Theorem mulasssrg 7699

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 7668	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 7687	. 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 7687	. 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 7687	. 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 7687	. 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 7513	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
7	6	ad2ant2r 501	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
8		mulclpr 7513	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
9	8	ad2ant2l 500	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
10		addclpr 7478	. . . 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 409	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
12		mulclpr 7513	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
13	12	ad2ant2rl 503	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
14		mulclpr 7513	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
15	14	ad2ant2lr 502	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
16		addclpr 7478	. . . 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 409	. . 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 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. ) )
19		mulclpr 7513	. . . . 5 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
20	19	ad2ant2r 501	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. v ) e. P. )
21		mulclpr 7513	. . . . 5 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
22	21	ad2ant2l 500	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. u ) e. P. )
23		addclpr 7478	. . . 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 409	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
25		mulclpr 7513	. . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
26	25	ad2ant2rl 503	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. u ) e. P. )
27		mulclpr 7513	. . . . 5 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
28	27	ad2ant2lr 502	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. v ) e. P. )
29		addclpr 7478	. . . 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 409	. . 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 304	. 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 7521	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
33	32	adantl 275	. . 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 7529	. . . . . 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 275	. . . . 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 987	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> f e. P. )
37		simp2 988	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> g e. P. )
38		simp3 989	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> h e. P. )
39		addclpr 7478	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) e. P. )
40	39	adantl 275	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) e. P. )
41		mulcomprg 7521	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r .P. s ) = ( s .P. r ) )
42	41	adantl 275	. . . . 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 6018	. . . 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 275	. . 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 7522	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f .P. g ) .P. h ) = ( f .P. ( g .P. h ) ) )
46	45	adantl 275	. . 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 7513	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) e. P. )
48	47	adantl 275	. . 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 1011	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
50		simp1r 1012	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
51		simp2l 1013	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
52		simp2r 1014	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
53		simp3l 1015	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
54		simp3r 1016	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
55		addcomprg 7519	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
56	55	adantl 275	. . 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 7520	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
58	57	adantl 275	. . 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 7478	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
60	59	adantl 275	. . 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 6034	. 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 6034	. 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 6611	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 103   /\ w3a 968   = wceq 1343   e. wcel 2136  (class class class)co 5842   P.cnp 7232   +P. cpp 7234   .P. cmp 7235   ~R cer 7237   R.cnr 7238   .R cmr 7243

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-imp 7410  df-enr 7667  df-nr 7668  df-mr 7670

This theorem is referenced by:  axmulass  7814