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Theorem recexprlemm 7130
Description: 𝐵 is inhabited. Lemma for recexpr 7144. (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 6981 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6984 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
3 recclnq 6898 . . . . . . 7 (𝑥Q → (*Q𝑥) ∈ Q)
4 nsmallnqq 6918 . . . . . . 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 6900 . . . . . . . . . . . 12 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
87eleq1d 2153 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (2nd𝐴) ↔ 𝑥 ∈ (2nd𝐴)))
98anbi2d 452 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴))))
10 breq2 3826 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑥)))
11 fveq2 5270 . . . . . . . . . . . . . 14 (𝑦 = (*Q𝑥) → (*Q𝑦) = (*Q‘(*Q𝑥)))
1211eleq1d 2153 . . . . . . . . . . . . 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 2700 . . . . . . . . . . 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 7128 . . . . . . . . 9 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
1916, 18syl6ibr 160 . . . . . . . 8 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
2019expcomd 1373 . . . . . . 7 (𝑥Q → (𝑥 ∈ (2nd𝐴) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵))))
2120imp 122 . . . . . 6 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵)))
2221reximdv 2470 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (∃𝑞Q 𝑞 <Q (*Q𝑥) → ∃𝑞Q 𝑞 ∈ (1st𝐵)))
236, 22mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
2423rexlimiva 2480 . . 3 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
251, 2, 243syl 17 . 2 (𝐴P → ∃𝑞Q 𝑞 ∈ (1st𝐵))
26 prml 6983 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
27 1nq 6872 . . . . . . . 8 1QQ
28 addclnq 6881 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → ((*Q𝑥) +Q 1Q) ∈ Q)
293, 27, 28sylancl 404 . . . . . . 7 (𝑥Q → ((*Q𝑥) +Q 1Q) ∈ Q)
30 ltaddnq 6913 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
313, 27, 30sylancl 404 . . . . . . 7 (𝑥Q → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
32 breq2 3826 . . . . . . . 8 (𝑟 = ((*Q𝑥) +Q 1Q) → ((*Q𝑥) <Q 𝑟 ↔ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)))
3332rspcev 2715 . . . . . . 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 2153 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (1st𝐴) ↔ 𝑥 ∈ (1st𝐴)))
3736anbi2d 452 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴))))
38 breq1 3825 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑦 <Q 𝑟 ↔ (*Q𝑥) <Q 𝑟))
3911eleq1d 2153 . . . . . . . . . . . . 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 2700 . . . . . . . . . . 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 7129 . . . . . . . . 9 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
4543, 44syl6ibr 160 . . . . . . . 8 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
4645expcomd 1373 . . . . . . 7 (𝑥Q → (𝑥 ∈ (1st𝐴) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵))))
4746imp 122 . . . . . 6 ((𝑥Q𝑥 ∈ (1st𝐴)) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵)))
4847reximdv 2470 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → (∃𝑟Q (*Q𝑥) <Q 𝑟 → ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
4935, 48mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5049rexlimiva 2480 . . 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 1287  wex 1424  wcel 1436  {cab 2071  wrex 2356  cop 3434   class class class wbr 3822  cfv 4983  (class class class)co 5615  1st c1st 5868  2nd c2nd 5869  Qcnq 6786  1Qc1q 6787   +Q cplq 6788  *Qcrq 6790   <Q cltq 6791  Pcnp 6797
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-eprel 4092  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-lti 6813  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858  df-ltnqqs 6859  df-inp 6972
This theorem is referenced by:  recexprlempr  7138
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