Theorem joinlmuladdmuld 7817

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

Step	Hyp	Ref	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 )
4	1, 2, 3	adddird 7815	. 2 |- ( ph -> ( ( A + C ) x. B ) = ( ( A x. B ) + ( C x. B ) ) )
5		joinlmuladdmuld.4	. 2 |- ( ph -> ( ( A x. B ) + ( C x. B ) ) = D )
6	4, 5	eqtrd 2173	1 |- ( ph -> ( ( A + C ) x. B ) = D )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481  (class class class)co 5782   CCcc 7642   + caddc 7647   x. cmul 7649

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-addcl 7740  ax-mulcom 7745  ax-distr 7748

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139  df-ov 5785

This theorem is referenced by:  div4p1lem1div2  8997  arisum  11299  tangtx  12967