Theorem distrsrg 7455

Description: Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)

Assertion

Ref	Expression
distrsrg	|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) )

Proof of Theorem distrsrg

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 7423	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 7441	. 2 |- ( ( ( 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 7442	. 2 |- ( ( ( 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 7442	. 2 |- ( ( ( 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 7442	. 2 |- ( ( ( 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 7441	. 2 |- ( ( ( ( ( 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 7246	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
8	7	ad2ant2r 496	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z +P. v ) e. P. )
9		addclpr 7246	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
10	9	ad2ant2l 495	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. u ) e. P. )
11	8, 10	jca 302	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z +P. v ) e. P. /\ ( w +P. u ) e. P. ) )
12		mulclpr 7281	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
13	12	ad2ant2r 496	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
14		mulclpr 7281	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
15	14	ad2ant2l 495	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
16		addclpr 7246	. . . 4 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
17	13, 15, 16	syl2anc 406	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
18		mulclpr 7281	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
19	18	ad2ant2rl 498	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
20		mulclpr 7281	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
21	20	ad2ant2lr 497	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
22		addclpr 7246	. . . 4 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
23	19, 21, 22	syl2anc 406	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
24	17, 23	jca 302	. 2 |- ( ( ( 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. ) )
25		mulclpr 7281	. . . . 5 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
26	25	ad2ant2r 496	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. v ) e. P. )
27		mulclpr 7281	. . . . 5 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
28	27	ad2ant2l 495	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. u ) e. P. )
29		addclpr 7246	. . . 4 |- ( ( ( x .P. v ) e. P. /\ ( y .P. u ) e. P. ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
30	26, 28, 29	syl2anc 406	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
31		mulclpr 7281	. . . . 5 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
32	31	ad2ant2rl 498	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. u ) e. P. )
33		mulclpr 7281	. . . . 5 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
34	33	ad2ant2lr 497	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. v ) e. P. )
35		addclpr 7246	. . . 4 |- ( ( ( x .P. u ) e. P. /\ ( y .P. v ) e. P. ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
36	32, 34, 35	syl2anc 406	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. u ) +P. ( y .P. v ) ) e. P. )
37	30, 36	jca 302	. 2 |- ( ( ( 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. ) )
38		simp1l 973	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
39		simp2l 975	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
40		simp3l 977	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
41		distrprg 7297	. . . . 5 |- ( ( x e. P. /\ z e. P. /\ v e. P. ) -> ( x .P. ( z +P. v ) ) = ( ( x .P. z ) +P. ( x .P. v ) ) )
42	38, 39, 40, 41	syl3anc 1184	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. ( z +P. v ) ) = ( ( x .P. z ) +P. ( x .P. v ) ) )
43		simp1r 974	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
44		simp2r 976	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
45		simp3r 978	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
46		distrprg 7297	. . . . 5 |- ( ( y e. P. /\ w e. P. /\ u e. P. ) -> ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) ) )
47	43, 44, 45, 46	syl3anc 1184	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. ( w +P. u ) ) = ( ( y .P. w ) +P. ( y .P. u ) ) )
48	42, 47	oveq12d 5724	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( 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 ) ) ) )
49	38, 39, 12	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. z ) e. P. )
50	38, 40, 25	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. v ) e. P. )
51	43, 44, 14	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. w ) e. P. )
52		addcomprg 7287	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 273	. . . 4 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
54		addassprg 7288	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
55	54	adantl 273	. . . 4 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
56	43, 45, 27	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. u ) e. P. )
57		addclpr 7246	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
58	57	adantl 273	. . . 4 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) e. P. )
59	49, 50, 51, 53, 55, 56, 58	caov4d 5887	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( 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 ) ) ) )
60	48, 59	eqtrd 2132	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( 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 ) ) ) )
61		distrprg 7297	. . . . 5 |- ( ( x e. P. /\ w e. P. /\ u e. P. ) -> ( x .P. ( w +P. u ) ) = ( ( x .P. w ) +P. ( x .P. u ) ) )
62	38, 44, 45, 61	syl3anc 1184	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. ( w +P. u ) ) = ( ( x .P. w ) +P. ( x .P. u ) ) )
63		distrprg 7297	. . . . 5 |- ( ( y e. P. /\ z e. P. /\ v e. P. ) -> ( y .P. ( z +P. v ) ) = ( ( y .P. z ) +P. ( y .P. v ) ) )
64	43, 39, 40, 63	syl3anc 1184	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. ( z +P. v ) ) = ( ( y .P. z ) +P. ( y .P. v ) ) )
65	62, 64	oveq12d 5724	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( 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 ) ) ) )
66	38, 44, 18	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. w ) e. P. )
67	38, 45, 31	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. u ) e. P. )
68	43, 39, 20	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. z ) e. P. )
69	43, 40, 33	syl2anc 406	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. v ) e. P. )
70	66, 67, 68, 53, 55, 69, 58	caov4d 5887	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( 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 ) ) ) )
71	65, 70	eqtrd 2132	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( 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 ) ) ) )
72	1, 2, 3, 4, 5, 6, 11, 24, 37, 60, 71	ecovidi 6471	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( A .R ( B +R C ) ) = ( ( A .R B ) +R ( A .R C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 930   = wceq 1299   e. wcel 1448  (class class class)co 5706   P.cnp 7000   +P. cpp 7002   .P. cmp 7003   ~R cer 7005   R.cnr 7006   +R cplr 7010   .R cmr 7011

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-imp 7178  df-enr 7422  df-nr 7423  df-plr 7424  df-mr 7425

This theorem is referenced by:  pn0sr  7467  axmulass  7558  axdistr  7559