Theorem distrsrg 7789

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 7757	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 7775	. 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 7776	. 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 7776	. 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 7776	. 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 7775	. 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 7567	. . . 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 7567	. . . 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 7602	. . . . 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 7602	. . . . 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 7567	. . . 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 7602	. . . . 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 7602	. . . . 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 7567	. . . 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 7602	. . . . 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 7602	. . . . 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 7567	. . . 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 7602	. . . . 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 7602	. . . . 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 7567	. . . 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 1023	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
39		simp2l 1025	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
40		simp3l 1027	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
41		distrprg 7618	. . . . 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 1249	. . . 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 1024	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
44		simp2r 1026	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
45		simp3r 1028	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
46		distrprg 7618	. . . . 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 1249	. . . 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 5915	. . 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 7608	. . . . 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 7609	. . . . 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 7567	. . . . 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 6082	. . 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 2222	. 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 7618	. . . . 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 1249	. . . 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 7618	. . . . 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 1249	. . . 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 5915	. . 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 6082	. . 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 2222	. 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 6674	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 980   = wceq 1364   e. wcel 2160  (class class class)co 5897   P.cnp 7321   +P. cpp 7323   .P. cmp 7324   ~R cer 7326   R.cnr 7327   +R cplr 7331   .R cmr 7332

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-iplp 7498  df-imp 7499  df-enr 7756  df-nr 7757  df-plr 7758  df-mr 7759

This theorem is referenced by:  pn0sr  7801  axmulass  7903  axdistr  7904