Theorem distrsr 9912

Description: Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
distrsr	|- ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) )

Proof of Theorem distrsr

Dummy variables f g h u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 9878	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 9896	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R +R [ <. v , u >. ] ~R ) = [ <. ( z +P. v ) , ( w +P. u ) >. ] ~R )
3		mulsrpr 9897	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( ( z +P. v ) e. P. /\ ( w +P. u ) e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. ( z +P. v ) , ( w +P. u ) >. ] ~R ) = [ <. ( ( x .P. ( z +P. v ) ) +P. ( y .P. ( w +P. u ) ) ) , ( ( x .P. ( w +P. u ) ) +P. ( y .P. ( z +P. v ) ) ) >. ] ~R )
4		mulsrpr 9897	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R )
5		mulsrpr 9897	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. v , u >. ] ~R ) = [ <. ( ( x .P. v ) +P. ( y .P. u ) ) , ( ( x .P. u ) +P. ( y .P. v ) ) >. ] ~R )
6		addsrpr 9896	. . 3 |- ( ( ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) /\ ( ( ( x .P. v ) +P. ( y .P. u ) ) e. P. /\ ( ( x .P. u ) +P. ( y .P. v ) ) e. P. ) ) -> ( [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R +R [ <. ( ( x .P. v ) +P. ( y .P. u ) ) , ( ( x .P. u ) +P. ( y .P. v ) ) >. ] ~R ) = [ <. ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( ( x .P. v ) +P. ( y .P. u ) ) ) , ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( ( x .P. u ) +P. ( y .P. v ) ) ) >. ] ~R )
7		addclpr 9840	. . . . 5 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
8		addclpr 9840	. . . . 5 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
9	7, 8	anim12i 590	. . . 4 |- ( ( ( z e. P. /\ v e. P. ) /\ ( w e. P. /\ u e. P. ) ) -> ( ( z +P. v ) e. P. /\ ( w +P. u ) e. P. ) )
10	9	an4s 869	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z +P. v ) e. P. /\ ( w +P. u ) e. P. ) )
11		mulclpr 9842	. . . . . 6 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
12		mulclpr 9842	. . . . . 6 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
13		addclpr 9840	. . . . . 6 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
14	11, 12, 13	syl2an 494	. . . . 5 |- ( ( ( x e. P. /\ z e. P. ) /\ ( y e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
15	14	an4s 869	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
16		mulclpr 9842	. . . . . 6 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
17		mulclpr 9842	. . . . . 6 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
18		addclpr 9840	. . . . . 6 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
19	16, 17, 18	syl2an 494	. . . . 5 |- ( ( ( x e. P. /\ w e. P. ) /\ ( y e. P. /\ z e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
20	19	an42s 870	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
21	15, 20	jca 554	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) )
22		mulclpr 9842	. . . . . 6 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
23		mulclpr 9842	. . . . . 6 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
24		addclpr 9840	. . . . . 6 |- ( ( ( x .P. v ) e. P. /\ ( y .P. u ) e. P. ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
25	22, 23, 24	syl2an 494	. . . . 5 |- ( ( ( x e. P. /\ v e. P. ) /\ ( y e. P. /\ u e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
