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Theorem genpmu 6644
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 6601 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 6604 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 2nd `  A ) )
3 rexex 2383 . . . 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 265 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 2nd `  A
) )
6 prop 6601 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prmu 6604 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 2nd `  B ) )
8 rexex 2383 . . . . 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 466 . . 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 6641 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 2nd `  A
)  /\  g  e.  ( 2nd `  B ) )  ->  ( f G g )  e.  ( 2nd `  ( A F B ) ) ) )
1413imp 119 . . . . 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 6608 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
161, 15sylan 271 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
17 elprnqu 6608 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
186, 17sylan 271 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 325 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 530 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5680 . . . . . . 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 107 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2120 . . . . . 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 2675 . . . . 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 386 . . 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 1792 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
295, 28exlimddv 1792 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 101    /\ w3a 894    = wceq 1257   E.wex 1395    e. wcel 1407   E.wrex 2322   {crab 2325   <.cop 3403   ` cfv 4927  (class class class)co 5537    |-> cmpt2 5539   1stc1st 5790   2ndc2nd 5791   Q.cnq 6406   P.cnp 6417
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-qs 6140  df-ni 6430  df-nqqs 6474  df-inp 6592
This theorem is referenced by:  addclpr  6663  mulclpr  6698
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