Theorem mulasssrg 8021

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 7990	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 8009	. 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 8009	. 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 8009	. 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 8009	. 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 7835	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
7	6	ad2ant2r 509	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
8		mulclpr 7835	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
9	8	ad2ant2l 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
10		addclpr 7800	. . . 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 411	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
12		mulclpr 7835	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
13	12	ad2ant2rl 511	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
14		mulclpr 7835	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
15	14	ad2ant2lr 510	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
16		addclpr 7800	. . . 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 411	. . 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 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. ) )
19		mulclpr 7835	. . . . 5 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
20	19	ad2ant2r 509	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. v ) e. P. )
21		mulclpr 7835	. . . . 5 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
22	21	ad2ant2l 508	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. u ) e. P. )
23		addclpr 7800	. . . 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 411	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
25		mulclpr 7835	. . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
26	25	ad2ant2rl 511	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. u ) e. P. )
27		mulclpr 7835	. . . . 5 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
28	27	ad2ant2lr 510	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. v ) e. P. )
29		addclpr 7800	. . . 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 411	. . 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 306	. 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 7843	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
33	32	adantl 277	. . 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 7851	. . . . . 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 277	. . . . 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 1024	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> f e. P. )
37		simp2 1025	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> g e. P. )
38		simp3 1026	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> h e. P. )
39		addclpr 7800	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) e. P. )
40	39	adantl 277	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) e. P. )
41		mulcomprg 7843	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r .P. s ) = ( s .P. r ) )
42	41	adantl 277	. . . . 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 6209	. . . 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 277	. . 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 7844	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f .P. g ) .P. h ) = ( f .P. ( g .P. h ) ) )
46	45	adantl 277	. . 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 7835	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) e. P. )
48	47	adantl 277	. . 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 1048	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
50		simp1r 1049	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
51		simp2l 1050	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
52		simp2r 1051	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
53		simp3l 1052	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
54		simp3r 1053	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
55		addcomprg 7841	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
56	55	adantl 277	. . 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 7842	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
58	57	adantl 277	. . 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 7800	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
60	59	adantl 277	. . 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 6225	. 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 6225	. 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 6857	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 104   /\ w3a 1005   = wceq 1398   e. wcel 2202  (class class class)co 6028   P.cnp 7554   +P. cpp 7556   .P. cmp 7557   ~R cer 7559   R.cnr 7560   .R cmr 7565

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-iplp 7731  df-imp 7732  df-enr 7989  df-nr 7990  df-mr 7992

This theorem is referenced by:  axmulass  8136