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Theorem genpdisj 6845
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 6834 . . . . . . . 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 2391 . . . . . . . 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 6835 . . . . . . . 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 2391 . . . . . . . 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 1852 . . . . . 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 551 . . . . . . . . . . . . 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 6797 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
15 prltlu 6809 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A
) )  ->  a  <Q  c )
16153expib 1142 . . . . . . . . . . . . . . . 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 6797 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 prltlu 6809 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B )  /\  d  e.  ( 2nd `  B
) )  ->  b  <Q  d )
20193expib 1142 . . . . . . . . . . . . . . . 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 563 . . . . . . . . . . . . . 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 6832 . . . . . . . . . . . . . 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 2101 . . . . . . . . . . 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 6720 . . . . . . . . . . 11  |-  <Q  Or  Q.
36 ltrelnq 6687 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
3735, 36soirri 4769 . . . . . . . . . 10  |-  -.  (
a G b ) 
<Q  ( a G b )
38 breq2 3809 . . . . . . . . . 10  |-  ( ( a G b )  =  ( c G d )  ->  (
( a G b )  <Q  ( a G b )  <->  ( a G b )  <Q 
( c G d ) ) )
3937, 38mtbii 632 . . . . . . . . 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 1305 . . . . . . 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 1820 . . . . 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 1820 . . . 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 1304 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
4746ralrimivw 2440 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 920    = wceq 1285   F. wfal 1290   E.wex 1422    e. wcel 1434   A.wral 2353   E.wrex 2354   {crab 2357   <.cop 3419   class class class wbr 3805   ` cfv 4952  (class class class)co 5564    |-> cmpt2 5566   1stc1st 5817   2ndc2nd 5818   Q.cnq 6602    <Q cltq 6607   P.cnp 6613
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 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-eprel 4072  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6626  df-mi 6628  df-lti 6629  df-enq 6669  df-nqqs 6670  df-ltnqqs 6675  df-inp 6788
This theorem is referenced by:  addclpr  6859  mulclpr  6894
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