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

 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

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
 Copyright terms: Public domain W3C validator