ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpmu Unicode version

Theorem genpmu 7338
Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1  |-  F  =  ( 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 G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
Assertion
Ref Expression
genpmu  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
Distinct variable groups:    x, y, z, w, v, q, A   
x, B, y, z, w, v, q    x, G, y, z, w, v, q    F, q
Allowed substitution hints:    F( x, y, z, w, v)

Proof of Theorem genpmu
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7295 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 7298 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 2nd `  A ) )
3 rexex 2479 . . . 4  |-  ( E. f  e.  Q.  f  e.  ( 2nd `  A
)  ->  E. f 
f  e.  ( 2nd `  A ) )
41, 2, 33syl 17 . . 3  |-  ( A  e.  P.  ->  E. f 
f  e.  ( 2nd `  A ) )
54adantr 274 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 2nd `  A
) )
6 prop 7295 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prmu 7298 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 2nd `  B ) )
8 rexex 2479 . . . . 5  |-  ( E. g  e.  Q.  g  e.  ( 2nd `  B
)  ->  E. g 
g  e.  ( 2nd `  B ) )
96, 7, 83syl 17 . . . 4  |-  ( B  e.  P.  ->  E. g 
g  e.  ( 2nd `  B ) )
109ad2antlr 480 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. g  g  e.  ( 2nd `  B
) )
11 genpelvl.1 . . . . . . 7  |-  F  =  ( 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 G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
12 genpelvl.2 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
1311, 12genppreclu 7335 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 2nd `  A
)  /\  g  e.  ( 2nd `  B ) )  ->  ( f G g )  e.  ( 2nd `  ( A F B ) ) ) )
1413imp 123 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f G g )  e.  ( 2nd `  ( A F B ) ) )
15 elprnqu 7302 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
161, 15sylan 281 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
17 elprnqu 7302 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
186, 17sylan 281 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 336 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 577 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5925 . . . . . . 7  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f G g )  e.  Q. )
2220, 21syl 14 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f G g )  e.  Q. )
23 simpr 109 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2208 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B
) ) )  /\  q  =  ( f G g ) )  ->  ( q  e.  ( 2nd `  ( A F B ) )  <-> 
( f G g )  e.  ( 2nd `  ( A F B ) ) ) )
2522, 24rspcedv 2793 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
( f G g )  e.  ( 2nd `  ( A F B ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) ) )
2614, 25mpd 13 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
2726anassrs 397 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A
) )  /\  g  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
2810, 27exlimddv 1870 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
295, 28exlimddv 1870 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   {crab 2420   <.cop 3530   ` cfv 5123  (class class class)co 5774    e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   Q.cnq 7100   P.cnp 7111
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7124  df-nqqs 7168  df-inp 7286
This theorem is referenced by:  addclpr  7357  mulclpr  7392
  Copyright terms: Public domain W3C validator