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Theorem genpdisj 7082
Description: The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-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. )
genpdisj.ord  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
genpdisj.com  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
Assertion
Ref Expression
genpdisj  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  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 genpdisj
Dummy variables  a  b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genpelvl.1 . . . . . . . . 9  |-  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 ) ) } >. )
2 genpelvl.2 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
31, 2genpelvl 7071 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 1st `  ( A F B ) )  <->  E. a  e.  ( 1st `  A ) E. b  e.  ( 1st `  B ) q  =  ( a G b ) ) )
4 r2ex 2398 . . . . . . . 8  |-  ( E. a  e.  ( 1st `  A ) E. b  e.  ( 1st `  B
) q  =  ( a G b )  <->  E. a E. b ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) )
53, 4syl6bb 194 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 1st `  ( A F B ) )  <->  E. a E. b ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) ) ) )
61, 2genpelvu 7072 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  <->  E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B ) q  =  ( c G d ) ) )
7 r2ex 2398 . . . . . . . 8  |-  ( E. c  e.  ( 2nd `  A ) E. d  e.  ( 2nd `  B
) q  =  ( c G d )  <->  E. c E. d ( ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )
86, 7syl6bb 194 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( q  e.  ( 2nd `  ( A F B ) )  <->  E. c E. d ( ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )
95, 8anbi12d 457 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) )  <->  ( E. a E. b ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  E. c E. d ( ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B
) )  /\  q  =  ( c G d ) ) ) ) )
10 ee4anv 1857 . . . . . 6  |-  ( E. a E. b E. c E. d ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )  <->  ( E. a E. b ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B
) )  /\  q  =  ( a G b ) )  /\  E. c E. d ( ( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )
119, 10syl6bbr 196 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) )  <->  E. a E. b E. c E. d ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) ) )
1211biimpa 290 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  ->  E. a E. b E. c E. d ( ( ( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B
) )  /\  q  =  ( a G b ) )  /\  ( ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )
13 an4 553 . . . . . . . . . . . . 13  |-  ( ( ( a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A ) )  /\  ( b  e.  ( 1st `  B
)  /\  d  e.  ( 2nd `  B ) ) )  <->  ( (
a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B
) )  /\  (
c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B
) ) ) )
14 prop 7034 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
15 prltlu 7046 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A
) )  ->  a  <Q  c )
16153expib 1146 . . . . . . . . . . . . . . . 16  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  ( ( a  e.  ( 1st `  A
)  /\  c  e.  ( 2nd `  A ) )  ->  a  <Q  c ) )
1714, 16syl 14 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  (
( a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A ) )  ->  a  <Q  c ) )
18 prop 7034 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 prltlu 7046 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B )  /\  d  e.  ( 2nd `  B
) )  ->  b  <Q  d )
20193expib 1146 . . . . . . . . . . . . . . . 16  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  ( ( b  e.  ( 1st `  B
)  /\  d  e.  ( 2nd `  B ) )  ->  b  <Q  d ) )
2118, 20syl 14 . . . . . . . . . . . . . . 15  |-  ( B  e.  P.  ->  (
( b  e.  ( 1st `  B )  /\  d  e.  ( 2nd `  B ) )  ->  b  <Q  d ) )
2217, 21im2anan9 565 . . . . . . . . . . . . . 14  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( a  e.  ( 1st `  A
)  /\  c  e.  ( 2nd `  A ) )  /\  ( b  e.  ( 1st `  B
)  /\  d  e.  ( 2nd `  B ) ) )  ->  (
a  <Q  c  /\  b  <Q  d ) ) )
23 genpdisj.ord . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z G x )  <Q 
( z G y ) ) )
24 genpdisj.com . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )
2523, 24genplt2i 7069 . . . . . . . . . . . . . 14  |-  ( ( a  <Q  c  /\  b  <Q  d )  -> 
( a G b )  <Q  ( c G d ) )
2622, 25syl6 33 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( a  e.  ( 1st `  A
)  /\  c  e.  ( 2nd `  A ) )  /\  ( b  e.  ( 1st `  B
)  /\  d  e.  ( 2nd `  B ) ) )  ->  (
a G b ) 
<Q  ( c G d ) ) )
2713, 26syl5bir 151 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) ) )  ->  (
a G b ) 
<Q  ( c G d ) ) )
2827imp 122 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) ) ) )  -> 
( a G b )  <Q  ( c G d ) )
2928adantlr 461 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( a  e.  ( 1st `  A )  /\  b  e.  ( 1st `  B ) )  /\  ( c  e.  ( 2nd `  A
)  /\  d  e.  ( 2nd `  B ) ) ) )  -> 
( a G b )  <Q  ( c G d ) )
3029adantrlr 469 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B
) ) ) )  ->  ( a G b )  <Q  (
c G d ) )
3130adantrrr 471 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )  ->  ( a G b )  <Q  (
c G d ) )
32 eqtr2 2106 . . . . . . . . . . 11  |-  ( ( q  =  ( a G b )  /\  q  =  ( c G d ) )  ->  ( a G b )  =  ( c G d ) )
3332ad2ant2l 492 . . . . . . . . . 10  |-  ( ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )  -> 
( a G b )  =  ( c G d ) )
3433adantl 271 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )  ->  ( a G b )  =  ( c G d ) )
35 ltsonq 6957 . . . . . . . . . . 11  |-  <Q  Or  Q.
36 ltrelnq 6924 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
3735, 36soirri 4826 . . . . . . . . . 10  |-  -.  (
a G b ) 
<Q  ( a G b )
38 breq2 3849 . . . . . . . . . 10  |-  ( ( a G b )  =  ( c G d )  ->  (
( a G b )  <Q  ( a G b )  <->  ( a G b )  <Q 
( c G d ) ) )
3937, 38mtbii 634 . . . . . . . . 9  |-  ( ( a G b )  =  ( c G d )  ->  -.  ( a G b )  <Q  ( c G d ) )
4034, 39syl 14 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )  ->  -.  ( a G b )  <Q 
( c G d ) )
4131, 40pm2.21fal 1309 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  /\  (
( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) ) )  -> F.  )
4241ex 113 . . . . . 6  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  ->  (
( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )  -> F.  ) )
4342exlimdvv 1825 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  ->  ( E. c E. d ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )  -> F.  ) )
4443exlimdvv 1825 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  ->  ( E. a E. b E. c E. d ( ( ( a  e.  ( 1st `  A
)  /\  b  e.  ( 1st `  B ) )  /\  q  =  ( a G b ) )  /\  (
( c  e.  ( 2nd `  A )  /\  d  e.  ( 2nd `  B ) )  /\  q  =  ( c G d ) ) )  -> F.  ) )
4512, 44mpd 13 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )  -> F.  )
4645inegd 1308 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
4746ralrimivw 2447 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289   F. wfal 1294   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652    |-> cmpt2 5654   1stc1st 5909   2ndc2nd 5910   Q.cnq 6839    <Q cltq 6844   P.cnp 6850
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-mi 6865  df-lti 6866  df-enq 6906  df-nqqs 6907  df-ltnqqs 6912  df-inp 7025
This theorem is referenced by:  addclpr  7096  mulclpr  7131
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