Theorem distrsrg 7721

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 7689	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		addsrpr 7707	. 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 7708	. 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 7708	. 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 7708	. 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 7707	. 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 7499	. . . 4 |- ( ( z e. P. /\ v e. P. ) -> ( z +P. v ) e. P. )
8	7	ad2ant2r 506	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z +P. v ) e. P. )
9		addclpr 7499	. . . 4 |- ( ( w e. P. /\ u e. P. ) -> ( w +P. u ) e. P. )
10	9	ad2ant2l 505	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w +P. u ) e. P. )
11	8, 10	jca 304	. 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 7534	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
13	12	ad2ant2r 506	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
14		mulclpr 7534	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
15	14	ad2ant2l 505	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
16		addclpr 7499	. . . 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 409	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
18		mulclpr 7534	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
19	18	ad2ant2rl 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
20		mulclpr 7534	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
21	20	ad2ant2lr 507	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
22		addclpr 7499	. . . 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 409	. . 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 304	. 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 7534	. . . . 5 |- ( ( x e. P. /\ v e. P. ) -> ( x .P. v ) e. P. )
26	25	ad2ant2r 506	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. v ) e. P. )
27		mulclpr 7534	. . . . 5 |- ( ( y e. P. /\ u e. P. ) -> ( y .P. u ) e. P. )
28	27	ad2ant2l 505	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. u ) e. P. )
29		addclpr 7499	. . . 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 409	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( x .P. v ) +P. ( y .P. u ) ) e. P. )
31		mulclpr 7534	. . . . 5 |- ( ( x e. P. /\ u e. P. ) -> ( x .P. u ) e. P. )
32	31	ad2ant2rl 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( x .P. u ) e. P. )
33		mulclpr 7534	. . . . 5 |- ( ( y e. P. /\ v e. P. ) -> ( y .P. v ) e. P. )
34	33	ad2ant2lr 507	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( y .P. v ) e. P. )
35		addclpr 7499	. . . 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 409	. . 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 304	. 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 1016	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
39		simp2l 1018	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
40		simp3l 1020	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
41		distrprg 7550	. . . . 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 1233	. . . 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 1017	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
44		simp2r 1019	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
45		simp3r 1021	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
46		distrprg 7550	. . . . 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 1233	. . . 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 5871	. . 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 409	. . . 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 409	. . . 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 409	. . . 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 7540	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
53	52	adantl 275	. . . 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 7541	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
55	54	adantl 275	. . . 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 409	. . . 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 7499	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
58	57	adantl 275	. . . 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 6037	. . 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 2203	. 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 7550	. . . . 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 1233	. . . 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 7550	. . . . 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 1233	. . . 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 5871	. . 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 409	. . . 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 409	. . . 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 409	. . . 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 409	. . . 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 6037	. . 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 2203	. 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 6625	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 973   = wceq 1348   e. wcel 2141  (class class class)co 5853   P.cnp 7253   +P. cpp 7255   .P. cmp 7256   ~R cer 7258   R.cnr 7259   +R cplr 7263   .R cmr 7264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431  df-enr 7688  df-nr 7689  df-plr 7690  df-mr 7691

This theorem is referenced by:  pn0sr  7733  axmulass  7835  axdistr  7836