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Theorem mulclpr 6668
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.
StepHypRef Expression
1 df-imp 6565 . . . 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
) ) } >. )
21genpelxp 6607 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  ( ~P Q.  X.  ~P Q. ) )
3 mulclnq 6472 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
41, 3genpml 6613 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A  .P.  B
) ) )
51, 3genpmu 6614 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. r  e.  Q.  r  e.  ( 2nd `  ( A  .P.  B
) ) )
62, 4, 5jca32 293 . 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 6497 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  .Q  x )  <Q  (
z  .Q  y ) ) )
8 mulcomnqg 6479 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  =  ( y  .Q  x ) )
9 mulnqprl 6664 . . . . 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 ) ) ) )
101, 3, 7, 8, 9genprndl 6617 . . . 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 6665 . . . . 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
) ) ) )
121, 3, 7, 8, 11genprndu 6618 . . . 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 ) ) ) ) )
1310, 12jca 290 . . 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 ) ) ) ) ) )
141, 3, 7, 8genpdisj 6619 . . 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 6667 . . 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 ) ) ) ) )
1613, 14, 153jca 1084 . 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 6572 . 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 ) ) ) ) ) ) )
186, 16, 17sylanbrc 394 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    e. wcel 1393   A.wral 2306   E.wrex 2307   ~Pcpw 3359   class class class wbr 3764    X. cxp 4343   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6376    .Q cmq 6379    <Q cltq 6381   P.cnp 6387    .P. cmp 6390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-imp 6565
This theorem is referenced by:  mulnqprlemfl  6671  mulnqprlemfu  6672  mulnqpr  6673  mulassprg  6677  distrlem1prl  6678  distrlem1pru  6679  distrlem4prl  6680  distrlem4pru  6681  distrlem5prl  6682  distrlem5pru  6683  distrprg  6684  1idpr  6688  recexprlemex  6733  ltmprr  6738  mulcmpblnrlemg  6823  mulcmpblnr  6824  mulclsr  6837  mulcomsrg  6840  mulasssrg  6841  distrsrg  6842  m1m1sr  6844  1idsr  6851  00sr  6852  recexgt0sr  6856  mulgt0sr  6860  mulextsr1lem  6862  mulextsr1  6863  recidpirq  6932
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