Theorem joinlmuladdmuld 8100

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 8098	. 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 2238	1 |- ( ph -> ( ( A + C ) x. B ) = D )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176  (class class class)co 5944   CCcc 7923   + caddc 7928   x. cmul 7930

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-addcl 8021  ax-mulcom 8026  ax-distr 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  div4p1lem1div2  9291  arisum  11809  tangtx  15310  binom4  15451