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Theorem genpdisj 6678
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 6667 . . . . . . . 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 2361 . . . . . . . 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 189 . . . . . . 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 6668 . . . . . . . 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 2361 . . . . . . . 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 189 . . . . . . 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 450 . . . . . 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 1825 . . . . . 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 191 . . . . 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 284 . . . 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 528 . . . . . . . . . . . . 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 6630 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
15 prltlu 6642 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A
) )  ->  a  <Q  c )
16153expib 1118 . . . . . . . . . . . . . . . 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 6630 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 prltlu 6642 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B )  /\  d  e.  ( 2nd `  B
) )  ->  b  <Q  d )
20193expib 1118 . . . . . . . . . . . . . . . 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 540 . . . . . . . . . . . . . 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 6665 . . . . . . . . . . . . . 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 146 . . . . . . . . . . . 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 119 . . . . . . . . . . 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 454 . . . . . . . . . 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 462 . . . . . . . . 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 464 . . . . . . . 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 2074 . . . . . . . . . . 11  |-  ( ( q  =  ( a G b )  /\  q  =  ( c G d ) )  ->  ( a G b )  =  ( c G d ) )
3332ad2ant2l 485 . . . . . . . . . 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 266 . . . . . . . . 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 6553 . . . . . . . . . . 11  |-  <Q  Or  Q.
36 ltrelnq 6520 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
3735, 36soirri 4746 . . . . . . . . . 10  |-  -.  (
a G b ) 
<Q  ( a G b )
38 breq2 3795 . . . . . . . . . 10  |-  ( ( a G b )  =  ( c G d )  ->  (
( a G b )  <Q  ( a G b )  <->  ( a G b )  <Q 
( c G d ) ) )
3937, 38mtbii 609 . . . . . . . . 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 1280 . . . . . . 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 112 . . . . . 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 1793 . . . . 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 1793 . . . 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 1279 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
4746ralrimivw 2410 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 101    <-> wb 102    /\ w3a 896    = wceq 1259   F. wfal 1264   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327   <.cop 3405   class class class wbr 3791   ` cfv 4929  (class class class)co 5539    |-> cmpt2 5541   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435    <Q cltq 6440   P.cnp 6446
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 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-lti 6462  df-enq 6502  df-nqqs 6503  df-ltnqqs 6508  df-inp 6621
This theorem is referenced by:  addclpr  6692  mulclpr  6727
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