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Theorem mulclpr 7344
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 7241 . . . 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 7283 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  ( ~P Q.  X.  ~P Q. ) )
3 mulclnq 7148 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
41, 3genpml 7289 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A  .P.  B
) ) )
51, 3genpmu 7290 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. r  e.  Q.  r  e.  ( 2nd `  ( A  .P.  B
) ) )
62, 4, 5jca32 306 . 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 7173 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  .Q  x )  <Q  (
z  .Q  y ) ) )
8 mulcomnqg 7155 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  =  ( y  .Q  x ) )
9 mulnqprl 7340 . . . . 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 7293 . . . 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 7341 . . . . 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 7294 . . . 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 302 . . 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 7295 . . 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 7343 . . 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 1144 . 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 7248 . 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 411 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 103    <-> wb 104    \/ wo 680    /\ w3a 945    e. wcel 1463   A.wral 2391   E.wrex 2392   ~Pcpw 3478   class class class wbr 3897    X. cxp 4505   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    .Q cmq 7055    <Q cltq 7057   P.cnp 7063    .P. cmp 7066
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-imp 7241
This theorem is referenced by:  mulnqprlemfl  7347  mulnqprlemfu  7348  mulnqpr  7349  mulassprg  7353  distrlem1prl  7354  distrlem1pru  7355  distrlem4prl  7356  distrlem4pru  7357  distrlem5prl  7358  distrlem5pru  7359  distrprg  7360  1idpr  7364  recexprlemex  7409  ltmprr  7414  mulcmpblnrlemg  7512  mulcmpblnr  7513  mulclsr  7526  mulcomsrg  7529  mulasssrg  7530  distrsrg  7531  m1m1sr  7533  1idsr  7540  00sr  7541  recexgt0sr  7545  mulgt0sr  7550  mulextsr1lem  7552  mulextsr1  7553  recidpirq  7630
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