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Theorem genpn0 8627
Description: The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
genp.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
Assertion
Ref Expression
genpn0  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, w, v, G, y, z
Allowed substitution hints:    A( w, v)    B( w, v)    F( x, y, z, w, v)

Proof of Theorem genpn0
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prn0 8613 . . . 4  |-  ( A  e.  P.  ->  A  =/=  (/) )
2 n0 3464 . . . 4  |-  ( A  =/=  (/)  <->  E. f  f  e.  A )
31, 2sylib 188 . . 3  |-  ( A  e.  P.  ->  E. f 
f  e.  A )
4 prn0 8613 . . . 4  |-  ( B  e.  P.  ->  B  =/=  (/) )
5 n0 3464 . . . 4  |-  ( B  =/=  (/)  <->  E. g  g  e.  B )
64, 5sylib 188 . . 3  |-  ( B  e.  P.  ->  E. g 
g  e.  B )
73, 6anim12i 549 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. f  f  e.  A  /\  E. g  g  e.  B
) )
8 genp.1 . . . . . . . . 9  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )
9 genp.2 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
108, 9genpprecl 8625 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  A  /\  g  e.  B )  ->  (
f G g )  e.  ( A F B ) ) )
11 ne0i 3461 . . . . . . . . 9  |-  ( ( f G g )  e.  ( A F B )  ->  ( A F B )  =/=  (/) )
12 0pss 3492 . . . . . . . . 9  |-  ( (/)  C.  ( A F B )  <->  ( A F B )  =/=  (/) )
1311, 12sylibr 203 . . . . . . . 8  |-  ( ( f G g )  e.  ( A F B )  ->  (/)  C.  ( A F B ) )
1410, 13syl6 29 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  A  /\  g  e.  B )  ->  (/)  C.  ( A F B ) ) )
1514exp3acom23 1362 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( g  e.  B  ->  ( f  e.  A  -> 
(/)  C.  ( A F B ) ) ) )
1615exlimdv 1664 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. g  g  e.  B  ->  (
f  e.  A  ->  (/)  C.  ( A F B ) ) ) )
1716com23 72 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( f  e.  A  ->  ( E. g  g  e.  B  ->  (/)  C.  ( A F B ) ) ) )
1817exlimdv 1664 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. f  f  e.  A  ->  ( E. g  g  e.  B  ->  (/)  C.  ( A F B ) ) ) )
1918imp3a 420 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( E. f 
f  e.  A  /\  E. g  g  e.  B
)  ->  (/)  C.  ( A F B ) ) )
207, 19mpd 14 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
(/)  C.  ( A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544    C. wpss 3153   (/)c0 3455  (class class class)co 5858    e. cmpt2 5860   Q.cnq 8474   P.cnp 8481
This theorem is referenced by:  genpcl  8632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ni 8496  df-nq 8536  df-np 8605
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