Theorem mulclpr 7600

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 7497	. . . 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 7539	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. ( ~P Q. X. ~P Q. ) )
3		mulclnq 7404	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
4	1, 3	genpml 7545	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A .P. B ) ) )
5	1, 3	genpmu 7546	. . 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 7429	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z .Q x ) <Q ( z .Q y ) ) )
8		mulcomnqg 7411	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
9		mulnqprl 7596	. . . . 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 7549	. . . 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 7597	. . . . 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 7550	. . . 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 7551	. . 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 7599	. . 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 1179	. 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 7504	. 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 709   /\ w3a 980   e. wcel 2160   A.wral 2468   E.wrex 2469   ~Pcpw 3590   class class class wbr 4018   X. cxp 4642   ` cfv 5235  (class class class)co 5895   1stc1st 6162   2ndc2nd 6163   Q.cnq 7308   .Q cmq 7311   <Q cltq 7313   P.cnp 7319   .P. cmp 7322

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-imp 7497

This theorem is referenced by:  mulnqprlemfl  7603  mulnqprlemfu  7604  mulnqpr  7605  mulassprg  7609  distrlem1prl  7610  distrlem1pru  7611  distrlem4prl  7612  distrlem4pru  7613  distrlem5prl  7614  distrlem5pru  7615  distrprg  7616  1idpr  7620  recexprlemex  7665  ltmprr  7670  mulcmpblnrlemg  7768  mulcmpblnr  7769  mulclsr  7782  mulcomsrg  7785  mulasssrg  7786  distrsrg  7787  m1m1sr  7789  1idsr  7796  00sr  7797  recexgt0sr  7801  mulgt0sr  7806  mulextsr1lem  7808  mulextsr1  7809  recidpirq  7886