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Theorem joinlmuladdmuld 7936
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 7934 . 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 2203 1  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141  (class class class)co 5851   CCcc 7761    + caddc 7766    x. cmul 7768
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-addcl 7859  ax-mulcom 7864  ax-distr 7867
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5854
This theorem is referenced by:  div4p1lem1div2  9120  arisum  11450  tangtx  13514  binom4  13652
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