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Theorem genpmu 6616
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 6573 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 6576 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 2nd `  A ) )
3 rexex 2368 . . . 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 261 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 2nd `  A
) )
6 prop 6573 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prmu 6576 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 2nd `  B ) )
8 rexex 2368 . . . . 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 458 . . 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 6613 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 2nd `  A
)  /\  g  e.  ( 2nd `  B ) )  ->  ( f G g )  e.  ( 2nd `  ( A F B ) ) ) )
1413imp 115 . . . . 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 6580 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
161, 15sylan 267 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
17 elprnqu 6580 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
186, 17sylan 267 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 321 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 522 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5655 . . . . . . 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 103 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2106 . . . . . 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 2660 . . . . 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 380 . . 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 1778 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
295, 28exlimddv 1778 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 97    /\ w3a 885    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   {crab 2310   <.cop 3378   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   P.cnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6402  df-nqqs 6446  df-inp 6564
This theorem is referenced by:  addclpr  6635  mulclpr  6670
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