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

Theorem recexprlempr 7662
Description: 𝐵 is a positive real. Lemma for recexpr 7668. (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
recexprlempr (𝐴P𝐵P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlempr
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 recexpr.1 . . . 4 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlemm 7654 . . 3 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
3 ltrelnq 7395 . . . . . . . . . . 11 <Q ⊆ (Q × Q)
43brel 4696 . . . . . . . . . 10 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
54simpld 112 . . . . . . . . 9 (𝑥 <Q 𝑦𝑥Q)
65adantr 276 . . . . . . . 8 ((𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
76exlimiv 1609 . . . . . . 7 (∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) → 𝑥Q)
87abssi 3245 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q
9 nqex 7393 . . . . . . 7 Q ∈ V
109elpw2 4175 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ⊆ Q)
118, 10mpbir 146 . . . . 5 {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ 𝒫 Q
123brel 4696 . . . . . . . . . 10 (𝑦 <Q 𝑥 → (𝑦Q𝑥Q))
1312simprd 114 . . . . . . . . 9 (𝑦 <Q 𝑥𝑥Q)
1413adantr 276 . . . . . . . 8 ((𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
1514exlimiv 1609 . . . . . . 7 (∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴)) → 𝑥Q)
1615abssi 3245 . . . . . 6 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q
179elpw2 4175 . . . . . 6 ({𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ 𝒫 Q ↔ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ⊆ Q)
1816, 17mpbir 146 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ 𝒫 Q
19 opelxpi 4676 . . . . 5 (({𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))} ∈ 𝒫 Q ∧ {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))} ∈ 𝒫 Q) → ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q))
2011, 18, 19mp2an 426 . . . 4 ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩ ∈ (𝒫 Q × 𝒫 Q)
211, 20eqeltri 2262 . . 3 𝐵 ∈ (𝒫 Q × 𝒫 Q)
222, 21jctil 312 . 2 (𝐴P → (𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵))))
231recexprlemrnd 7659 . . 3 (𝐴P → (∀𝑞Q (𝑞 ∈ (1st𝐵) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))))
241recexprlemdisj 7660 . . 3 (𝐴P → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)))
251recexprlemloc 7661 . . 3 (𝐴P → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))
2623, 24, 253jca 1179 . 2 (𝐴P → ((∀𝑞Q (𝑞 ∈ (1st𝐵) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵)))))
27 elnp1st2nd 7506 . 2 (𝐵P ↔ ((𝐵 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐵) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐵))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐵) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐵)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐵) ∧ 𝑞 ∈ (2nd𝐵)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐵) ∨ 𝑟 ∈ (2nd𝐵))))))
2822, 26, 27sylanbrc 417 1 (𝐴P𝐵P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  w3a 980   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wral 2468  wrex 2469  wss 3144  𝒫 cpw 3590  cop 3610   class class class wbr 4018   × cxp 4642  cfv 5235  1st c1st 6164  2nd c2nd 6165  Qcnq 7310  *Qcrq 7314   <Q cltq 7315  Pcnp 7321
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-inp 7496
This theorem is referenced by:  recexprlem1ssl  7663  recexprlem1ssu  7664  recexprlemss1l  7665  recexprlemss1u  7666  recexprlemex  7667  recexpr  7668
  Copyright terms: Public domain W3C validator