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Theorem genpdisj 6378
 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 6366 . . . . . . . 8 ((A P B P) → (𝑞 (1st ‘(A𝐹B)) ↔ 𝑎 (1stA)𝑏 (1stB)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2322 . . . . . . . 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 6367 . . . . . . . 8 ((A P B P) → (𝑞 (2nd ‘(A𝐹B)) ↔ 𝑐 (2ndA)𝑑 (2ndB)𝑞 = (𝑐𝐺𝑑)))
7 r2ex 2322 . . . . . . . 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 445 . . . . . 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 1791 . . . . . 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 507 . . . . . . . . . . . . 13 (((𝑎 (1stA) 𝑐 (2ndA)) (𝑏 (1stB) 𝑑 (2ndB))) ↔ ((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB))))
14 prop 6329 . . . . . . . . . . . . . . . 16 (A P → ⟨(1stA), (2ndA)⟩ P)
15 prltlu 6341 . . . . . . . . . . . . . . . . 17 ((⟨(1stA), (2ndA)⟩ P 𝑎 (1stA) 𝑐 (2ndA)) → 𝑎 <Q 𝑐)
16153expib 1093 . . . . . . . . . . . . . . . 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 6329 . . . . . . . . . . . . . . . 16 (B P → ⟨(1stB), (2ndB)⟩ P)
19 prltlu 6341 . . . . . . . . . . . . . . . . 17 ((⟨(1stB), (2ndB)⟩ P 𝑏 (1stB) 𝑑 (2ndB)) → 𝑏 <Q 𝑑)
20193expib 1093 . . . . . . . . . . . . . . . 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 517 . . . . . . . . . . . . . 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 6364 . . . . . . . . . . . . . 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 449 . . . . . . . . . 10 ((((A P B P) (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B)))) ((𝑎 (1stA) 𝑏 (1stB)) (𝑐 (2ndA) 𝑑 (2ndB)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3029adantrlr 457 . . . . . . . . 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 459 . . . . . . . 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 2040 . . . . . . . . . . 11 ((𝑞 = (𝑎𝐺𝑏) 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3332ad2ant2l 465 . . . . . . . . . 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 6257 . . . . . . . . . . 11 <Q Or Q
36 ltrelnq 6224 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
3735, 36soirri 4646 . . . . . . . . . 10 ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38 breq2 3742 . . . . . . . . . 10 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
3937, 38mtbii 586 . . . . . . . . 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 1249 . . . . . . 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 1759 . . . . 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 1759 . . . 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 1248 . 2 ((A P B P) → ¬ (𝑞 (1st ‘(A𝐹B)) 𝑞 (2nd ‘(A𝐹B))))
4746ralrimivw 2371 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 873   = wceq 1228   ⊥ wfal 1233  ∃wex 1362   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285  {crab 2288  ⟨cop 3353   class class class wbr 3738  ‘cfv 4829  (class class class)co 5436   ↦ cmpt2 5438  1st c1st 5688  2nd c2nd 5689  Qcnq 6138
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