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Theorem genpdisj 6775
 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 6764 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2387 . . . . . . . 8 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4syl6bb 194 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
61, 2genpelvu 6765 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑)))
7 r2ex 2387 . . . . . . . 8 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))
86, 7syl6bb 194 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
95, 8anbi12d 457 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
10 ee4anv 1851 . . . . . 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 196 . . . . 5 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
1211biimpa 290 . . . 4 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
13 an4 551 . . . . . . . . . . . . 13 (((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) ∧ (𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵))) ↔ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))))
14 prop 6727 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
15 prltlu 6739 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) → 𝑎 <Q 𝑐)
16153expib 1142 . . . . . . . . . . . . . . . 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 6727 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 prltlu 6739 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵)) → 𝑏 <Q 𝑑)
20193expib 1142 . . . . . . . . . . . . . . . 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 563 . . . . . . . . . . . . . 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 6762 . . . . . . . . . . . . . 14 ((𝑎 <Q 𝑐𝑏 <Q 𝑑) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2622, 25syl6 33 . . . . . . . . . . . . 13 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) ∧ (𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2713, 26syl5bir 151 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2827imp 122 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2928adantlr 461 . . . . . . . . . 10 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3029adantrlr 469 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3130adantrrr 471 . . . . . . . 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 2100 . . . . . . . . . . 11 ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3332ad2ant2l 492 . . . . . . . . . 10 ((((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3433adantl 271 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35 ltsonq 6650 . . . . . . . . . . 11 <Q Or Q
36 ltrelnq 6617 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
3735, 36soirri 4749 . . . . . . . . . 10 ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38 breq2 3797 . . . . . . . . . 10 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
3937, 38mtbii 632 . . . . . . . . 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 1305 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ⊥)
4241ex 113 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ((((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4342exlimdvv 1819 . . . . 5 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4443exlimdvv 1819 . . . 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 1304 . 2 ((𝐴P𝐵P) → ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
4746ralrimivw 2436 1 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 920   = wceq 1285  ⊥wfal 1290  ∃wex 1422   ∈ wcel 1434  ∀wral 2349  ∃wrex 2350  {crab 2353  ⟨cop 3409   class class class wbr 3793  ‘cfv 4932  (class class class)co 5543   ↦ cmpt2 5545  1st c1st 5796  2nd c2nd 5797  Qcnq 6532
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