Theorem joinlmuladdmuli 42519

Description: Join AB+CB into (A+C) on LHS. (Contributed by David A. Wheeler, 26-Oct-2019.)

Hypotheses

Ref	Expression
joinlmuladdmuli.1	|- A e. CC
joinlmuladdmuli.2	|- B e. CC
joinlmuladdmuli.3	|- C e. CC
joinlmuladdmuli.4	|- ( ( A x. B ) + ( C x. B ) ) = D

Assertion

Ref	Expression
joinlmuladdmuli	|- ( ( A + C ) x. B ) = D

Proof of Theorem joinlmuladdmuli

Step	Hyp	Ref	Expression
1		joinlmuladdmuli.1	. . . 4 |- A e. CC
2	1	a1i 11	. . 3 |- ( T. -> A e. CC )
3		joinlmuladdmuli.2	. . . 4 |- B e. CC
4	3	a1i 11	. . 3 |- ( T. -> B e. CC )
5		joinlmuladdmuli.3	. . . 4 |- C e. CC
6	5	a1i 11	. . 3 |- ( T. -> C e. CC )
7		joinlmuladdmuli.4	. . . 4 |- ( ( A x. B ) + ( C x. B ) ) = D
8	7	a1i 11	. . 3 |- ( T. -> ( ( A x. B ) + ( C x. B ) ) = D )
9	2, 4, 6, 8	joinlmuladdmuld 10067	. 2 |- ( T. -> ( ( A + C ) x. B ) = D )
10	9	trud 1493	1 |- ( ( A + C ) x. B ) = D

Syntax hints:   = wceq 1483   T. wtru 1484   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-addcl 9996  ax-mulcom 10000  ax-distr 10003

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)