Theorem mulcomsrg 7940

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 7910	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 7929	. 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 7929	. 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 7763	. . . 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 7763	. . . 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 6018	. 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 7763	. . . . 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 7763	. . . . 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 6018	. . 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 7755	. . . . . 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 7755	. . . . . 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 7761	. . . 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 2262	. 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 6788	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200  (class class class)co 6000   P.cnp 7474   +P. cpp 7476   .P. cmp 7477   ~R cer 7479   R.cnr 7480   .R cmr 7485

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-2o 6561  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-rq 7535  df-ltnqqs 7536  df-enq0 7607  df-nq0 7608  df-0nq0 7609  df-plq0 7610  df-mq0 7611  df-inp 7649  df-iplp 7651  df-imp 7652  df-enr 7909  df-nr 7910  df-mr 7912

This theorem is referenced by:  mulresr  8021  axmulcom  8054  axmulass  8056  axcnre  8064