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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  S  S  G  G
caovdilemd.distr  S  S  S  F G  G F G
caovdilemd.ass  S  S  S  G G  G G
caovdilemd.cl  S  S  G  S
caovdilemd.a  S
caovdilemd.b  S
caovdilemd.c  C  S
caovdilemd.d  D  S
caovdilemd.h  H  S
caovdl2.6  R  S
caovdl2.com  S  S  F  F
caovdl2.ass  S  S  S  F F  F F
caovdl2.cl  S  S  F  S
Assertion
Ref Expression
caovlem2d  G C F G D G H F G D F G C G R  G C G H F D G R F G C G R F D G H
Distinct variable groups:   ,,,   ,,,   , C,,   , D,,   ,,,   , F,,   , G,,   , H,,   , R,,   , S,,

Proof of Theorem caovlem2d
Dummy variables  s  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caovdilemd.cl . . . 4  S  S  G  S
2 caovdilemd.a . . . 4  S
3 caovdilemd.c . . . . 5  C  S
4 caovdilemd.h . . . . 5  H  S
51, 3, 4caovcld 5596 . . . 4  C G H  S
61, 2, 5caovcld 5596 . . 3  G C G H  S
7 caovdilemd.b . . . 4  S
8 caovdilemd.d . . . . 5  D  S
91, 8, 4caovcld 5596 . . . 4  D G H  S
101, 7, 9caovcld 5596 . . 3  G D G H  S
11 caovdl2.6 . . . . 5  R  S
121, 8, 11caovcld 5596 . . . 4  D G R  S
131, 2, 12caovcld 5596 . . 3  G D G R  S
14 caovdl2.com . . 3  S  S  F  F
15 caovdl2.ass . . 3  S  S  S  F F  F F
161, 3, 11caovcld 5596 . . . 4  C G R  S
171, 7, 16caovcld 5596 . . 3  G C G R  S
18 caovdl2.cl . . 3  S  S  F  S
196, 10, 13, 14, 15, 17, 18caov42d 5629 . 2  G C G H F G D G H F G D G R F G C G R  G C G H F G D G R F G C G R F G D G H
20 caovdilemd.com . . . 4  S  S  G  G
21 caovdilemd.distr . . . 4  S  S  S  F G  G F G
22 caovdilemd.ass . . . 4  S  S  S  G G  G G
2320, 21, 22, 1, 2, 7, 3, 8, 4caovdilemd 5634 . . 3  G C F G D G H  G C G H F G D G H
2420, 21, 22, 1, 2, 7, 8, 3, 11caovdilemd 5634 . . 3  G D F G C G R  G D G R F G C G R
2523, 24oveq12d 5473 . 2  G C F G D G H F G D F G C G R  G C G H F G D G H F G D G R F G C G R
26 simpr1 909 . . . . . . 7  S  S  S  S
2718caovclg 5595 . . . . . . . . 9  r  S  s  S  r F s  S
2827caovclg 5595 . . . . . . . 8  S  S  F  S
29283adantr1 1062 . . . . . . 7  S  S  S  F  S
3026, 29jca 290 . . . . . 6  S  S  S  S  F  S
3120caovcomg 5598 . . . . . . 7  r  S  s  S  r G s  s G r
3231caovcomg 5598 . . . . . 6  S  F  S  G F  F G
3330, 32syldan 266 . . . . 5  S  S  S  G F  F G
34 3anrot 889 . . . . . 6  S  S  S  S  S  S
3521caovdirg 5620 . . . . . . 7  r  S  s  S  t  S  r F s G t  r G t F s G t
3635caovdirg 5620 . . . . . 6  S  S  S  F G  G F G
3734, 36sylan2b 271 . . . . 5  S  S  S  F G  G F G
3820eqcomd 2042 . . . . . . 7  S  S  G  G
39383adantr3 1064 . . . . . 6  S  S  S  G  G
4031caovcomg 5598 . . . . . . . 8  S  S  G  G
4140ancom2s 500 . . . . . . 7  S  S  G  G
42413adantr2 1063 . . . . . 6  S  S  S  G  G
4339, 42oveq12d 5473 . . . . 5  S  S  S  G F G  G F G
4433, 37, 433eqtrd 2073 . . . 4  S  S  S  G F  G F G
4544, 2, 5, 12caovdid 5618 . . 3  G C G H F D G R  G C G H F G D G R
4644, 7, 16, 9caovdid 5618 . . 3  G C G R F D G H  G C G R F G D G H
4745, 46oveq12d 5473 . 2  G C G H F D G R F G C G R F D G H  G C G H F G D G R F G C G R F G D G H
4819, 25, 473eqtr4d 2079 1  G C F G D G H F G D F G C G R  G C G H F D G R F G C G R F D G H
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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