Theorem mulclpr 7486

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 7383	. . . 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 7425	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. ( ~P Q. X. ~P Q. ) )
3		mulclnq 7290	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
4	1, 3	genpml 7431	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A .P. B ) ) )
5	1, 3	genpmu 7432	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. r e. Q. r e. ( 2nd ` ( A .P. B ) ) )
6	2, 4, 5	jca32 308	. 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 7315	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z .Q x ) <Q ( z .Q y ) ) )
8		mulcomnqg 7297	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
9		mulnqprl 7482	. . . . 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 7435	. . . 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 7483	. . . . 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 7436	. . . 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 304	. . 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 7437	. . 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 7485	. . 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 1162	. 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 7390	. 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 414	1 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. P. )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 963   e. wcel 2128   A.wral 2435   E.wrex 2436   ~Pcpw 3543   class class class wbr 3965   X. cxp 4583   ` cfv 5169  (class class class)co 5821   1stc1st 6083   2ndc2nd 6084   Q.cnq 7194   .Q cmq 7197   <Q cltq 7199   P.cnp 7205   .P. cmp 7208

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-1st 6085  df-2nd 6086  df-recs 6249  df-irdg 6314  df-1o 6360  df-2o 6361  df-oadd 6364  df-omul 6365  df-er 6477  df-ec 6479  df-qs 6483  df-ni 7218  df-pli 7219  df-mi 7220  df-lti 7221  df-plpq 7258  df-mpq 7259  df-enq 7261  df-nqqs 7262  df-plqqs 7263  df-mqqs 7264  df-1nqqs 7265  df-rq 7266  df-ltnqqs 7267  df-enq0 7338  df-nq0 7339  df-0nq0 7340  df-plq0 7341  df-mq0 7342  df-inp 7380  df-imp 7383

This theorem is referenced by:  mulnqprlemfl  7489  mulnqprlemfu  7490  mulnqpr  7491  mulassprg  7495  distrlem1prl  7496  distrlem1pru  7497  distrlem4prl  7498  distrlem4pru  7499  distrlem5prl  7500  distrlem5pru  7501  distrprg  7502  1idpr  7506  recexprlemex  7551  ltmprr  7556  mulcmpblnrlemg  7654  mulcmpblnr  7655  mulclsr  7668  mulcomsrg  7671  mulasssrg  7672  distrsrg  7673  m1m1sr  7675  1idsr  7682  00sr  7683  recexgt0sr  7687  mulgt0sr  7692  mulextsr1lem  7694  mulextsr1  7695  recidpirq  7772