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Theorem joinlmuladdmuld 7981
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 7979 . 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 2210 1  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148  (class class class)co 5872   CCcc 7806    + caddc 7811    x. cmul 7813
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-addcl 7904  ax-mulcom 7909  ax-distr 7912
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875
This theorem is referenced by:  div4p1lem1div2  9168  arisum  11499  tangtx  14130  binom4  14268
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