Theorem distrsrg 8076

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 8044	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 8062	. 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 8063	. 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 8063	. 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 8063	. 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 8062	. 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 7854	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
8	7	ad2ant2r 509	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z +P. v ) e. P. )
9		addclpr 7854	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
10	9	ad2ant2l 508	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. u ) e. P. )
11	8, 10	jca 306	. 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 7889	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
13	12	ad2ant2r 509	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
14		mulclpr 7889	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
15	14	ad2ant2l 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
16		addclpr 7854	. . . 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 411	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
18		mulclpr 7889	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
19	18	ad2ant2rl 511	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
20		mulclpr 7889	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
21	20	ad2ant2lr 510	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
22		addclpr 7854	. . . 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 411	. . 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 306	. 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 7889	. . . . 5 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
26	25	ad2ant2r 509	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. v ) e. P. )
27		mulclpr 7889	. . . . 5 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
28	27	ad2ant2l 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. u ) e. P. )
29		addclpr 7854	. . . 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 411	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
31		mulclpr 7889	. . . . 5 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
32	31	ad2ant2rl 511	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. u ) e. P. )
33		mulclpr 7889	. . . . 5 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
34	33	ad2ant2lr 510	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. v ) e. P. )
35		addclpr 7854	. . . 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 411	. . 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 306	. 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 1048	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
39		simp2l 1050	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
40		simp3l 1052	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
41		distrprg 7905	. . . . 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 1274	. . . 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 1049	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
44		simp2r 1051	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
45		simp3r 1053	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
46		distrprg 7905	. . . . 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 1274	. . . 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 6070	. . 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 411	. . . 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 411	. . . 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 411	. . . 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 7895	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 277	. . . 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 7896	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
55	54	adantl 277	. . . 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 411	. . . 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 7854	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
58	57	adantl 277	. . . 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 6241	. . 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 2267	. 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 7905	. . . . 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 1274	. . . 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 7905	. . . . 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 1274	. . . 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 6070	. . 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 411	. . . 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 411	. . . 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 411	. . . 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 411	. . . 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 6241	. . 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 2267	. 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 6883	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 104   /\ w3a 1005   = wceq 1398   e. wcel 2205  (class class class)co 6052   P.cnp 7608   +P. cpp 7610   .P. cmp 7611   ~R cer 7613   R.cnr 7614   +R cplr 7618   .R cmr 7619

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-enq0 7741  df-nq0 7742  df-0nq0 7743  df-plq0 7744  df-mq0 7745  df-inp 7783  df-iplp 7785  df-imp 7786  df-enr 8043  df-nr 8044  df-plr 8045  df-mr 8046

This theorem is referenced by:  pn0sr  8088  axmulass  8190  axdistr  8191