Theorem joinlmuladdmuld 7947

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 7945	. 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 2203	1 |- ( ph -> ( ( A + C ) x. B ) = D )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   + caddc 7777   x. cmul 7779

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 7870  ax-mulcom 7875  ax-distr 7878

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 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by:  div4p1lem1div2  9131  arisum  11461  tangtx  13553  binom4  13691