Theorem mulclsr 8087

Description: Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)

Assertion

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

Proof of Theorem mulclsr

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

Step	Hyp	Ref	Expression
1		df-nr 8060	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		oveq1 6067	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = ( A .R [ <. z , w >. ] ~R ) )
3	2	eleq1d 2303	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) e. ( ( P. X. P. ) /. ~R ) <-> ( A .R [ <. z , w >. ] ~R ) e. ( ( P. X. P. ) /. ~R ) ) )
4		oveq2 6068	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A .R [ <. z , w >. ] ~R ) = ( A .R B ) )
5	4	eleq1d 2303	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( A .R [ <. z , w >. ] ~R ) e. ( ( P. X. P. ) /. ~R ) <-> ( A .R B ) e. ( ( P. X. P. ) /. ~R ) ) )
6		mulsrpr 8079	. . . 4 |- ( ( ( 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 )
7		mulclpr 7905	. . . . . . . 8 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
8		mulclpr 7905	. . . . . . . 8 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
9		addclpr 7870	. . . . . . . 8 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
10	7, 8, 9	syl2an 289	. . . . . . 7 |- ( ( ( x e. P. /\ z e. P. ) /\ ( y e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
11	10	an4s 592	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
12		mulclpr 7905	. . . . . . . 8 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
13		mulclpr 7905	. . . . . . . 8 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
14		addclpr 7870	. . . . . . . 8 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
15	12, 13, 14	syl2an 289	. . . . . . 7 |- ( ( ( x e. P. /\ w e. P. ) /\ ( y e. P. /\ z e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
16	15	an42s 593	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
17	11, 16	jca 306	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) )
18		opelxpi 4788	. . . . 5 |- ( ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) -> <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. e. ( P. X. P. ) )
19		enrex 8070	. . . . . 6 |- ~R e. _V
20	19	ecelqsi 6838	. . . . 5 |- ( <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. e. ( P. X. P. ) -> [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
21	17, 18, 20	3syl 17	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R e. ( ( P. X. P. ) /. ~R ) )
22	6, 21	eqeltrd 2311	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) e. ( ( P. X. P. ) /. ~R ) )
23	1, 3, 5, 22	2ecoptocl 6872	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) e. ( ( P. X. P. ) /. ~R ) )
24	23, 1	eleqtrrdi 2328	1 |- ( ( A e. R. /\ B e. R. ) -> ( A .R B ) e. R. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   <.cop 3698   X. cxp 4754  (class class class)co 6060   [cec 6780   /.cqs 6781   P.cnp 7624   +P. cpp 7626   .P. cmp 7627   ~R cer 7629   R.cnr 7630   .R cmr 7635

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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717

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 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-eprel 4416  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-irdg 6616  df-1o 6662  df-2o 6663  df-oadd 6666  df-omul 6667  df-er 6782  df-ec 6784  df-qs 6788  df-ni 7637  df-pli 7638  df-mi 7639  df-lti 7640  df-plpq 7677  df-mpq 7678  df-enq 7680  df-nqqs 7681  df-plqqs 7682  df-mqqs 7683  df-1nqqs 7684  df-rq 7685  df-ltnqqs 7686  df-enq0 7757  df-nq0 7758  df-0nq0 7759  df-plq0 7760  df-mq0 7761  df-inp 7799  df-iplp 7801  df-imp 7802  df-enr 8059  df-nr 8060  df-mr 8062

This theorem is referenced by:  pn0sr  8104  negexsr  8105  caucvgsrlemoffval  8129  caucvgsrlemofff  8130  map2psrprg  8138  mulcnsr  8168  mulresr  8171  mulcnsrec  8176  axmulcl  8199  axmulrcl  8200  axmulcom  8204  axmulass  8206  axdistr  8207  axrnegex  8212