26	25	an4s 869	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
27		mulclpr 9842	. . . . . 6 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
28		mulclpr 9842	. . . . . 6 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
29		addclpr 9840	. . . . . 6 |- ( ( ( x .P. u ) e. P. /\ ( y .P. v ) e. P. ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
30	27, 28, 29	syl2an 494	. . . . 5 |- ( ( ( x e. P. /\ u e. P. ) /\ ( y e. P. /\ v e. P. ) ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
31	30	an42s 870	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
32	26, 31	jca 554	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( x .P. v ) +P. ( y .P. u ) ) e. P. /\ ( ( x .P. u ) +P. ( y .P. v ) ) e. P. ) )
33		distrpr 9850	. . . . 5 |- ( x .P. ( z +P. v ) ) = ( ( x .P. z ) +P. ( x .P. v ) )
34		distrpr 9850	. . . . 5 |- ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) )
35	33, 34	oveq12i 6662	. . . 4 |- ( ( x .P. ( z +P. v ) ) +P. ( y .P. ( w +P. u ) ) ) = ( ( ( x .P. z ) +P. ( x .P. v ) ) +P. ( ( y .P. w ) +P. ( y .P. u ) ) )
36		ovex 6678	. . . . 5 |- ( x .P. z ) e. _V
37		ovex 6678	. . . . 5 |- ( x .P. v ) e. _V
38		ovex 6678	. . . . 5 |- ( y .P. w ) e. _V
39		addcompr 9843	. . . . 5 |- ( f +P. g ) = ( g +P. f )
40		addasspr 9844	. . . . 5 |- ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) )
41		ovex 6678	. . . . 5 |- ( y .P. u ) e. _V
42	36, 37, 38, 39, 40, 41	caov4 6865	. . . 4 |- ( ( ( x .P. z ) +P. ( x .P. v ) ) +P. ( ( y .P. w ) +P. ( y .P. u ) ) ) = ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( ( x .P. v ) +P. ( y .P. u ) ) )
43	35, 42	eqtri 2644	. . 3 |- ( ( x .P. ( z +P. v ) ) +P. ( y .P. ( w +P. u ) ) ) = ( ( ( x .P. z ) +P. ( y .P. w ) ) +P. ( ( x .P. v ) +P. ( y .P. u ) ) )
44		distrpr 9850	. . . . 5 |- ( x .P. ( w +P. u ) ) = ( ( x .P. w ) +P. ( x .P. u ) )
45		distrpr 9850	. . . . 5 |- ( y .P. ( z +P. v ) ) = ( ( y .P. z ) +P. ( y .P. v ) )
46	44, 45	oveq12i 6662	. . . 4 |- ( ( x .P. ( w +P. u ) ) +P. ( y .P. ( z +P. v ) ) ) = ( ( ( x .P. w ) +P. ( x .P. u ) ) +P. ( ( y .P. z ) +P. ( y .P. v ) ) )
47		ovex 6678	. . . . 5 |- ( x .P. w ) e. _V
48		ovex 6678	. . . . 5 |- ( x .P. u ) e. _V
49		ovex 6678	. . . . 5 |- ( y .P. z ) e. _V
50		ovex 6678	. . . . 5 |- ( y .P. v ) e. _V
51	47, 48, 49, 39, 40, 50	caov4 6865	. . . 4 |- ( ( ( x .P. w ) +P. ( x .P. u ) ) +P. ( ( y .P. z ) +P. ( y .P. v ) ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( ( x .P. u ) +P. ( y .P. v ) ) )
52	46, 51	eqtri 2644	. . 3 |- ( ( x .P. ( w +P. u ) ) +P. ( y .P. ( z +P. v ) ) ) = ( ( ( x .P. w ) +P. ( y .P. z ) ) +P. ( ( x .P. u ) +P. ( y .P. v ) ) )
53	1, 2, 3, 4, 5, 6, 10, 21, 32, 43, 52	ecovdi 7856	. 2 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) )
54		dmaddsr 9906	. . 3 |- dom +R = ( R. X. R. )
55		0nsr 9900	. . 3 |- -. (/) e. R.
56		dmmulsr 9907	. . 3 |- dom .R = ( R. X. R. )
57	54, 55, 56	ndmovdistr 6823	. 2 |- ( -. ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) )
58	53, 57	pm2.61i 176	1 |- ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) )

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990  (class class class)co 6650   P.cnp 9681   +P. cpp 9683   .P. cmp 9684   ~R cer 9686   R.cnr 9687   +R cplr 9691   .R cmr 9692

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880

This theorem is referenced by:  pn0sr  9922  axmulass  9978  axdistr  9979