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Theorem genpdisj 6506
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  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
genpelvl.2  Q.  Q.  G  Q.
genpdisj.ord  Q.  Q.  Q.  <Q  G  <Q  G
genpdisj.com  Q.  Q.  G  G
Assertion
Ref Expression
genpdisj  P.  P.  q  Q.  q  1st `  F  q  2nd `  F
Distinct variable groups:   ,,,,, q,   ,,,,,, q   , G,,,,, q    F, q
Allowed substitution hints:    F(,,,,)

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  P. ,  P.  |->  <. {  Q.  |  Q.  Q.  1st `  1st `  G } ,  {  Q.  |  Q.  Q.  2nd `  2nd `  G } >.
2 genpelvl.2 . . . . . . . . 9  Q.  Q.  G  Q.
31, 2genpelvl 6495 . . . . . . . 8  P.  P.  q  1st `  F  a  1st `  b  1st `  q  a G b
4 r2ex 2338 . . . . . . . 8  a  1st `  b  1st `  q 
a G b  a b a  1st `  b  1st `  q 
a G b
53, 4syl6bb 185 . . . . . . 7  P.  P.  q  1st `  F  a b a  1st `  b  1st `  q 
a G b
61, 2genpelvu 6496 . . . . . . . 8  P.  P.  q  2nd `  F  c  2nd `  d  2nd `  q  c G d
7 r2ex 2338 . . . . . . . 8  c  2nd `  d  2nd `  q 
c G d  c d c  2nd `  d  2nd `  q 
c G d
86, 7syl6bb 185 . . . . . . 7  P.  P.  q  2nd `  F  c d c  2nd `  d  2nd `  q 
c G d
95, 8anbi12d 442 . . . . . 6  P.  P.  q  1st `  F  q  2nd `  F  a b a  1st `  b  1st `  q  a G b  c d c  2nd `  d  2nd `  q  c G d
10 ee4anv 1806 . . . . . 6  a b c d a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d  a b a  1st `  b  1st `  q  a G b  c d c  2nd `  d  2nd `  q  c G d
119, 10syl6bbr 187 . . . . 5  P.  P.  q  1st `  F  q  2nd `  F  a b c d a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
1211biimpa 280 . . . 4  P.  P.  q  1st `  F  q  2nd `  F  a b c d a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
13 an4 520 . . . . . . . . . . . . 13  a  1st `  c  2nd `  b  1st `  d  2nd `  a  1st `  b  1st `  c  2nd `  d  2nd `
14 prop 6458 . . . . . . . . . . . . . . . 16  P.  <. 1st `  ,  2nd `  >.  P.
15 prltlu 6470 . . . . . . . . . . . . . . . . 17 
<. 1st `  ,  2nd `  >.  P.  a  1st `  c  2nd `  a  <Q  c
16153expib 1106 . . . . . . . . . . . . . . . 16  <. 1st `  ,  2nd `  >.  P.  a  1st `  c  2nd `  a  <Q  c
1714, 16syl 14 . . . . . . . . . . . . . . 15  P.  a  1st `  c  2nd `  a  <Q  c
18 prop 6458 . . . . . . . . . . . . . . . 16  P.  <. 1st `  ,  2nd `  >.  P.
19 prltlu 6470 . . . . . . . . . . . . . . . . 17 
<. 1st `  ,  2nd `  >.  P.  b  1st `  d  2nd `  b  <Q  d
20193expib 1106 . . . . . . . . . . . . . . . 16  <. 1st `  ,  2nd `  >.  P.  b  1st `  d  2nd `  b  <Q  d
2118, 20syl 14 . . . . . . . . . . . . . . 15  P.  b  1st `  d  2nd `  b  <Q  d
2217, 21im2anan9 530 . . . . . . . . . . . . . 14  P.  P.  a  1st `  c  2nd `  b  1st `  d  2nd ` 
a  <Q  c  b  <Q  d
23 genpdisj.ord . . . . . . . . . . . . . . 15  Q.  Q.  Q.  <Q  G  <Q  G
24 genpdisj.com . . . . . . . . . . . . . . 15  Q.  Q.  G  G
2523, 24genplt2i 6493 . . . . . . . . . . . . . 14  a  <Q  c  b  <Q  d  a G b  <Q  c G d
2622, 25syl6 29 . . . . . . . . . . . . 13  P.  P.  a  1st `  c  2nd `  b  1st `  d  2nd ` 
a G b 
<Q  c G d
2713, 26syl5bir 142 . . . . . . . . . . . 12  P.  P.  a  1st `  b  1st `  c  2nd `  d  2nd ` 
a G b 
<Q  c G d
2827imp 115 . . . . . . . . . . 11  P.  P.  a  1st `  b  1st `  c  2nd `  d  2nd `  a G b  <Q  c G d
2928adantlr 446 . . . . . . . . . 10  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  c  2nd `  d  2nd `  a G b  <Q  c G d
3029adantrlr 454 . . . . . . . . 9  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b 
c  2nd `  d  2nd `  a G b  <Q  c G d
3130adantrrr 456 . . . . . . . 8  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d 
a G b 
<Q  c G d
32 eqtr2 2055 . . . . . . . . . . 11  q  a G b  q  c G d  a G b  c G d
3332ad2ant2l 477 . . . . . . . . . 10  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d  a G b  c G d
3433adantl 262 . . . . . . . . 9  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d 
a G b  c G d
35 ltsonq 6382 . . . . . . . . . . 11  <Q  Or  Q.
36 ltrelnq 6349 . . . . . . . . . . 11  <Q  C_  Q.  X.  Q.
3735, 36soirri 4662 . . . . . . . . . 10 
a G b 
<Q  a G b
38 breq2 3759 . . . . . . . . . 10  a G b  c G d  a G b  <Q  a G b  a G b  <Q  c G d
3937, 38mtbii 598 . . . . . . . . 9  a G b  c G d  a G b  <Q  c G d
4034, 39syl 14 . . . . . . . 8  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d  a G b  <Q  c G d
4131, 40pm2.21fal 1263 . . . . . . 7  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
4241ex 108 . . . . . 6  P.  P.  q  1st `  F  q  2nd `  F  a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
4342exlimdvv 1774 . . . . 5  P.  P.  q  1st `  F  q  2nd `  F  c d a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
4443exlimdvv 1774 . . . 4  P.  P.  q  1st `  F  q  2nd `  F  a b c d a  1st `  b  1st `  q  a G b  c  2nd `  d  2nd `  q  c G d
4512, 44mpd 13 . . 3  P.  P.  q  1st `  F  q  2nd `  F
4645inegd 1262 . 2  P.  P.  q  1st `  F  q  2nd `  F
4746ralrimivw 2387 1  P.  P.  q  Q.  q  1st `  F  q  2nd `  F
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wfal 1247  wex 1378   wcel 1390  wral 2300  wrex 2301   {crab 2304   <.cop 3370   class class class wbr 3755   ` cfv 4845  (class class class)co 5455    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708   Q.cnq 6264    <Q cltq 6269   P.cnp 6275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-mi 6290  df-lti 6291  df-enq 6331  df-nqqs 6332  df-ltnqqs 6337  df-inp 6449
This theorem is referenced by:  addclpr  6520  mulclpr  6553
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