Theorem joinlmuladdmuld 8049

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 8047	. 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 2226	1 |- ( ph -> ( ( A + C ) x. B ) = D )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164  (class class class)co 5919   CCcc 7872   + caddc 7877   x. cmul 7879

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-addcl 7970  ax-mulcom 7975  ax-distr 7978

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-iota 5216  df-fv 5263  df-ov 5922

This theorem is referenced by:  div4p1lem1div2  9239  arisum  11644  tangtx  15014  binom4  15152