Theorem joinlmuladdmuld 8054

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 8052	. 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 2229	1 |- ( ph -> ( ( A + C ) x. B ) = D )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7877   + caddc 7882   x. cmul 7884

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-addcl 7975  ax-mulcom 7980  ax-distr 7983

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  div4p1lem1div2  9245  arisum  11663  tangtx  15074  binom4  15215