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Theorem genpdisj 6619
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 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpdisj.ord ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
genpdisj.com ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
Assertion
Ref Expression
genpdisj ((𝐴P𝐵P) → ∀𝑞Q ¬ (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 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
2 genpelvl.2 . . . . . . . . 9 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpelvl 6608 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2344 . . . . . . . 8 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 185 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
61, 2genpelvu 6609 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑)))
7 r2ex 2344 . . . . . . . 8 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))
86, 7syl6bb 185 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
95, 8anbi12d 442 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
10 ee4anv 1809 . . . . . 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 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
1211biimpa 280 . . . 4 (((𝐴P𝐵P) ∧ (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 6571 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
15 prltlu 6583 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) → 𝑎 <Q 𝑐)
16153expib 1107 . . . . . . . . . . . . . . . 16 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) → 𝑎 <Q 𝑐))
1714, 16syl 14 . . . . . . . . . . . . . . 15 (𝐴P → ((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) → 𝑎 <Q 𝑐))
18 prop 6571 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 prltlu 6583 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵)) → 𝑏 <Q 𝑑)
20193expib 1107 . . . . . . . . . . . . . . . 16 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ((𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵)) → 𝑏 <Q 𝑑))
2118, 20syl 14 . . . . . . . . . . . . . . 15 (𝐵P → ((𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵)) → 𝑏 <Q 𝑑))
2217, 21im2anan9 530 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) ∧ (𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎 <Q 𝑐𝑏 <Q 𝑑)))
23 genpdisj.ord . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q𝑧Q) → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦)))
24 genpdisj.com . . . . . . . . . . . . . . 15 ((𝑥Q𝑦Q) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥))
2523, 24genplt2i 6606 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2622, 25syl6 29 . . . . . . . . . . . . 13 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) ∧ (𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2713, 26syl5bir 142 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2827imp 115 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2928adantlr 446 . . . . . . . . . 10 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3029adantrlr 454 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3130adantrrr 456 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
32 eqtr2 2058 . . . . . . . . . . 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 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35 ltsonq 6494 . . . . . . . . . . 11 <Q Or Q
36 ltrelnq 6461 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
3735, 36soirri 4719 . . . . . . . . . 10 ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38 breq2 3768 . . . . . . . . . 10 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
3937, 38mtbii 599 . . . . . . . . 9 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
4034, 39syl 14 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ¬ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
4131, 40pm2.21fal 1264 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ⊥)
4241ex 108 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ((((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4342exlimdvv 1777 . . . . 5 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4443exlimdvv 1777 . . . 4 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4512, 44mpd 13 . . 3 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ⊥)
4645inegd 1263 . 2 ((𝐴P𝐵P) → ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
4746ralrimivw 2393 1 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wfal 1248  wex 1381  wcel 1393  wral 2306  wrex 2307  {crab 2310  cop 3378   class class class wbr 3764  cfv 4902  (class class class)co 5512  cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6376   <Q cltq 6381  Pcnp 6387
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-mi 6402  df-lti 6403  df-enq 6443  df-nqqs 6444  df-ltnqqs 6449  df-inp 6562
This theorem is referenced by:  addclpr  6633  mulclpr  6668
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