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Theorem recexprlemm 7374
Description: 𝐵 is inhabited. Lemma for recexpr 7388. (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 7225 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7228 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
3 recclnq 7142 . . . . . . 7 (𝑥Q → (*Q𝑥) ∈ Q)
4 nsmallnqq 7162 . . . . . . 7 ((*Q𝑥) ∈ Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
53, 4syl 14 . . . . . 6 (𝑥Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
65adantr 272 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 <Q (*Q𝑥))
7 recrecnq 7144 . . . . . . . . . . . 12 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
87eleq1d 2181 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (2nd𝐴) ↔ 𝑥 ∈ (2nd𝐴)))
98anbi2d 457 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴))))
10 breq2 3897 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑥)))
11 fveq2 5373 . . . . . . . . . . . . . 14 (𝑦 = (*Q𝑥) → (*Q𝑦) = (*Q‘(*Q𝑥)))
1211eleq1d 2181 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)))
1310, 12anbi12d 462 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴))))
1413spcegv 2743 . . . . . . . . . . 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 169 . . . . . . . . 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 7372 . . . . . . . . 9 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
1916, 18syl6ibr 161 . . . . . . . 8 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
2019expcomd 1398 . . . . . . 7 (𝑥Q → (𝑥 ∈ (2nd𝐴) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵))))
2120imp 123 . . . . . 6 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵)))
2221reximdv 2505 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (∃𝑞Q 𝑞 <Q (*Q𝑥) → ∃𝑞Q 𝑞 ∈ (1st𝐵)))
236, 22mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
2423rexlimiva 2516 . . 3 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
251, 2, 243syl 17 . 2 (𝐴P → ∃𝑞Q 𝑞 ∈ (1st𝐵))
26 prml 7227 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
27 1nq 7116 . . . . . . . 8 1QQ
28 addclnq 7125 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → ((*Q𝑥) +Q 1Q) ∈ Q)
293, 27, 28sylancl 407 . . . . . . 7 (𝑥Q → ((*Q𝑥) +Q 1Q) ∈ Q)
30 ltaddnq 7157 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
313, 27, 30sylancl 407 . . . . . . 7 (𝑥Q → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
32 breq2 3897 . . . . . . . 8 (𝑟 = ((*Q𝑥) +Q 1Q) → ((*Q𝑥) <Q 𝑟 ↔ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)))
3332rspcev 2758 . . . . . . 7 ((((*Q𝑥) +Q 1Q) ∈ Q ∧ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3429, 31, 33syl2anc 406 . . . . . 6 (𝑥Q → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3534adantr 272 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
367eleq1d 2181 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (1st𝐴) ↔ 𝑥 ∈ (1st𝐴)))
3736anbi2d 457 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴))))
38 breq1 3896 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑦 <Q 𝑟 ↔ (*Q𝑥) <Q 𝑟))
3911eleq1d 2181 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (1st𝐴)))
4038, 39anbi12d 462 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴))))
4140spcegv 2743 . . . . . . . . . . 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 169 . . . . . . . . 9 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
4417recexprlemelu 7373 . . . . . . . . 9 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
4543, 44syl6ibr 161 . . . . . . . 8 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
4645expcomd 1398 . . . . . . 7 (𝑥Q → (𝑥 ∈ (1st𝐴) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵))))
4746imp 123 . . . . . 6 ((𝑥Q𝑥 ∈ (1st𝐴)) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵)))
4847reximdv 2505 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → (∃𝑟Q (*Q𝑥) <Q 𝑟 → ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
4935, 48mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5049rexlimiva 2516 . . 3 (∃𝑥Q 𝑥 ∈ (1st𝐴) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
511, 26, 503syl 17 . 2 (𝐴P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5225, 51jca 302 1 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wex 1449  wcel 1461  {cab 2099  wrex 2389  cop 3494   class class class wbr 3893  cfv 5079  (class class class)co 5726  1st c1st 5988  2nd c2nd 5989  Qcnq 7030  1Qc1q 7031   +Q cplq 7032  *Qcrq 7034   <Q cltq 7035  Pcnp 7041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-inp 7216
This theorem is referenced by:  recexprlempr  7382
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