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Theorem genpdisj 7497
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 7486 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏)))
4 r2ex 2495 . . . . . . . 8 (∃𝑎 ∈ (1st𝐴)∃𝑏 ∈ (1st𝐵)𝑞 = (𝑎𝐺𝑏) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)))
53, 4bitrdi 196 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ ∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏))))
61, 2genpelvu 7487 . . . . . . . 8 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑)))
7 r2ex 2495 . . . . . . . 8 (∃𝑐 ∈ (2nd𝐴)∃𝑑 ∈ (2nd𝐵)𝑞 = (𝑐𝐺𝑑) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))
86, 7bitrdi 196 . . . . . . 7 ((𝐴P𝐵P) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
95, 8anbi12d 473 . . . . . 6 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ (∃𝑎𝑏((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ∃𝑐𝑑((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
10 ee4anv 1932 . . . . . 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, 10bitr4di 198 . . . . 5 ((𝐴P𝐵P) → ((𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))) ↔ ∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))))
1211biimpa 296 . . . 4 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ∃𝑎𝑏𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))))
13 an4 586 . . . . . . . . . . . . 13 (((𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) ∧ (𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵))) ↔ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))))
14 prop 7449 . . . . . . . . . . . . . . . 16 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
15 prltlu 7461 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑎 ∈ (1st𝐴) ∧ 𝑐 ∈ (2nd𝐴)) → 𝑎 <Q 𝑐)
16153expib 1206 . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . 16 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
19 prltlu 7461 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑏 ∈ (1st𝐵) ∧ 𝑑 ∈ (2nd𝐵)) → 𝑏 <Q 𝑑)
20193expib 1206 . . . . . . . . . . . . . . . 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 598 . . . . . . . . . . . . . 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 7484 . . . . . . . . . . . . . 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 153 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
2827imp 124 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
2928adantlr 477 . . . . . . . . . 10 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ ((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3029adantrlr 485 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ (𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)))) → (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑))
3130adantrrr 487 . . . . . . . 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 2194 . . . . . . . . . . 11 ((𝑞 = (𝑎𝐺𝑏) ∧ 𝑞 = (𝑐𝐺𝑑)) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3332ad2ant2l 508 . . . . . . . . . 10 ((((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
3433adantl 277 . . . . . . . . 9 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → (𝑎𝐺𝑏) = (𝑐𝐺𝑑))
35 ltsonq 7372 . . . . . . . . . . 11 <Q Or Q
36 ltrelnq 7339 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
3735, 36soirri 5015 . . . . . . . . . 10 ¬ (𝑎𝐺𝑏) <Q (𝑎𝐺𝑏)
38 breq2 4002 . . . . . . . . . 10 ((𝑎𝐺𝑏) = (𝑐𝐺𝑑) → ((𝑎𝐺𝑏) <Q (𝑎𝐺𝑏) ↔ (𝑎𝐺𝑏) <Q (𝑐𝐺𝑑)))
3937, 38mtbii 674 . . . . . . . . 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 1373 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) ∧ (((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑)))) → ⊥)
4241ex 115 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → ((((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4342exlimdvv 1895 . . . . 5 (((𝐴P𝐵P) ∧ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))) → (∃𝑐𝑑(((𝑎 ∈ (1st𝐴) ∧ 𝑏 ∈ (1st𝐵)) ∧ 𝑞 = (𝑎𝐺𝑏)) ∧ ((𝑐 ∈ (2nd𝐴) ∧ 𝑑 ∈ (2nd𝐵)) ∧ 𝑞 = (𝑐𝐺𝑑))) → ⊥))
4443exlimdvv 1895 . . . 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 1372 . 2 ((𝐴P𝐵P) → ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
4746ralrimivw 2549 1 ((𝐴P𝐵P) → ∀𝑞Q ¬ (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ∧ 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wfal 1358  wex 1490  wcel 2146  wral 2453  wrex 2454  {crab 2457  cop 3592   class class class wbr 3998  cfv 5208  (class class class)co 5865  cmpo 5867  1st c1st 6129  2nd c2nd 6130  Qcnq 7254   <Q cltq 7259  Pcnp 7265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-mi 7280  df-lti 7281  df-enq 7321  df-nqqs 7322  df-ltnqqs 7327  df-inp 7440
This theorem is referenced by:  addclpr  7511  mulclpr  7546
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