MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mulclpr Unicode version

Theorem mulclpr 8886
Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
mulclpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )

Proof of Theorem mulclpr
Dummy variables  x  y  z  w  v 
f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mp 8850 . 2  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  .Q  z ) } )
2 mulclnq 8813 . 2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3 ltmnq 8838 . 2  |-  ( h  e.  Q.  ->  (
f  <Q  g  <->  ( h  .Q  f )  <Q  (
h  .Q  g ) ) )
4 mulcomnq 8819 . 2  |-  ( x  .Q  y )  =  ( y  .Q  x
)
5 mulclprlem 8885 . 2  |-  ( ( ( ( A  e. 
P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  (
g  .Q  h )  ->  x  e.  ( A  .P.  B ) ) )
61, 2, 3, 4, 5genpcl 8874 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  .P.  B
)  e.  P. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725  (class class class)co 6072    .Q cmq 8720   P.cnp 8723    .P. cmp 8726
This theorem is referenced by:  mulasspr  8890  distrlem1pr  8891  distrlem4pr  8892  distrlem5pr  8893  mulcmpblnr  8938  mulclsr  8948  mulasssr  8954  distrsr  8955  m1m1sr  8957  1idsr  8962  00sr  8963  recexsrlem  8967  mulgt0sr  8969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720  df-er 6896  df-ni 8738  df-mi 8740  df-lti 8741  df-mpq 8775  df-ltpq 8776  df-enq 8777  df-nq 8778  df-erq 8779  df-mq 8781  df-1nq 8782  df-rq 8783  df-ltnq 8784  df-np 8847  df-mp 8850
  Copyright terms: Public domain W3C validator