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Theorem genpml 7472
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 7430 . . . 4  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prml 7432 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. f  e.  Q.  f  e.  ( 1st `  A ) )
3 rexex 2516 . . . 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 274 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. f  f  e.  ( 1st `  A
) )
6 prop 7430 . . . . 5  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
7 prml 7432 . . . . 5  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. g  e.  Q.  g  e.  ( 1st `  B ) )
8 rexex 2516 . . . . 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 486 . . 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 7469 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( f  e.  ( 1st `  A
)  /\  g  e.  ( 1st `  B ) )  ->  ( f G g )  e.  ( 1st `  ( A F B ) ) ) )
1413imp 123 . . . . 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 7436 . . . . . . . . . 10  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
161, 15sylan 281 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
17 elprnql 7436 . . . . . . . . . 10  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
186, 17sylan 281 . . . . . . . . 9  |-  ( ( B  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
1916, 18anim12i 336 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  g  e.  ( 1st `  B ) ) )  ->  ( f  e. 
Q.  /\  g  e.  Q. ) )
2019an4s 583 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B ) ) )  ->  (
f  e.  Q.  /\  g  e.  Q. )
)
2112caovcl 6005 . . . . . . 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 109 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
f  e.  ( 1st `  A )  /\  g  e.  ( 1st `  B
) ) )  /\  q  =  ( f G g ) )  ->  q  =  ( f G g ) )
2423eleq1d 2239 . . . . . 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 2838 . . . . 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 398 . . 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 1891 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  f  e.  ( 1st `  A ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
295, 28exlimddv 1891 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 103    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3584   ` cfv 5196  (class class class)co 5851    e. cmpo 5853   1stc1st 6115   2ndc2nd 6116   Q.cnq 7235   P.cnp 7246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-qs 6517  df-ni 7259  df-nqqs 7303  df-inp 7421
This theorem is referenced by:  addclpr  7492  mulclpr  7527
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