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Theorem addlocpr 6378
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 6344 to both A and B, and uses nqtri3or 6242 rather than prloc 6332 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 6253 . . . . . 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 904 . . . 4 (((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) → 𝑝 Q (𝑞 +Q 𝑝) = 𝑟)
4 halfnqq 6254 . . . . . 6 (𝑝 Q Q ( +Q ) = 𝑝)
54ad2antrl 460 . . . . 5 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) (𝑝 Q (𝑞 +Q 𝑝) = 𝑟)) → Q ( +Q ) = 𝑝)
6 prop 6316 . . . . . . . . . 10 (A P → ⟨(1stA), (2ndA)⟩ P)
7 prarloc 6344 . . . . . . . . . 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 1048 . . . . . . 7 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) Q) → 𝑑 (1stA)u (2ndA)u <Q (𝑑 +Q ))
1110ad2ant2r 463 . . . . . 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 6316 . . . . . . . . . . . . . 14 (B P → ⟨(1stB), (2ndB)⟩ P)
13 prarloc 6344 . . . . . . . . . . . . . 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 1048 . . . . . . . . . . 11 ((((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) Q) → 𝑒 (1stB)𝑡 (2ndB)𝑡 <Q (𝑒 +Q ))
1716ad2ant2r 463 . . . . . . . . . 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 925 . . . . . . . . . . . . . 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 927 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 5433 . . . . . . . . . . . . . . . 16 (( +Q ) = 𝑝 → (𝑞 +Q ( +Q )) = (𝑞 +Q 𝑝))
2928eqeq1d 2022 . . . . . . . . . . . . . . 15 (( +Q ) = 𝑝 → ((𝑞 +Q ( +Q )) = 𝑟 ↔ (𝑞 +Q 𝑝) = 𝑟))
3029ad2antll 461 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 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 473 . . . . . . . . . . . 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 474 . . . . . . . . . . . 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 475 . . . . . . . . . . . 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 469 . . . . . . . . . . . 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 6377 . . . . . . . . . . 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 2410 . . . . . . . . 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 2410 . . . . . 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 2407 . . . 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 2407 . . 3 (((A P B P) (𝑞 Q 𝑟 Q) 𝑞 <Q 𝑟) → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B))))
50493expia 1087 . 2 (((A P B P) (𝑞 Q 𝑟 Q)) → (𝑞 <Q 𝑟 → (𝑞 (1st ‘(A +P B)) 𝑟 (2nd ‘(A +P B)))))
5150ralrimivva 2371 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 613   w3a 867   = wceq 1224   wcel 1367  wral 2276  wrex 2277  cop 3343   class class class wbr 3728  cfv 4818  (class class class)co 5425  1st c1st 5677  2nd c2nd 5678  Qcnq 6127   +Q cplq 6129   <Q cltq 6132  Pcnp 6138   +P cpp 6140
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 529  ax-in2 530  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-13 1378  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-coll 3836  ax-sep 3839  ax-nul 3847  ax-pow 3891  ax-pr 3908  ax-un 4109  ax-setind 4193  ax-iinf 4227
This theorem depends on definitions:  df-bi 110  df-dc 727  df-3or 868  df-3an 869  df-tru 1227  df-fal 1230  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ne 2180  df-ral 2281  df-rex 2282  df-reu 2283  df-rab 2285  df-v 2529  df-sbc 2734  df-csb 2822  df-dif 2889  df-un 2891  df-in 2893  df-ss 2900  df-nul 3194  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-uni 3545  df-int 3580  df-iun 3623  df-br 3729  df-opab 3783  df-mpt 3784  df-tr 3819  df-eprel 3990  df-id 3994  df-po 3997  df-iso 3998  df-iord 4042  df-on 4044  df-suc 4047  df-iom 4230  df-xp 4267  df-rel 4268  df-cnv 4269  df-co 4270  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274  df-iota 4783  df-fun 4820  df-fn 4821  df-f 4822  df-f1 4823  df-fo 4824  df-f1o 4825  df-fv 4826  df-ov 5428  df-oprab 5429  df-mpt2 5430  df-1st 5679  df-2nd 5680  df-recs 5831  df-irdg 5867  df-1o 5905  df-2o 5906  df-oadd 5909  df-omul 5910  df-er 6006  df-ec 6008  df-qs 6012  df-ni 6151  df-pli 6152  df-mi 6153  df-lti 6154  df-plpq 6190  df-mpq 6191  df-enq 6193  df-nqqs 6194  df-plqqs 6195  df-mqqs 6196  df-1nqqs 6197  df-rq 6198  df-ltnqqs 6199  df-enq0 6266  df-nq0 6267  df-0nq0 6268  df-plq0 6269  df-mq0 6270  df-inp 6307  df-iplp 6309
This theorem is referenced by:  addclpr  6379
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