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Theorem genpmu 6770
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 6727 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 6730 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 2nd `  A ) )
3 rexex 2411 . . . 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 270 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 2nd `  A
) )
6 prop 6727 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prmu 6730 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 2nd `  B ) )
8 rexex 2411 . . . . 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 473 . . 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 6767 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 2nd `  A
)  /\  g  e.  ( 2nd `  B ) )  ->  ( f G g )  e.  ( 2nd `  ( A F B ) ) ) )
1413imp 122 . . . . 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 6734 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
161, 15sylan 277 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
17 elprnqu 6734 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
186, 17sylan 277 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 331 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 553 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5686 . . . . . . 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 108 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 2nd `  A )  /\  g  e.  ( 2nd `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2148 . . . . . 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 2706 . . . . 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 392 . . 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 1820 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
295, 28exlimddv 1820 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 102    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   E.wrex 2350   {crab 2353   <.cop 3409   ` cfv 4932  (class class class)co 5543    |-> cmpt2 5545   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532   P.cnp 6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-qs 6178  df-ni 6556  df-nqqs 6600  df-inp 6718
This theorem is referenced by:  addclpr  6789  mulclpr  6824
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