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Theorem joinlmuladdmuld 8016
Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)
Hypotheses
Ref Expression
joinlmuladdmuld.1  |-  ( ph  ->  A  e.  CC )
joinlmuladdmuld.2  |-  ( ph  ->  B  e.  CC )
joinlmuladdmuld.3  |-  ( ph  ->  C  e.  CC )
joinlmuladdmuld.4  |-  ( ph  ->  ( ( A  x.  B )  +  ( C  x.  B ) )  =  D )
Assertion
Ref Expression
joinlmuladdmuld  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )

Proof of Theorem joinlmuladdmuld
StepHypRef Expression
1 joinlmuladdmuld.1 . . 3  |-  ( ph  ->  A  e.  CC )
2 joinlmuladdmuld.3 . . 3  |-  ( ph  ->  C  e.  CC )
3 joinlmuladdmuld.2 . . 3  |-  ( ph  ->  B  e.  CC )
41, 2, 3adddird 8014 . 2  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  ( ( A  x.  B )  +  ( C  x.  B ) ) )
5 joinlmuladdmuld.4 . 2  |-  ( ph  ->  ( ( A  x.  B )  +  ( C  x.  B ) )  =  D )
64, 5eqtrd 2222 1  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160  (class class class)co 5897   CCcc 7840    + caddc 7845    x. cmul 7847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcl 7938  ax-mulcom 7943  ax-distr 7946
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900
This theorem is referenced by:  div4p1lem1div2  9203  arisum  11541  tangtx  14736  binom4  14874
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