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Theorem caovlem2d 5724
 Description: Rearrangement of expression involving multiplication () and addition (). (Contributed by Jim Kingdon, 3-Jan-2020.)
Hypotheses
Ref Expression
caovdilemd.com
caovdilemd.distr
caovdilemd.ass
caovdilemd.cl
caovdilemd.a
caovdilemd.b
caovdilemd.c
caovdilemd.d
caovdilemd.h
caovdl2.6
caovdl2.com
caovdl2.ass
caovdl2.cl
Assertion
Ref Expression
caovlem2d
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,

Proof of Theorem caovlem2d
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caovdilemd.cl . . . 4
2 caovdilemd.a . . . 4
3 caovdilemd.c . . . . 5
4 caovdilemd.h . . . . 5
51, 3, 4caovcld 5685 . . . 4
61, 2, 5caovcld 5685 . . 3
7 caovdilemd.b . . . 4
8 caovdilemd.d . . . . 5
91, 8, 4caovcld 5685 . . . 4
101, 7, 9caovcld 5685 . . 3
11 caovdl2.6 . . . . 5
121, 8, 11caovcld 5685 . . . 4
131, 2, 12caovcld 5685 . . 3
14 caovdl2.com . . 3
15 caovdl2.ass . . 3
161, 3, 11caovcld 5685 . . . 4
171, 7, 16caovcld 5685 . . 3
18 caovdl2.cl . . 3
196, 10, 13, 14, 15, 17, 18caov42d 5718 . 2
20 caovdilemd.com . . . 4
21 caovdilemd.distr . . . 4
22 caovdilemd.ass . . . 4
2320, 21, 22, 1, 2, 7, 3, 8, 4caovdilemd 5723 . . 3
2420, 21, 22, 1, 2, 7, 8, 3, 11caovdilemd 5723 . . 3
2523, 24oveq12d 5561 . 2
26 simpr1 945 . . . . . . 7
2718caovclg 5684 . . . . . . . . 9
2827caovclg 5684 . . . . . . . 8
29283adantr1 1098 . . . . . . 7
3026, 29jca 300 . . . . . 6
3120caovcomg 5687 . . . . . . 7
3231caovcomg 5687 . . . . . 6
3330, 32syldan 276 . . . . 5
34 3anrot 925 . . . . . 6
3521caovdirg 5709 . . . . . . 7
3635caovdirg 5709 . . . . . 6
3734, 36sylan2b 281 . . . . 5
3820eqcomd 2087 . . . . . . 7
39383adantr3 1100 . . . . . 6
4031caovcomg 5687 . . . . . . . 8
4140ancom2s 531 . . . . . . 7
42413adantr2 1099 . . . . . 6
4339, 42oveq12d 5561 . . . . 5
4433, 37, 433eqtrd 2118 . . . 4
4544, 2, 5, 12caovdid 5707 . . 3
4644, 7, 16, 9caovdid 5707 . . 3
4745, 46oveq12d 5561 . 2
4819, 25, 473eqtr4d 2124 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   w3a 920   wceq 1285   wcel 1434  (class class class)co 5543 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546 This theorem is referenced by:  mulasssrg  6997
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