Theorem int-rightdistd 38483

Description: AdditionMultiplicationRightDistribution generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)

Hypotheses

Ref	Expression
int-rightdistd.1	|- ( ph -> B e. RR )
int-rightdistd.2	|- ( ph -> C e. RR )
int-rightdistd.3	|- ( ph -> D e. RR )
int-rightdistd.4	|- ( ph -> A = B )

Assertion

Ref	Expression
int-rightdistd	|- ( ph -> ( B x. ( C + D ) ) = ( ( A x. C ) + ( A x. D ) ) )

Proof of Theorem int-rightdistd

Step	Hyp	Ref	Expression
1		int-rightdistd.1	. . . 4 |- ( ph -> B e. RR )
2	1	recnd 10068	. . 3 |- ( ph -> B e. CC )
3		int-rightdistd.2	. . . . 5 |- ( ph -> C e. RR )
4	3	recnd 10068	. . . 4 |- ( ph -> C e. CC )
5		int-rightdistd.3	. . . . 5 |- ( ph -> D e. RR )
6	5	recnd 10068	. . . 4 |- ( ph -> D e. CC )
7	4, 6	addcld 10059	. . 3 |- ( ph -> ( C + D ) e. CC )
8	2, 7	mulcomd 10061	. 2 |- ( ph -> ( B x. ( C + D ) ) = ( ( C + D ) x. B ) )
9	4, 2	mulcomd 10061	. . . . 5 |- ( ph -> ( C x. B ) = ( B x. C ) )
10		int-rightdistd.4	. . . . . . 7 |- ( ph -> A = B )
11	10	eqcomd 2628	. . . . . 6 |- ( ph -> B = A )
12	11	oveq1d 6665	. . . . 5 |- ( ph -> ( B x. C ) = ( A x. C ) )
13	9, 12	eqtrd 2656	. . . 4 |- ( ph -> ( C x. B ) = ( A x. C ) )
14	6, 2	mulcomd 10061	. . . . 5 |- ( ph -> ( D x. B ) = ( B x. D ) )
15	11	oveq1d 6665	. . . . 5 |- ( ph -> ( B x. D ) = ( A x. D ) )
16	14, 15	eqtrd 2656	. . . 4 |- ( ph -> ( D x. B ) = ( A x. D ) )
17	13, 16	oveq12d 6668	. . 3 |- ( ph -> ( ( C x. B ) + ( D x. B ) ) = ( ( A x. C ) + ( A x. D ) ) )
18	4, 2, 6, 17	joinlmuladdmuld 10067	. 2 |- ( ph -> ( ( C + D ) x. B ) = ( ( A x. C ) + ( A x. D ) ) )
19	8, 18	eqtrd 2656	1 |- ( ph -> ( B x. ( C + D ) ) = ( ( A x. C ) + ( A x. D ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990  (class class class)co 6650   RRcr 9935   + 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-resscn 9993  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)