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Theorem recexprlemloc 6729
 Description: 𝐵 is located. Lemma for recexpr 6736. (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 6573 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 6586 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
31, 2sylan 267 . . . . . . . 8 ((𝐴P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
43adantlr 446 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
5 simprr 484 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q 𝑢)
6 elprnql 6579 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
71, 6sylan 267 . . . . . . . . . . . . 13 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
87ad2ant2r 478 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
98adantlr 446 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
10 recrecnq 6492 . . . . . . . . . . 11 (𝑢Q → (*Q‘(*Q𝑢)) = 𝑢)
119, 10syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) = 𝑢)
125, 11breqtrrd 3790 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q (*Q‘(*Q𝑢)))
13 recclnq 6490 . . . . . . . . . . 11 (𝑢Q → (*Q𝑢) ∈ Q)
149, 13syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) ∈ Q)
15 ltrelnq 6463 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
1615brel 4392 . . . . . . . . . . . . 13 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
1716adantl 262 . . . . . . . . . . . 12 ((𝐴P𝑞 <Q 𝑟) → (𝑞Q𝑟Q))
1817ad2antrr 457 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (𝑞Q𝑟Q))
1918simprd 107 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟Q)
20 ltrnqg 6518 . . . . . . . . . 10 (((*Q𝑢) ∈ Q𝑟Q) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2114, 19, 20syl2anc 391 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2212, 21mpbird 156 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) <Q 𝑟)
23 simprl 483 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢 ∈ (1st𝐴))
2411, 23eqeltrd 2114 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) ∈ (1st𝐴))
25 breq1 3767 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → (𝑦 <Q 𝑟 ↔ (*Q𝑢) <Q 𝑟))
26 fveq2 5178 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑢) → (*Q𝑦) = (*Q‘(*Q𝑢)))
2726eleq1d 2106 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑢)) ∈ (1st𝐴)))
2825, 27anbi12d 442 . . . . . . . . . . 11 (𝑦 = (*Q𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴))))
2928spcegv 2641 . . . . . . . . . 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 6721 . . . . . . . . . 10 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
3229, 31syl6ibr 151 . . . . . . . . 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 409 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd𝐵))
354, 34rexlimddv 2437 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵))
3635olcd 653 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
37 prnminu 6587 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
381, 37sylan 267 . . . . . . . 8 ((𝐴P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
3938adantlr 446 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
40 elprnqu 6580 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
411, 40sylan 267 . . . . . . . . . . . . 13 ((𝐴P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4241adantlr 446 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4342ad2ant2r 478 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣Q)
44 recrecnq 6492 . . . . . . . . . . 11 (𝑣Q → (*Q‘(*Q𝑣)) = 𝑣)
4543, 44syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) = 𝑣)
46 simprr 484 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 <Q (*Q𝑞))
4745, 46eqbrtrd 3784 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) <Q (*Q𝑞))
4817ad2antrr 457 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞Q𝑟Q))
4948simpld 105 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞Q)
50 recclnq 6490 . . . . . . . . . . 11 (𝑣Q → (*Q𝑣) ∈ Q)
5143, 50syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q𝑣) ∈ Q)
52 ltrnqg 6518 . . . . . . . . . 10 ((𝑞Q ∧ (*Q𝑣) ∈ Q) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5349, 51, 52syl2anc 391 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5447, 53mpbird 156 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 <Q (*Q𝑣))
55 simprl 483 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 ∈ (2nd𝐴))
5645, 55eqeltrd 2114 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) ∈ (2nd𝐴))
57 breq2 3768 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑣)))
58 fveq2 5178 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑣) → (*Q𝑦) = (*Q‘(*Q𝑣)))
5958eleq1d 2106 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)))
6057, 59anbi12d 442 . . . . . . . . . . 11 (𝑦 = (*Q𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴))))
6160spcegv 2641 . . . . . . . . . 10 ((*Q𝑣) ∈ Q → ((𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
6230recexprlemell 6720 . . . . . . . . . 10 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
6361, 62syl6ibr 151 . . . . . . . . 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 409 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 ∈ (1st𝐵))
6639, 65rexlimddv 2437 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵))
6766orcd 652 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
68 ltrnqi 6519 . . . . . 6 (𝑞 <Q 𝑟 → (*Q𝑟) <Q (*Q𝑞))
69 prloc 6589 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) <Q (*Q𝑞)) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
701, 68, 69syl2an 273 . . . . 5 ((𝐴P𝑞 <Q 𝑟) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
7136, 67, 70mpjaodan 711 . . . 4 ((𝐴P𝑞 <Q 𝑟) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
7271ex 108 . . 3 (𝐴P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7372ralrimivw 2393 . 2 (𝐴P → ∀𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7473ralrimivw 2393 1 (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766  Qcnq 6378  *Qcrq 6382
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