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Theorem genpdisj 7324
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 7313 . . . . . . . 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 2453 . . . . . . . 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 195 . . . . . . 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 7314 . . . . . . . 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 2453 . . . . . . . 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 195 . . . . . . 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 464 . . . . . 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 1904 . . . . . 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 197 . . . . 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 294 . . . 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 575 . . . . . . . . . . . . 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 7276 . . . . . . . . . . . . . . . 16  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
15 prltlu 7288 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  a  e.  ( 1st `  A )  /\  c  e.  ( 2nd `  A
) )  ->  a  <Q  c )
16153expib 1184 . . . . . . . . . . . . . . . 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 7276 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
19 prltlu 7288 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  b  e.  ( 1st `  B )  /\  d  e.  ( 2nd `  B
) )  ->  b  <Q  d )
20193expib 1184 . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . 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 7311 . . . . . . . . . . . . . 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 152 . . . . . . . . . . . 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 123 . . . . . . . . . . 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 468 . . . . . . . . . 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 476 . . . . . . . . 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 478 . . . . . . . 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 2156 . . . . . . . . . . 11  |-  ( ( q  =  ( a G b )  /\  q  =  ( c G d ) )  ->  ( a G b )  =  ( c G d ) )
3332ad2ant2l 499 . . . . . . . . . 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 275 . . . . . . . . 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 7199 . . . . . . . . . . 11  |-  <Q  Or  Q.
36 ltrelnq 7166 . . . . . . . . . . 11  |-  <Q  C_  ( Q.  X.  Q. )
3735, 36soirri 4928 . . . . . . . . . 10  |-  -.  (
a G b ) 
<Q  ( a G b )
38 breq2 3928 . . . . . . . . . 10  |-  ( ( a G b )  =  ( c G d )  ->  (
( a G b )  <Q  ( a G b )  <->  ( a G b )  <Q 
( c G d ) ) )
3937, 38mtbii 663 . . . . . . . . 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 1351 . . . . . . 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 114 . . . . . 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 1869 . . . . 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 1869 . . . 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 1350 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
4746ralrimivw 2504 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 103    <-> wb 104    /\ w3a 962    = wceq 1331   F. wfal 1336   E.wex 1468    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767    e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081    <Q cltq 7086   P.cnp 7092
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-enq 7148  df-nqqs 7149  df-ltnqqs 7154  df-inp 7267
This theorem is referenced by:  addclpr  7338  mulclpr  7373
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