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Theorem genpml 6673
Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-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
genpml  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( 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 genpml
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6631 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prml 6633 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 1st `  A ) )
3 rexex 2385 . . . 4  |-  ( E. f  e.  Q.  f  e.  ( 1st `  A
)  ->  E. f 
f  e.  ( 1st `  A ) )
41, 2, 33syl 17 . . 3  |-  ( A  e.  P.  ->  E. f 
f  e.  ( 1st `  A ) )
54adantr 265 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 1st `  A
) )
6 prop 6631 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prml 6633 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 1st `  B ) )
8 rexex 2385 . . . . 5  |-  ( E. g  e.  Q.  g  e.  ( 1st `  B
)  ->  E. g 
g  e.  ( 1st `  B ) )
96, 7, 83syl 17 . . . 4  |-  ( B  e.  P.  ->  E. g 
g  e.  ( 1st `  B ) )
109ad2antlr 466 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 1st `  A ) )  ->  E. g  g  e.  ( 1st `  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, 12genpprecll 6670 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 1st `  A
)  /\  g  e.  ( 1st `  B ) )  ->  ( f G g )  e.  ( 1st `  ( A F B ) ) ) )
1413imp 119 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  (
f G g )  e.  ( 1st `  ( A F B ) ) )
15 elprnql 6637 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
161, 15sylan 271 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
17 elprnql 6637 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
186, 17sylan 271 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 325 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 1st `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 530 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 5683 . . . . . . 7  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f G g )  e.  Q. )
2220, 21syl 14 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  (
f G g )  e.  Q. )
23 simpr 107 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2122 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B
) ) )  /\  q  =  ( f G g ) )  ->  ( q  e.  ( 1st `  ( A F B ) )  <-> 
( f G g )  e.  ( 1st `  ( A F B ) ) ) )
2522, 24rspcedv 2677 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  (
( f G g )  e.  ( 1st `  ( A F B ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) ) )
2614, 25mpd 13 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
2726anassrs 386 . . 3  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  f  e.  ( 1st `  A
) )  /\  g  e.  ( 1st `  B
) )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
2810, 27exlimddv 1794 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 1st `  A ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
295, 28exlimddv 1794 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {crab 2327   <.cop 3406   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   P.cnp 6447
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 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-qs 6143  df-ni 6460  df-nqqs 6504  df-inp 6622
This theorem is referenced by:  addclpr  6693  mulclpr  6728
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