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Theorem mulclpr 6728
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 6625 . . . 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 6667 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  ( ~P Q.  X.  ~P Q. ) )
3 mulclnq 6532 . . . 4  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
41, 3genpml 6673 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A  .P.  B
) ) )
51, 3genpmu 6674 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. r  e.  Q.  r  e.  ( 2nd `  ( A  .P.  B
) ) )
62, 4, 5jca32 297 . 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 6557 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  .Q  x )  <Q  (
z  .Q  y ) ) )
8 mulcomnqg 6539 . . . . 5  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  =  ( y  .Q  x ) )
9 mulnqprl 6724 . . . . 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 6677 . . . 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 6725 . . . . 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 6678 . . . 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 294 . . 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 6679 . . 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 6727 . . 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 1095 . 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 6632 . 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 402 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   A.wral 2323   E.wrex 2324   ~Pcpw 3387   class class class wbr 3792    X. cxp 4371   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    .Q cmq 6439    <Q cltq 6441   P.cnp 6447    .P. cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-imp 6625
This theorem is referenced by:  mulnqprlemfl  6731  mulnqprlemfu  6732  mulnqpr  6733  mulassprg  6737  distrlem1prl  6738  distrlem1pru  6739  distrlem4prl  6740  distrlem4pru  6741  distrlem5prl  6742  distrlem5pru  6743  distrprg  6744  1idpr  6748  recexprlemex  6793  ltmprr  6798  mulcmpblnrlemg  6883  mulcmpblnr  6884  mulclsr  6897  mulcomsrg  6900  mulasssrg  6901  distrsrg  6902  m1m1sr  6904  1idsr  6911  00sr  6912  recexgt0sr  6916  mulgt0sr  6920  mulextsr1lem  6922  mulextsr1  6923  recidpirq  6992
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