 . . . . . 6 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (z𝐺x) = (x𝐺z))
4339, 42oveq12d 5473 . . . . 5 ((φ (x 𝑆 y 𝑆 z 𝑆)) → ((y𝐺x)𝐹(z𝐺x)) = ((x𝐺y)𝐹(x𝐺z)))
4433, 37, 433eqtrd 2073 . . . 4 ((φ (x 𝑆 y 𝑆 z 𝑆)) → (x𝐺(y𝐹z)) = ((x𝐺y)𝐹(x𝐺z)))
4544, 2, 5, 12caovdid 5618 . . 3 (φ → (A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅))) = ((A𝐺(𝐶𝐺𝐻))𝐹(A𝐺(𝐷𝐺𝑅))))
4644, 7, 16, 9caovdid 5618 . . 3 (φ → (B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻))) = ((B𝐺(𝐶𝐺𝑅))𝐹(B𝐺(𝐷𝐺𝐻))))
4745, 46oveq12d 5473 . 2 (φ → ((A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))) = (((A𝐺(𝐶𝐺𝐻))𝐹(A𝐺(𝐷𝐺𝑅)))𝐹((B𝐺(𝐶𝐺𝑅))𝐹(B𝐺(𝐷𝐺𝐻)))))
4819, 25, 473eqtr4d 2079 1 (φ → ((((A𝐺𝐶)𝐹(B𝐺𝐷))𝐺𝐻)𝐹(((A𝐺𝐷)𝐹(B𝐺𝐶))𝐺𝑅)) = ((A𝐺((𝐶𝐺𝐻)𝐹(𝐷𝐺𝑅)))𝐹(B𝐺((𝐶𝐺𝑅)𝐹(𝐷𝐺𝐻)))))
 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:  mulasssrg  6686
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