Theorem mulclpr 7886

Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)

Assertion

Ref	Expression
mulclpr	|- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )

Proof of Theorem mulclpr

Dummy variables q r t u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-imp 7783	. . . 4 |- .P. = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y .Q z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y .Q z ) ) } >. )
2	1	genpelxp 7825	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. ( ~P Q. X. ~P Q. ) )
3		mulclnq 7690	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
4	1, 3	genpml 7831	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A .P. B ) ) )
5	1, 3	genpmu 7832	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. r e. Q. r e. ( 2nd ` ( A .P. B ) ) )
6	2, 4, 5	jca32 310	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( A .P. B ) e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` ( A .P. B ) ) /\ E. r e. Q. r e. ( 2nd ` ( A .P. B ) ) ) ) )
7		ltmnqg 7715	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z .Q x ) <Q ( z .Q y ) ) )
8		mulcomnqg 7697	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
9		mulnqprl 7882	. . . . 5 |- ( ( ( ( A e. P. /\ u e. ( 1st ` A ) ) /\ ( B e. P. /\ t e. ( 1st ` B ) ) ) /\ x e. Q. ) -> ( x <Q ( u .Q t ) -> x e. ( 1st ` ( A .P. B ) ) ) )
10	1, 3, 7, 8, 9	genprndl 7835	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. ( q e. ( 1st ` ( A .P. B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A .P. B ) ) ) ) )
11		mulnqpru 7883	. . . . 5 |- ( ( ( ( A e. P. /\ u e. ( 2nd ` A ) ) /\ ( B e. P. /\ t e. ( 2nd ` B ) ) ) /\ x e. Q. ) -> ( ( u .Q t ) <Q x -> x e. ( 2nd ` ( A .P. B ) ) ) )
12	1, 3, 7, 8, 11	genprndu 7836	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> A. r e. Q. ( r e. ( 2nd ` ( A .P. B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A .P. B ) ) ) ) )
13	10, 12	jca 306	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A. q e. Q. ( q e. ( 1st ` ( A .P. B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A .P. B ) ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` ( A .P. B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A .P. B ) ) ) ) ) )
14	1, 3, 7, 8	genpdisj 7837	. . 3 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. -. ( q e. ( 1st ` ( A .P. B ) ) /\ q e. ( 2nd ` ( A .P. B ) ) ) )
15		mullocpr 7885	. . 3 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) )
16	13, 14, 15	3jca 1204	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( ( A. q e. Q. ( q e. ( 1st ` ( A .P. B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A .P. B ) ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` ( A .P. B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A .P. B ) ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` ( A .P. B ) ) /\ q e. ( 2nd ` ( A .P. B ) ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) ) )
17		elnp1st2nd 7790	. 2 |- ( ( A .P. B ) e. P. <-> ( ( ( A .P. B ) e. ( ~P Q. X. ~P Q. ) /\ ( E. q e. Q. q e. ( 1st ` ( A .P. B ) ) /\ E. r e. Q. r e. ( 2nd ` ( A .P. B ) ) ) ) /\ ( ( A. q e. Q. ( q e. ( 1st ` ( A .P. B ) ) <-> E. r e. Q. ( q <Q r /\ r e. ( 1st ` ( A .P. B ) ) ) ) /\ A. r e. Q. ( r e. ( 2nd ` ( A .P. B ) ) <-> E. q e. Q. ( q <Q r /\ q e. ( 2nd ` ( A .P. B ) ) ) ) ) /\ A. q e. Q. -. ( q e. ( 1st ` ( A .P. B ) ) /\ q e. ( 2nd ` ( A .P. B ) ) ) /\ A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A .P. B ) ) \/ r e. ( 2nd ` ( A .P. B ) ) ) ) ) ) )
18	6, 16, 17	sylanbrc 417	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   e. wcel 2203   A.wral 2520   E.wrex 2521   ~Pcpw 3668   class class class wbr 4108   X. cxp 4746   ` cfv 5351  (class class class)co 6049   1stc1st 6331   2ndc2nd 6332   Q.cnq 7594   .Q cmq 7597   <Q cltq 7599   P.cnp 7605   .P. cmp 7608

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-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7618  df-pli 7619  df-mi 7620  df-lti 7621  df-plpq 7658  df-mpq 7659  df-enq 7661  df-nqqs 7662  df-plqqs 7663  df-mqqs 7664  df-1nqqs 7665  df-rq 7666  df-ltnqqs 7667  df-enq0 7738  df-nq0 7739  df-0nq0 7740  df-plq0 7741  df-mq0 7742  df-inp 7780  df-imp 7783

This theorem is referenced by:  mulnqprlemfl  7889  mulnqprlemfu  7890  mulnqpr  7891  mulassprg  7895  distrlem1prl  7896  distrlem1pru  7897  distrlem4prl  7898  distrlem4pru  7899  distrlem5prl  7900  distrlem5pru  7901  distrprg  7902  1idpr  7906  recexprlemex  7951  ltmprr  7956  mulcmpblnrlemg  8054  mulcmpblnr  8055  mulclsr  8068  mulcomsrg  8071  mulasssrg  8072  distrsrg  8073  m1m1sr  8075  1idsr  8082  00sr  8083  recexgt0sr  8087  mulgt0sr  8092  mulextsr1lem  8094  mulextsr1  8095  recidpirq  8172