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

Theorem recexprlemloc 7093
Description: 𝐵 is located. Lemma for recexpr 7100. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemloc (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
Distinct variable groups:   𝑟,𝑞,𝑥,𝑦,𝐴   𝐵,𝑞,𝑟,𝑥,𝑦

Proof of Theorem recexprlemloc
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6937 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 6950 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
31, 2sylan 277 . . . . . . . 8 ((𝐴P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
43adantlr 461 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
5 simprr 499 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q 𝑢)
6 elprnql 6943 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
71, 6sylan 277 . . . . . . . . . . . . 13 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
87ad2ant2r 493 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
98adantlr 461 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
10 recrecnq 6856 . . . . . . . . . . 11 (𝑢Q → (*Q‘(*Q𝑢)) = 𝑢)
119, 10syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) = 𝑢)
125, 11breqtrrd 3837 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q (*Q‘(*Q𝑢)))
13 recclnq 6854 . . . . . . . . . . 11 (𝑢Q → (*Q𝑢) ∈ Q)
149, 13syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) ∈ Q)
15 ltrelnq 6827 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
1615brel 4448 . . . . . . . . . . . . 13 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
1716adantl 271 . . . . . . . . . . . 12 ((𝐴P𝑞 <Q 𝑟) → (𝑞Q𝑟Q))
1817ad2antrr 472 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (𝑞Q𝑟Q))
1918simprd 112 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟Q)
20 ltrnqg 6882 . . . . . . . . . 10 (((*Q𝑢) ∈ Q𝑟Q) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2114, 19, 20syl2anc 403 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2212, 21mpbird 165 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) <Q 𝑟)
23 simprl 498 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢 ∈ (1st𝐴))
2411, 23eqeltrd 2159 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) ∈ (1st𝐴))
25 breq1 3814 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → (𝑦 <Q 𝑟 ↔ (*Q𝑢) <Q 𝑟))
26 fveq2 5253 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑢) → (*Q𝑦) = (*Q‘(*Q𝑢)))
2726eleq1d 2151 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑢)) ∈ (1st𝐴)))
2825, 27anbi12d 457 . . . . . . . . . . 11 (𝑦 = (*Q𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴))))
2928spcegv 2697 . . . . . . . . . 10 ((*Q𝑢) ∈ Q → (((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
30 recexpr.1 . . . . . . . . . . 11 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
3130recexprlemelu 7085 . . . . . . . . . 10 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
3229, 31syl6ibr 160 . . . . . . . . 9 ((*Q𝑢) ∈ Q → (((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
3314, 32syl 14 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
3422, 24, 33mp2and 424 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd𝐵))
354, 34rexlimddv 2487 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵))
3635olcd 686 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
37 prnminu 6951 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
381, 37sylan 277 . . . . . . . 8 ((𝐴P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
3938adantlr 461 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
40 elprnqu 6944 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
411, 40sylan 277 . . . . . . . . . . . . 13 ((𝐴P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4241adantlr 461 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4342ad2ant2r 493 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣Q)
44 recrecnq 6856 . . . . . . . . . . 11 (𝑣Q → (*Q‘(*Q𝑣)) = 𝑣)
4543, 44syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) = 𝑣)
46 simprr 499 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 <Q (*Q𝑞))
4745, 46eqbrtrd 3831 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) <Q (*Q𝑞))
4817ad2antrr 472 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞Q𝑟Q))
4948simpld 110 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞Q)
50 recclnq 6854 . . . . . . . . . . 11 (𝑣Q → (*Q𝑣) ∈ Q)
5143, 50syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q𝑣) ∈ Q)
52 ltrnqg 6882 . . . . . . . . . 10 ((𝑞Q ∧ (*Q𝑣) ∈ Q) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5349, 51, 52syl2anc 403 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5447, 53mpbird 165 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 <Q (*Q𝑣))
55 simprl 498 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 ∈ (2nd𝐴))
5645, 55eqeltrd 2159 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) ∈ (2nd𝐴))
57 breq2 3815 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑣)))
58 fveq2 5253 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑣) → (*Q𝑦) = (*Q‘(*Q𝑣)))
5958eleq1d 2151 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)))
6057, 59anbi12d 457 . . . . . . . . . . 11 (𝑦 = (*Q𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴))))
6160spcegv 2697 . . . . . . . . . 10 ((*Q𝑣) ∈ Q → ((𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
6230recexprlemell 7084 . . . . . . . . . 10 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
6361, 62syl6ibr 160 . . . . . . . . 9 ((*Q𝑣) ∈ Q → ((𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
6451, 63syl 14 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → ((𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
6554, 56, 64mp2and 424 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 ∈ (1st𝐵))
6639, 65rexlimddv 2487 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵))
6766orcd 685 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
68 ltrnqi 6883 . . . . . 6 (𝑞 <Q 𝑟 → (*Q𝑟) <Q (*Q𝑞))
69 prloc 6953 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) <Q (*Q𝑞)) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
701, 68, 69syl2an 283 . . . . 5 ((𝐴P𝑞 <Q 𝑟) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
7136, 67, 70mpjaodan 745 . . . 4 ((𝐴P𝑞 <Q 𝑟) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
7271ex 113 . . 3 (𝐴P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7372ralrimivw 2441 . 2 (𝐴P → ∀𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7473ralrimivw 2441 1 (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354  cop 3425   class class class wbr 3811  cfv 4969  1st c1st 5844  2nd c2nd 5845  Qcnq 6742  *Qcrq 6746   <Q cltq 6747  Pcnp 6753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-mi 6768  df-lti 6769  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-inp 6928
This theorem is referenced by:  recexprlempr  7094
  Copyright terms: Public domain W3C validator