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Theorem joinlmuladdmuld 7612
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 7610 . 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 2127 1  |-  ( ph  ->  ( ( A  +  C )  x.  B
)  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445  (class class class)co 5690   CCcc 7445    + caddc 7450    x. cmul 7452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-addcl 7538  ax-mulcom 7543  ax-distr 7546
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-iota 5014  df-fv 5057  df-ov 5693
This theorem is referenced by:  div4p1lem1div2  8767  arisum  11056
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