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Theorem recexprlemm 6750
Description: 𝐵 is inhabited. Lemma for recexpr 6764. (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 6601 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6604 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
3 recclnq 6518 . . . . . . 7 (𝑥Q → (*Q𝑥) ∈ Q)
4 nsmallnqq 6538 . . . . . . 7 ((*Q𝑥) ∈ Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
53, 4syl 14 . . . . . 6 (𝑥Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
65adantr 265 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 <Q (*Q𝑥))
7 recrecnq 6520 . . . . . . . . . . . 12 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
87eleq1d 2120 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (2nd𝐴) ↔ 𝑥 ∈ (2nd𝐴)))
98anbi2d 445 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴))))
10 breq2 3793 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑥)))
11 fveq2 5203 . . . . . . . . . . . . . 14 (𝑦 = (*Q𝑥) → (*Q𝑦) = (*Q‘(*Q𝑥)))
1211eleq1d 2120 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)))
1310, 12anbi12d 450 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴))))
1413spcegv 2656 . . . . . . . . . . 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 163 . . . . . . . . 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 6748 . . . . . . . . 9 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
1916, 18syl6ibr 155 . . . . . . . 8 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
2019expcomd 1344 . . . . . . 7 (𝑥Q → (𝑥 ∈ (2nd𝐴) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵))))
2120imp 119 . . . . . 6 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵)))
2221reximdv 2435 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (∃𝑞Q 𝑞 <Q (*Q𝑥) → ∃𝑞Q 𝑞 ∈ (1st𝐵)))
236, 22mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
2423rexlimiva 2443 . . 3 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
251, 2, 243syl 17 . 2 (𝐴P → ∃𝑞Q 𝑞 ∈ (1st𝐵))
26 prml 6603 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
27 1nq 6492 . . . . . . . 8 1QQ
28 addclnq 6501 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → ((*Q𝑥) +Q 1Q) ∈ Q)
293, 27, 28sylancl 398 . . . . . . 7 (𝑥Q → ((*Q𝑥) +Q 1Q) ∈ Q)
30 ltaddnq 6533 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
313, 27, 30sylancl 398 . . . . . . 7 (𝑥Q → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
32 breq2 3793 . . . . . . . 8 (𝑟 = ((*Q𝑥) +Q 1Q) → ((*Q𝑥) <Q 𝑟 ↔ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)))
3332rspcev 2671 . . . . . . 7 ((((*Q𝑥) +Q 1Q) ∈ Q ∧ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3429, 31, 33syl2anc 397 . . . . . 6 (𝑥Q → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3534adantr 265 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
367eleq1d 2120 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (1st𝐴) ↔ 𝑥 ∈ (1st𝐴)))
3736anbi2d 445 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴))))
38 breq1 3792 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑦 <Q 𝑟 ↔ (*Q𝑥) <Q 𝑟))
3911eleq1d 2120 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (1st𝐴)))
4038, 39anbi12d 450 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴))))
4140spcegv 2656 . . . . . . . . . . 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 163 . . . . . . . . 9 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴))))
4417recexprlemelu 6749 . . . . . . . . 9 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
4543, 44syl6ibr 155 . . . . . . . 8 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
4645expcomd 1344 . . . . . . 7 (𝑥Q → (𝑥 ∈ (1st𝐴) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵))))
4746imp 119 . . . . . 6 ((𝑥Q𝑥 ∈ (1st𝐴)) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵)))
4847reximdv 2435 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → (∃𝑟Q (*Q𝑥) <Q 𝑟 → ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
4935, 48mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5049rexlimiva 2443 . . 3 (∃𝑥Q 𝑥 ∈ (1st𝐴) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
511, 26, 503syl 17 . 2 (𝐴P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5225, 51jca 294 1 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wex 1395  wcel 1407  {cab 2040  wrex 2322  cop 3403   class class class wbr 3789  cfv 4927  (class class class)co 5537  1st c1st 5790  2nd c2nd 5791  Qcnq 6406  1Qc1q 6407   +Q cplq 6408  *Qcrq 6410   <Q cltq 6411  Pcnp 6417
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 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-ral 2326  df-rex 2327  df-reu 2328  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-inp 6592
This theorem is referenced by:  recexprlempr  6758
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