Theorem mulcomsrg 7976

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 7946	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 7965	. 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 7965	. 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 7799	. . . 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 7799	. . . 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 6035	. 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 7799	. . . . 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 7799	. . . . 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 6035	. . 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 7791	. . . . . 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 7791	. . . . . 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 7797	. . . 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 2264	. 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 6811	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202  (class class class)co 6017   P.cnp 7510   +P. cpp 7512   .P. cmp 7513   ~R cer 7515   R.cnr 7516   .R cmr 7521

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-iplp 7687  df-imp 7688  df-enr 7945  df-nr 7946  df-mr 7948

This theorem is referenced by:  mulresr  8057  axmulcom  8090  axmulass  8092  axcnre  8100