Theorem mulclpr 7852

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 7749	. . . 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 7791	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( A .P. B ) e. ( ~P Q. X. ~P Q. ) )
3		mulclnq 7656	. . . 4 |- ( ( y e. Q. /\ z e. Q. ) -> ( y .Q z ) e. Q. )
4	1, 3	genpml 7797	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. q e. Q. q e. ( 1st ` ( A .P. B ) ) )
5	1, 3	genpmu 7798	. . 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 7681	. . . . 5 |- ( ( x e. Q. /\ y e. Q. /\ z e. Q. ) -> ( x <Q y <-> ( z .Q x ) <Q ( z .Q y ) ) )
8		mulcomnqg 7663	. . . . 5 |- ( ( x e. Q. /\ y e. Q. ) -> ( x .Q y ) = ( y .Q x ) )
9		mulnqprl 7848	. . . . 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 7801	. . . 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 7849	. . . . 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 7802	. . . 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 7803	. . 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 7851	. . 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 7756	. 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 2202   A.wral 2511   E.wrex 2512   ~Pcpw 3656   class class class wbr 4093   X. cxp 4729   ` cfv 5333  (class class class)co 6028   1stc1st 6310   2ndc2nd 6311   Q.cnq 7560   .Q cmq 7563   <Q cltq 7565   P.cnp 7571   .P. cmp 7574

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-imp 7749

This theorem is referenced by:  mulnqprlemfl  7855  mulnqprlemfu  7856  mulnqpr  7857  mulassprg  7861  distrlem1prl  7862  distrlem1pru  7863  distrlem4prl  7864  distrlem4pru  7865  distrlem5prl  7866  distrlem5pru  7867  distrprg  7868  1idpr  7872  recexprlemex  7917  ltmprr  7922  mulcmpblnrlemg  8020  mulcmpblnr  8021  mulclsr  8034  mulcomsrg  8037  mulasssrg  8038  distrsrg  8039  m1m1sr  8041  1idsr  8048  00sr  8049  recexgt0sr  8053  mulgt0sr  8058  mulextsr1lem  8060  mulextsr1  8061  recidpirq  8138