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Theorem joinlmuladdmuld 7807
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 7805 . 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 2172 1  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480  (class class class)co 5774   CCcc 7632    + caddc 7637    x. cmul 7639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-addcl 7730  ax-mulcom 7735  ax-distr 7738
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777
This theorem is referenced by:  div4p1lem1div2  8987  arisum  11281  tangtx  12943
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