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Theorem recexprlemloc 6786
Description: 𝐵 is located. Lemma for recexpr 6793. (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 6630 . . . . . . . . 9 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prnmaxl 6643 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
31, 2sylan 271 . . . . . . . 8 ((𝐴P ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
43adantlr 454 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → ∃𝑢 ∈ (1st𝐴)(*Q𝑟) <Q 𝑢)
5 simprr 492 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q 𝑢)
6 elprnql 6636 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (1st𝐴)) → 𝑢Q)
71, 6sylan 271 . . . . . . . . . . . . 13 ((𝐴P𝑢 ∈ (1st𝐴)) → 𝑢Q)
87ad2ant2r 486 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
98adantlr 454 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢Q)
10 recrecnq 6549 . . . . . . . . . . 11 (𝑢Q → (*Q‘(*Q𝑢)) = 𝑢)
119, 10syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) = 𝑢)
125, 11breqtrrd 3817 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑟) <Q (*Q‘(*Q𝑢)))
13 recclnq 6547 . . . . . . . . . . 11 (𝑢Q → (*Q𝑢) ∈ Q)
149, 13syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) ∈ Q)
15 ltrelnq 6520 . . . . . . . . . . . . . 14 <Q ⊆ (Q × Q)
1615brel 4419 . . . . . . . . . . . . 13 (𝑞 <Q 𝑟 → (𝑞Q𝑟Q))
1716adantl 266 . . . . . . . . . . . 12 ((𝐴P𝑞 <Q 𝑟) → (𝑞Q𝑟Q))
1817ad2antrr 465 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (𝑞Q𝑟Q))
1918simprd 111 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟Q)
20 ltrnqg 6575 . . . . . . . . . 10 (((*Q𝑢) ∈ Q𝑟Q) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2114, 19, 20syl2anc 397 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → ((*Q𝑢) <Q 𝑟 ↔ (*Q𝑟) <Q (*Q‘(*Q𝑢))))
2212, 21mpbird 160 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q𝑢) <Q 𝑟)
23 simprl 491 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑢 ∈ (1st𝐴))
2411, 23eqeltrd 2130 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → (*Q‘(*Q𝑢)) ∈ (1st𝐴))
25 breq1 3794 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → (𝑦 <Q 𝑟 ↔ (*Q𝑢) <Q 𝑟))
26 fveq2 5205 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑢) → (*Q𝑦) = (*Q‘(*Q𝑢)))
2726eleq1d 2122 . . . . . . . . . . . 12 (𝑦 = (*Q𝑢) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑢)) ∈ (1st𝐴)))
2825, 27anbi12d 450 . . . . . . . . . . 11 (𝑦 = (*Q𝑢) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑢) <Q 𝑟 ∧ (*Q‘(*Q𝑢)) ∈ (1st𝐴))))
2928spcegv 2658 . . . . . . . . . 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 6778 . . . . . . . . . 10 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
3229, 31syl6ibr 155 . . . . . . . . 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 417 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) ∧ (𝑢 ∈ (1st𝐴) ∧ (*Q𝑟) <Q 𝑢)) → 𝑟 ∈ (2nd𝐵))
354, 34rexlimddv 2454 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵))
3635olcd 663 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑟) ∈ (1st𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
37 prnminu 6644 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
381, 37sylan 271 . . . . . . . 8 ((𝐴P ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
3938adantlr 454 . . . . . . 7 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → ∃𝑣 ∈ (2nd𝐴)𝑣 <Q (*Q𝑞))
40 elprnqu 6637 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
411, 40sylan 271 . . . . . . . . . . . . 13 ((𝐴P𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4241adantlr 454 . . . . . . . . . . . 12 (((𝐴P𝑞 <Q 𝑟) ∧ 𝑣 ∈ (2nd𝐴)) → 𝑣Q)
4342ad2ant2r 486 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣Q)
44 recrecnq 6549 . . . . . . . . . . 11 (𝑣Q → (*Q‘(*Q𝑣)) = 𝑣)
4543, 44syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) = 𝑣)
46 simprr 492 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 <Q (*Q𝑞))
4745, 46eqbrtrd 3811 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) <Q (*Q𝑞))
4817ad2antrr 465 . . . . . . . . . . 11 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞Q𝑟Q))
4948simpld 109 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞Q)
50 recclnq 6547 . . . . . . . . . . 11 (𝑣Q → (*Q𝑣) ∈ Q)
5143, 50syl 14 . . . . . . . . . 10 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q𝑣) ∈ Q)
52 ltrnqg 6575 . . . . . . . . . 10 ((𝑞Q ∧ (*Q𝑣) ∈ Q) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5349, 51, 52syl2anc 397 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (𝑞 <Q (*Q𝑣) ↔ (*Q‘(*Q𝑣)) <Q (*Q𝑞)))
5447, 53mpbird 160 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 <Q (*Q𝑣))
55 simprl 491 . . . . . . . . 9 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑣 ∈ (2nd𝐴))
5645, 55eqeltrd 2130 . . . . . . . 8 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → (*Q‘(*Q𝑣)) ∈ (2nd𝐴))
57 breq2 3795 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑣)))
58 fveq2 5205 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑣) → (*Q𝑦) = (*Q‘(*Q𝑣)))
5958eleq1d 2122 . . . . . . . . . . . 12 (𝑦 = (*Q𝑣) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)))
6057, 59anbi12d 450 . . . . . . . . . . 11 (𝑦 = (*Q𝑣) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴))))
6160spcegv 2658 . . . . . . . . . 10 ((*Q𝑣) ∈ Q → ((𝑞 <Q (*Q𝑣) ∧ (*Q‘(*Q𝑣)) ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
6230recexprlemell 6777 . . . . . . . . . 10 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
6361, 62syl6ibr 155 . . . . . . . . 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 417 . . . . . . 7 ((((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) ∧ (𝑣 ∈ (2nd𝐴) ∧ 𝑣 <Q (*Q𝑞))) → 𝑞 ∈ (1st𝐵))
6639, 65rexlimddv 2454 . . . . . 6 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵))
6766orcd 662 . . . . 5 (((𝐴P𝑞 <Q 𝑟) ∧ (*Q𝑞) ∈ (2nd𝐴)) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
68 ltrnqi 6576 . . . . . 6 (𝑞 <Q 𝑟 → (*Q𝑟) <Q (*Q𝑞))
69 prloc 6646 . . . . . 6 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P ∧ (*Q𝑟) <Q (*Q𝑞)) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
701, 68, 69syl2an 277 . . . . 5 ((𝐴P𝑞 <Q 𝑟) → ((*Q𝑟) ∈ (1st𝐴) ∨ (*Q𝑞) ∈ (2nd𝐴)))
7136, 67, 70mpjaodan 722 . . . 4 ((𝐴P𝑞 <Q 𝑟) → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))
7271ex 112 . . 3 (𝐴P → (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7372ralrimivw 2410 . 2 (𝐴P → ∀𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
7473ralrimivw 2410 1 (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wral 2323  wrex 2324  cop 3405   class class class wbr 3791  cfv 4929  1st c1st 5792  2nd c2nd 5793  Qcnq 6435  *Qcrq 6439   <Q cltq 6440  Pcnp 6446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-lti 6462  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621
This theorem is referenced by:  recexprlempr  6787
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