Theorem mulcomsrg 8117

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 8087	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 8106	. 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 8106	. 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 7940	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) = ( z .P. x ) )
5	4	ad2ant2r 513	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) = ( z .P. x ) )
6		mulcomprg 7940	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) = ( w .P. y ) )
7	6	ad2ant2l 512	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) = ( w .P. y ) )
8	5, 7	oveq12d 6096	. 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 7940	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) = ( w .P. x ) )
10	9	ad2ant2rl 515	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) = ( w .P. x ) )
11		mulcomprg 7940	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) = ( z .P. y ) )
12	11	ad2ant2lr 514	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) = ( z .P. y ) )
13	10, 12	oveq12d 6096	. . 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 7932	. . . . . 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 515	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( w .P. x ) e. P. )
17		mulclpr 7932	. . . . . 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 514	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( z .P. y ) e. P. )
20		addcomprg 7938	. . . 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 415	. . 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 2271	. 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 6910	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6078   P.cnp 7651   +P. cpp 7653   .P. cmp 7654   ~R cer 7656   R.cnr 7657   .R cmr 7662

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6081  df-oprab 6082  df-mpo 6083  df-1st 6367  df-2nd 6368  df-recs 6569  df-irdg 6634  df-1o 6680  df-2o 6681  df-oadd 6684  df-omul 6685  df-er 6800  df-ec 6802  df-qs 6806  df-ni 7664  df-pli 7665  df-mi 7666  df-lti 7667  df-plpq 7704  df-mpq 7705  df-enq 7707  df-nqqs 7708  df-plqqs 7709  df-mqqs 7710  df-1nqqs 7711  df-rq 7712  df-ltnqqs 7713  df-enq0 7784  df-nq0 7785  df-0nq0 7786  df-plq0 7787  df-mq0 7788  df-inp 7826  df-iplp 7828  df-imp 7829  df-enr 8086  df-nr 8087  df-mr 8089

This theorem is referenced by:  mulresr  8198  axmulcom  8231  axmulass  8233  axcnre  8241