Theorem mulcomsrg 7719

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 7689	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 7708	. 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 7708	. 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 7542	. . . 4 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) = ( z .P. x ) )
5	4	ad2ant2r 506	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) = ( z .P. x ) )
6		mulcomprg 7542	. . . 4 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) = ( w .P. y ) )
7	6	ad2ant2l 505	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) = ( w .P. y ) )
8	5, 7	oveq12d 5871	. 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 7542	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) = ( w .P. x ) )
10	9	ad2ant2rl 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) = ( w .P. x ) )
11		mulcomprg 7542	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) = ( z .P. y ) )
12	11	ad2ant2lr 507	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) = ( z .P. y ) )
13	10, 12	oveq12d 5871	. . 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 7534	. . . . . 6 |- ( ( w e. P. /\ x e. P. ) -> ( w .P. x ) e. P. )
15	14	ancoms 266	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( w .P. x ) e. P. )
16	15	ad2ant2rl 508	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( w .P. x ) e. P. )
17		mulclpr 7534	. . . . . 6 |- ( ( z e. P. /\ y e. P. ) -> ( z .P. y ) e. P. )
18	17	ancoms 266	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( z .P. y ) e. P. )
19	18	ad2ant2lr 507	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( z .P. y ) e. P. )
20		addcomprg 7540	. . . 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 409	. . 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 2203	. 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 6621	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) = ( B .R A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141  (class class class)co 5853   P.cnp 7253   +P. cpp 7255   .P. cmp 7256   ~R cer 7258   R.cnr 7259   .R cmr 7264

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-imp 7431  df-enr 7688  df-nr 7689  df-mr 7691

This theorem is referenced by:  mulresr  7800  axmulcom  7833  axmulass  7835  axcnre  7843