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Theorem addlocpr 6518
Description: Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6485 to both A and B, and uses nqtri3or 6380 rather than prloc 6473 to decide whether 𝑞 is too big to be in the lower cut of A +P B (and deduce that if it is, then 𝑟 must be in the upper cut). What the two proofs have in common is that they take the difference between 𝑞 and 𝑟 to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
Assertion
Ref Expression
addlocpr ((A P B P) → 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
Distinct variable groups:   A,𝑞,𝑟   B,𝑞,𝑟

Proof of Theorem addlocpr
Dummy variables 𝑑 𝑒 𝑝 𝑡 u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexnqq 6391 . . . . . 6 ((𝑞 Q 𝑟 Q) → (𝑞 <Q 𝑟𝑝 Q (𝑞 +Q 𝑝) = 𝑟))
21biimpa 280 . . . . 5 (((𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) → 𝑝 Q (𝑞 +Q 𝑝) = 𝑟)
323adant1 921 . . . 4 (((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) → 𝑝 Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 6393 . . . . . 6 (𝑝 Q Q ( +Q ) = 𝑝)
54ad2antrl 459 . . . . 5 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) → Q ( +Q ) = 𝑝)
6 prop 6457 . . . . . . . . . 10 (A P → ⟨(1stA), (2ndA)⟩ P)
7 prarloc 6485 . . . . . . . . . 10 ((⟨(1stA), (2ndA)⟩ P Q) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
86, 7sylan 267 . . . . . . . . 9 ((A P Q) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
98adantlr 446 . . . . . . . 8 (((A P B P) Q) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
1093ad2antl1 1065 . . . . . . 7 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) Q) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
1110ad2ant2r 478 . . . . . 6 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
12 prop 6457 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
13 prarloc 6485 . . . . . . . . . . . . . 14 ((⟨(1stB), (2ndB)⟩ P Q) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
1412, 13sylan 267 . . . . . . . . . . . . 13 ((B P Q) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
1514adantll 445 . . . . . . . . . . . 12 (((A P B P) Q) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
16153ad2antl1 1065 . . . . . . . . . . 11 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) Q) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 478 . . . . . . . . . 10 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
1817adantr 261 . . . . . . . . 9 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
19 simpll1 942 . . . . . . . . . . . . . 14 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → (A P B P))
2019ad2antrr 457 . . . . . . . . . . . . 13 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → (A P B P))
2120simpld 105 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → A P)
2220simprd 107 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → B P)
23 simpll3 944 . . . . . . . . . . . . 13 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → 𝑞 <Q 𝑟)
2423ad2antrr 457 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → 𝑞 <Q 𝑟)
25 simplrl 487 . . . . . . . . . . . . 13 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → Q)
2625adantr 261 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → Q)
27 simplrr 488 . . . . . . . . . . . . . 14 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → (𝑞 +Q 𝑝) = 𝑟)
28 oveq2 5463 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2045 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 460 . . . . . . . . . . . . . 14 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3127, 30mpbird 156 . . . . . . . . . . . . 13 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → (𝑞 +Q ( +Q )) = 𝑟)
3231ad2antrr 457 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → (𝑞 +Q ( +Q )) = 𝑟)
33 simprll 489 . . . . . . . . . . . . 13 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → 𝑑 (1stA))
3433adantr 261 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → 𝑑 (1stA))
35 simprlr 490 . . . . . . . . . . . . 13 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → u (2ndA))
3635adantr 261 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → u (2ndA))
37 simplrr 488 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → u <Q (𝑑 +Q ))
38 simprll 489 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → 𝑒 (1stB))
39 simprlr 490 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → 𝑡 (2ndB))
40 simprr 484 . . . . . . . . . . . 12 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → 𝑡 <Q (𝑒 +Q ))
4121, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40addlocprlem 6517 . . . . . . . . . . 11 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) ((𝑒 (1stB) 𝑡 (2ndB)) 𝑡 <Q (𝑒 +Q ))) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
4241expr 357 . . . . . . . . . 10 (((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) (𝑒 (1stB) 𝑡 (2ndB))) → (𝑡 <Q (𝑒 +Q ) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
4342rexlimdvva 2434 . . . . . . . . 9 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → (𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
4418, 43mpd 13 . . . . . . . 8 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) ((𝑑 (1stA) u (2ndA)) u <Q (𝑑 +Q ))) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
4544expr 357 . . . . . . 7 ((((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) (𝑑 (1stA) u (2ndA))) → (u <Q (𝑑 +Q ) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
4645rexlimdvva 2434 . . . . . 6 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → (𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
4711, 46mpd 13 . . . . 5 (((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) ( Q ( +Q ) = 𝑝)) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
485, 47rexlimddv 2431 . . . 4 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
493, 48rexlimddv 2431 . . 3 (((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
50493expia 1105 . 2 (((A P B P) (𝑞 Q 𝑟 Q)) → (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
5150ralrimivva 2395 1 ((A P B P) → 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  wral 2300  wrex 2301  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277
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-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450
This theorem is referenced by:  addclpr  6519
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