Theorem mulcomsrg 8088

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

Assertion

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

Proof of Theorem mulcomsrg

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 8058	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 8077	. 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 8077	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( [ <. z , w >. ] ~R .R [ <. x , y >. ] ~R ) = [ <. ( ( z .P. x ) +P. ( w .P. y ) ) , ( ( z .P. y ) +P. ( w .P. x ) ) >. ] ~R )
4		mulcomprg 7911	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) = ( z .P. x ) )
5	4	ad2ant2r 509	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) = ( z .P. x ) )
6		mulcomprg 7911	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) = ( w .P. y ) )
7	6	ad2ant2l 508	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) = ( w .P. y ) )
8	5, 7	oveq12d 6076	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) = ( ( z .P. x ) +P. ( w .P. y ) ) )
9		mulcomprg 7911	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) = ( w .P. x ) )
10	9	ad2ant2rl 511	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) = ( w .P. x ) )
11		mulcomprg 7911	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) = ( z .P. y ) )
12	11	ad2ant2lr 510	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) = ( z .P. y ) )
13	10, 12	oveq12d 6076	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) = ( ( w .P. x ) +P. ( z .P. y ) ) )
14		mulclpr 7903	. . . . . 6 |- ( ( w e. P. /\ x e. P. ) -> ( w .P. x ) e. P. )
15	14	ancoms 268	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( w .P. x ) e. P. )
16	15	ad2ant2rl 511	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( w .P. x ) e. P. )
17		mulclpr 7903	. . . . . 6 |- ( ( z e. P. /\ y e. P. ) -> ( z .P. y ) e. P. )
18	17	ancoms 268	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( z .P. y ) e. P. )
19	18	ad2ant2lr 510	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( z .P. y ) e. P. )
20		addcomprg 7909	. . . 4 |- ( ( ( w .P. x ) e. P. /\ ( z .P. y ) e. P. ) -> ( ( w .P. x ) +P. ( z .P. y ) ) = ( ( z .P. y ) +P. ( w .P. x ) ) )
21	16, 19, 20	syl2anc 411	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( w .P. x ) +P. ( z .P. y ) ) = ( ( z .P. y ) +P. ( w .P. x ) ) )
22	13, 21	eqtrd 2267	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) = ( ( z .P. y ) +P. ( w .P. x ) ) )
23	1, 2, 3, 8, 22	ecovicom 6890	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205  (class class class)co 6058   P.cnp 7622   +P. cpp 7624   .P. cmp 7625   ~R cer 7627   R.cnr 7628   .R cmr 7633

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-imp 7800  df-enr 8057  df-nr 8058  df-mr 8060

This theorem is referenced by:  mulresr  8169  axmulcom  8202  axmulass  8204  axcnre  8212