Theorem mulclpr 7513

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 7410	. . . 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 7452	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. ( ~P Q. X. ~P Q. ) )
3		mulclnq 7317	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
4	1, 3	genpml 7458	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A .P. B ) ) )
5	1, 3	genpmu 7459	. . 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 7342	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z .Q x ) <Q ( z .Q y ) ) )
8		mulcomnqg 7324	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
9		mulnqprl 7509	. . . . 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 7462	. . . 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 7510	. . . . 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 7463	. . . 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 7464	. . 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 7512	. . 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 1167	. 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 7417	. 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 968   e. wcel 2136   A.wral 2444   E.wrex 2445   ~Pcpw 3559   class class class wbr 3982   X. cxp 4602   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   .Q cmq 7224   <Q cltq 7226   P.cnp 7232   .P. cmp 7235

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-imp 7410

This theorem is referenced by:  mulnqprlemfl  7516  mulnqprlemfu  7517  mulnqpr  7518  mulassprg  7522  distrlem1prl  7523  distrlem1pru  7524  distrlem4prl  7525  distrlem4pru  7526  distrlem5prl  7527  distrlem5pru  7528  distrprg  7529  1idpr  7533  recexprlemex  7578  ltmprr  7583  mulcmpblnrlemg  7681  mulcmpblnr  7682  mulclsr  7695  mulcomsrg  7698  mulasssrg  7699  distrsrg  7700  m1m1sr  7702  1idsr  7709  00sr  7710  recexgt0sr  7714  mulgt0sr  7719  mulextsr1lem  7721  mulextsr1  7722  recidpirq  7799