Theorem mulasssrg 6935

Description: Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Assertion

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

Proof of Theorem mulasssrg

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

Step	Hyp	Ref	Expression
1		df-nr 6904	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 6923	. 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 )
3		mulsrpr 6923	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R .R [ <. v , u >. ] ~R ) = [ <. ( ( z .P. v ) +P. ( w .P. u ) ) , ( ( z .P. u ) +P. ( w .P. v ) ) >. ] ~R )
4		mulsrpr 6923	. 2 |- ( ( ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R .R [ <. v , u >. ] ~R ) = [ <. ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. v ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. u ) ) , ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. u ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. v ) ) >. ] ~R )
5		mulsrpr 6923	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( ( ( z .P. v ) +P. ( w .P. u ) ) e. P. /\ ( ( z .P. u ) +P. ( w .P. v ) ) e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. ( ( z .P. v ) +P. ( w .P. u ) ) , ( ( z .P. u ) +P. ( w .P. v ) ) >. ] ~R ) = [ <. ( ( x .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) +P. ( y .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) , ( ( x .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) +P. ( y .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) ) >. ] ~R )
6		mulclpr 6762	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
7	6	ad2ant2r 492	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
8		mulclpr 6762	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
9	8	ad2ant2l 491	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
10		addclpr 6727	. . . 4 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
11	7, 9, 10	syl2anc 403	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
12		mulclpr 6762	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
13	12	ad2ant2rl 494	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
14		mulclpr 6762	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
15	14	ad2ant2lr 493	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
16		addclpr 6727	. . . 4 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
17	13, 15, 16	syl2anc 403	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
18	11, 17	jca 300	. 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. ) )
19		mulclpr 6762	. . . . 5 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
20	19	ad2ant2r 492	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. v ) e. P. )
21		mulclpr 6762	. . . . 5 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
22	21	ad2ant2l 491	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. u ) e. P. )
23		addclpr 6727	. . . 4 |- ( ( ( z .P. v ) e. P. /\ ( w .P. u ) e. P. ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
24	20, 22, 23	syl2anc 403	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
25		mulclpr 6762	. . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
26	25	ad2ant2rl 494	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. u ) e. P. )
27		mulclpr 6762	. . . . 5 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
28	27	ad2ant2lr 493	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. v ) e. P. )
29		addclpr 6727	. . . 4 |- ( ( ( z .P. u ) e. P. /\ ( w .P. v ) e. P. ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
30	26, 28, 29	syl2anc 403	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
31	24, 30	jca 300	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( z .P. v ) +P. ( w .P. u ) ) e. P. /\ ( ( z .P. u ) +P. ( w .P. v ) ) e. P. ) )
32		mulcomprg 6770	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
33	32	adantl 271	. . 3 |- ( ( ( ( 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 ) )
34		distrprg 6778	. . . . . 6 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( r .P. ( s +P. t ) ) = ( ( r .P. s ) +P. ( r .P. t ) ) )
35	34	adantl 271	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( r .P. ( s +P. t ) ) = ( ( r .P. s ) +P. ( r .P. t ) ) )
36		simp1 938	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> f e. P. )
37		simp2 939	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> g e. P. )
38		simp3 940	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> h e. P. )
39		addclpr 6727	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) e. P. )
40	39	adantl 271	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) e. P. )
41		mulcomprg 6770	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r .P. s ) = ( s .P. r ) )
42	41	adantl 271	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r .P. s ) = ( s .P. r ) )
43	35, 36, 37, 38, 40, 42	caovdir2d 5697	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) .P. h ) = ( ( f .P. h ) +P. ( g .P. h ) ) )
44	43	adantl 271	. . 3 |- ( ( ( ( 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. h ) +P. ( g .P. h ) ) )
45		mulassprg 6771	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f .P. g ) .P. h ) = ( f .P. ( g .P. h ) ) )
46	45	adantl 271	. . 3 |- ( ( ( ( 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 ) ) )
47		mulclpr 6762	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) e. P. )
48	47	adantl 271	. . 3 |- ( ( ( ( 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. )
49		simp1l 962	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
50		simp1r 963	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
51		simp2l 964	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
52		simp2r 965	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
53		simp3l 966	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
54		simp3r 967	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
55		addcomprg 6768	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
56	55	adantl 271	. . 3 |- ( ( ( ( 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 ) )
57		addassprg 6769	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
58	57	adantl 271	. . 3 |- ( ( ( ( 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 ) ) )
59		addclpr 6727	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
60	59	adantl 271	. . 3 |- ( ( ( ( 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. )
61	33, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 58, 60	caovlem2d 5713	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. v ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. u ) ) = ( ( x .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) +P. ( y .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) )
62	33, 44, 46, 48, 49, 50, 51, 52, 54, 53, 56, 58, 60	caovlem2d 5713	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. u ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. v ) ) = ( ( x .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) +P. ( y .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) ) )
63	1, 2, 3, 4, 5, 18, 31, 61, 62	ecoviass 6239	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 919   = wceq 1284   e. wcel 1433  (class class class)co 5532   P.cnp 6481   +P. cpp 6483   .P. cmp 6484   ~R cer 6486   R.cnr 6487   .R cmr 6492

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-iplp 6658  df-imp 6659  df-enr 6903  df-nr 6904  df-mr 6906

This theorem is referenced by:  axmulass  7039