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

Theorem recexprlemm 7086
Description: 𝐵 is inhabited. 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
recexprlemm (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
Distinct variable groups:   𝑟,𝑞,𝑥,𝑦,𝐴   𝐵,𝑞,𝑟,𝑥,𝑦

Proof of Theorem recexprlemm
StepHypRef Expression
1 prop 6937 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6940 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
3 recclnq 6854 . . . . . . 7 (𝑥Q → (*Q𝑥) ∈ Q)
4 nsmallnqq 6874 . . . . . . 7 ((*Q𝑥) ∈ Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
53, 4syl 14 . . . . . 6 (𝑥Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
65adantr 270 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 <Q (*Q𝑥))
7 recrecnq 6856 . . . . . . . . . . . 12 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
87eleq1d 2151 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (2nd𝐴) ↔ 𝑥 ∈ (2nd𝐴)))
98anbi2d 452 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴))))
10 breq2 3815 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑥)))
11 fveq2 5253 . . . . . . . . . . . . . 14 (𝑦 = (*Q𝑥) → (*Q𝑦) = (*Q‘(*Q𝑥)))
1211eleq1d 2151 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)))
1310, 12anbi12d 457 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴))))
1413spcegv 2697 . . . . . . . . . . 11 ((*Q𝑥) ∈ Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
153, 14syl 14 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
169, 15sylbird 168 . . . . . . . . 9 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))))
17 recexpr.1 . . . . . . . . . 10 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
1817recexprlemell 7084 . . . . . . . . 9 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
1916, 18syl6ibr 160 . . . . . . . 8 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
2019expcomd 1371 . . . . . . 7 (𝑥Q → (𝑥 ∈ (2nd𝐴) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵))))
2120imp 122 . . . . . 6 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵)))
2221reximdv 2468 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (∃𝑞Q 𝑞 <Q (*Q𝑥) → ∃𝑞Q 𝑞 ∈ (1st𝐵)))
236, 22mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
2423rexlimiva 2478 . . 3 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
251, 2, 243syl 17 . 2 (𝐴P → ∃𝑞Q 𝑞 ∈ (1st𝐵))
26 prml 6939 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
27 1nq 6828 . . . . . . . 8 1QQ
28 addclnq 6837 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → ((*Q𝑥) +Q 1Q) ∈ Q)
293, 27, 28sylancl 404 . . . . . . 7 (𝑥Q → ((*Q𝑥) +Q 1Q) ∈ Q)
30 ltaddnq 6869 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
313, 27, 30sylancl 404 . . . . . . 7 (𝑥Q → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
32 breq2 3815 . . . . . . . 8 (𝑟 = ((*Q𝑥) +Q 1Q) → ((*Q𝑥) <Q 𝑟 ↔ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)))
3332rspcev 2712 . . . . . . 7 ((((*Q𝑥) +Q 1Q) ∈ Q ∧ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3429, 31, 33syl2anc 403 . . . . . 6 (𝑥Q → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3534adantr 270 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
367eleq1d 2151 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (1st𝐴) ↔ 𝑥 ∈ (1st𝐴)))
3736anbi2d 452 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴))))
38 breq1 3814 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑦 <Q 𝑟 ↔ (*Q𝑥) <Q 𝑟))
3911eleq1d 2151 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (1st𝐴)))
4038, 39anbi12d 457 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴))))
4140spcegv 2697 . . . . . . . . . . 11 ((*Q𝑥) ∈ Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
423, 41syl 14 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
4337, 42sylbird 168 . . . . . . . . 9 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
4417recexprlemelu 7085 . . . . . . . . 9 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
4543, 44syl6ibr 160 . . . . . . . 8 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
4645expcomd 1371 . . . . . . 7 (𝑥Q → (𝑥 ∈ (1st𝐴) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵))))
4746imp 122 . . . . . 6 ((𝑥Q𝑥 ∈ (1st𝐴)) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵)))
4847reximdv 2468 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → (∃𝑟Q (*Q𝑥) <Q 𝑟 → ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
4935, 48mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5049rexlimiva 2478 . . 3 (∃𝑥Q 𝑥 ∈ (1st𝐴) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
511, 26, 503syl 17 . 2 (𝐴P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5225, 51jca 300 1 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742  1Qc1q 6743   +Q cplq 6744  *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-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  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