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Theorem genpmu 7138
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 7095 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 7098 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 2nd `  A ) )
3 rexex 2423 . . . 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 271 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 2nd `  A
) )
6 prop 7095 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prmu 7098 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 2nd `  B ) )
8 rexex 2423 . . . . 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 474 . . 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 7135 . . . . . 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 7102 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
161, 15sylan 278 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
17 elprnqu 7102 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
186, 17sylan 278 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 332 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 556 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5813 . . . . . . 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 2157 . . . . . 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 2727 . . . . 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 393 . . 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 1827 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
295, 28exlimddv 1827 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 925    = wceq 1290   E.wex 1427    e. wcel 1439   E.wrex 2361   {crab 2364   <.cop 3453   ` cfv 5028  (class class class)co 5666    |-> cmpt2 5668   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   P.cnp 6911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6924  df-nqqs 6968  df-inp 7086
This theorem is referenced by:  addclpr  7157  mulclpr  7192
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