ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  genpdisj Structured version   GIF version

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 = (w P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
genpelvl.2 ((y Q z Q) → (y𝐺z) Q)
genpdisj.ord ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
genpdisj.com ((x Q y Q) → (x𝐺y) = (y𝐺x))
Assertion
Ref Expression
genpdisj ((A P B P) → 𝑞 Q ¬ (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))))
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 P, v P ↦ ⟨{x Qy Q z Q (y (1stw) z (1stv) x = (y𝐺z))}, {x Qy Q z Q (y (2ndw) z (2ndv) x = (y𝐺z))}⟩)
2 genpelvl.2 . . . . . . . . 9 ((y Q z Q) → (y𝐺z) Q)
31, 2genpelvl 6494 . . . . . . . 8 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2338 . . . . . . . 8 (𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . 7 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏))))
61, 2genpelvu 6495 . . . . . . . 8 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) ↔ 𝑐 (2ndA)𝑑 (2ndB)𝑞 = (𝑐𝐺𝑑)))
7 r2ex 2338 . . . . . . . 8 (𝑐 (2ndA)𝑑 (2ndB)𝑞 = (𝑐𝐺𝑑) ↔ 𝑐𝑑((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))
86, 7syl6bb 185 . . . . . . 7 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) ↔ 𝑐𝑑((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))))
95, 8anbi12d 442 . . . . . 6 ((A P B P) → ((𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))) ↔ (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) 𝑐𝑑((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))))
10 ee4anv 1806 . . . . . 6 (𝑎𝑏𝑐𝑑(((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))) ↔ (𝑎𝑏((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) 𝑐𝑑((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))))
119, 10syl6bbr 187 . . . . 5 ((A P B P) → ((𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))) ↔ 𝑎𝑏𝑐𝑑(((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))))
1211biimpa 280 . . . 4 (((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) → 𝑎𝑏𝑐𝑑(((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))))
13 an4 520 . . . . . . . . . . . . 13 (((𝑎 (1stA) 𝑐 (2ndA)) (𝑏 (1stB) 𝑑 (2ndB))) ↔ ((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB))))
14 prop 6457 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
15 prltlu 6469 . . . . . . . . . . . . . . . . 17 ((⟨(1stA), (2ndA)⟩ P 𝑎 (1stA) 𝑐 (2ndA)) → 𝑎 <Q 𝑐)
16153expib 1106 . . . . . . . . . . . . . . . 16 (⟨(1stA), (2ndA)⟩ P → ((𝑎 (1stA) 𝑐 (2ndA)) → 𝑎 <Q 𝑐))
1714, 16syl 14 . . . . . . . . . . . . . . 15 (A P → ((𝑎 (1stA) 𝑐 (2ndA)) → 𝑎 <Q 𝑐))
18 prop 6457 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
19 prltlu 6469 . . . . . . . . . . . . . . . . 17 ((⟨(1stB), (2ndB)⟩ P 𝑏 (1stB) 𝑑 (2ndB)) → 𝑏 <Q 𝑑)
20193expib 1106 . . . . . . . . . . . . . . . 16 (⟨(1stB), (2ndB)⟩ P → ((𝑏 (1stB) 𝑑 (2ndB)) → 𝑏 <Q 𝑑))
2118, 20syl 14 . . . . . . . . . . . . . . 15 (B P → ((𝑏 (1stB) 𝑑 (2ndB)) → 𝑏 <Q 𝑑))
2217, 21im2anan9 530 . . . . . . . . . . . . . 14 ((A P B P) → (((𝑎 (1stA) 𝑐 (2ndA)) (𝑏 (1stB) 𝑑 (2ndB))) → (𝑎 <Q 𝑐 𝑏 <Q 𝑑)))
23 genpdisj.ord . . . . . . . . . . . . . . 15 ((x Q y Q z Q) → (x <Q y ↔ (z𝐺x) <Q (z𝐺y)))
24 genpdisj.com . . . . . . . . . . . . . . 15 ((x Q y Q) → (x𝐺y) = (y𝐺x))
2523, 24genplt2i 6492 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐 𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2622, 25syl6 29 . . . . . . . . . . . . 13 ((A P B P) → (((𝑎 (1stA) 𝑐 (2ndA)) (𝑏 (1stB) 𝑑 (2ndB))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2713, 26syl5bir 142 . . . . . . . . . . . 12 ((A P B P) → (((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2827imp 115 . . . . . . . . . . 11 (((A P B P) ((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2928adantlr 446 . . . . . . . . . 10 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) ((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3029adantrlr 454 . . . . . . . . 9 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) (𝑐 (2ndA) 𝑑 (2ndB)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3130adantrrr 456 . . . . . . . 8 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
32 eqtr2 2055 . . . . . . . . . . 11 ((𝑞 = (𝑎𝐺𝑏) 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3332ad2ant2l 477 . . . . . . . . . 10 ((((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3433adantl 262 . . . . . . . . 9 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35 ltsonq 6382 . . . . . . . . . . 11 <Q Or Q
36 ltrelnq 6349 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
3735, 36soirri 4662 . . . . . . . . . 10 ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38 breq2 3759 . . . . . . . . . 10 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
3937, 38mtbii 598 . . . . . . . . 9 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
4034, 39syl 14 . . . . . . . 8 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
4131, 40pm2.21fal 1263 . . . . . . 7 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) (((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑)))) → ⊥ )
4241ex 108 . . . . . 6 (((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) → ((((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))) → ⊥ ))
4342exlimdvv 1774 . . . . 5 (((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) → (𝑐𝑑(((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))) → ⊥ ))
4443exlimdvv 1774 . . . 4 (((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) → (𝑎𝑏𝑐𝑑(((𝑎 (1stA) 𝑏 (1stB)) 𝑞 = (𝑎𝐺𝑏)) ((𝑐 (2ndA) 𝑑 (2ndB)) 𝑞 = (𝑐𝐺𝑑))) → ⊥ ))
4512, 44mpd 13 . . 3 (((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) → ⊥ )
4645inegd 1262 . 2 ((A P B P) → ¬ (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))))
4746ralrimivw 2387 1 ((A P B P) → 𝑞 Q ¬ (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))))
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  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   <Q cltq 6269  Pcnp 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-bnd 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 6448
This theorem is referenced by:  addclpr  6520  mulclpr  6551
  Copyright terms: Public domain W3C validator