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Theorem recexprlemm 7455
Description: 𝐵 is inhabited. Lemma for recexpr 7469. (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 7306 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 7309 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (2nd𝐴))
3 recclnq 7223 . . . . . . 7 (𝑥Q → (*Q𝑥) ∈ Q)
4 nsmallnqq 7243 . . . . . . 7 ((*Q𝑥) ∈ Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
53, 4syl 14 . . . . . 6 (𝑥Q → ∃𝑞Q 𝑞 <Q (*Q𝑥))
65adantr 274 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 <Q (*Q𝑥))
7 recrecnq 7225 . . . . . . . . . . . 12 (𝑥Q → (*Q‘(*Q𝑥)) = 𝑥)
87eleq1d 2209 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (2nd𝐴) ↔ 𝑥 ∈ (2nd𝐴)))
98anbi2d 460 . . . . . . . . . 10 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴))))
10 breq2 3940 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑞 <Q 𝑦𝑞 <Q (*Q𝑥)))
11 fveq2 5428 . . . . . . . . . . . . . 14 (𝑦 = (*Q𝑥) → (*Q𝑦) = (*Q‘(*Q𝑥)))
1211eleq1d 2209 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (2nd𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (2nd𝐴)))
1310, 12anbi12d 465 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)) ↔ (𝑞 <Q (*Q𝑥) ∧ (*Q‘(*Q𝑥)) ∈ (2nd𝐴))))
1413spcegv 2777 . . . . . . . . . . 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 7453 . . . . . . . . 9 (𝑞 ∈ (1st𝐵) ↔ ∃𝑦(𝑞 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴)))
1916, 18syl6ibr 161 . . . . . . . 8 (𝑥Q → ((𝑞 <Q (*Q𝑥) ∧ 𝑥 ∈ (2nd𝐴)) → 𝑞 ∈ (1st𝐵)))
2019expcomd 1418 . . . . . . 7 (𝑥Q → (𝑥 ∈ (2nd𝐴) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵))))
2120imp 123 . . . . . 6 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (𝑞 <Q (*Q𝑥) → 𝑞 ∈ (1st𝐵)))
2221reximdv 2536 . . . . 5 ((𝑥Q𝑥 ∈ (2nd𝐴)) → (∃𝑞Q 𝑞 <Q (*Q𝑥) → ∃𝑞Q 𝑞 ∈ (1st𝐵)))
236, 22mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (2nd𝐴)) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
2423rexlimiva 2547 . . 3 (∃𝑥Q 𝑥 ∈ (2nd𝐴) → ∃𝑞Q 𝑞 ∈ (1st𝐵))
251, 2, 243syl 17 . 2 (𝐴P → ∃𝑞Q 𝑞 ∈ (1st𝐵))
26 prml 7308 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑥Q 𝑥 ∈ (1st𝐴))
27 1nq 7197 . . . . . . . 8 1QQ
28 addclnq 7206 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → ((*Q𝑥) +Q 1Q) ∈ Q)
293, 27, 28sylancl 410 . . . . . . 7 (𝑥Q → ((*Q𝑥) +Q 1Q) ∈ Q)
30 ltaddnq 7238 . . . . . . . 8 (((*Q𝑥) ∈ Q ∧ 1QQ) → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
313, 27, 30sylancl 410 . . . . . . 7 (𝑥Q → (*Q𝑥) <Q ((*Q𝑥) +Q 1Q))
32 breq2 3940 . . . . . . . 8 (𝑟 = ((*Q𝑥) +Q 1Q) → ((*Q𝑥) <Q 𝑟 ↔ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)))
3332rspcev 2792 . . . . . . 7 ((((*Q𝑥) +Q 1Q) ∈ Q ∧ (*Q𝑥) <Q ((*Q𝑥) +Q 1Q)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3429, 31, 33syl2anc 409 . . . . . 6 (𝑥Q → ∃𝑟Q (*Q𝑥) <Q 𝑟)
3534adantr 274 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q (*Q𝑥) <Q 𝑟)
367eleq1d 2209 . . . . . . . . . . 11 (𝑥Q → ((*Q‘(*Q𝑥)) ∈ (1st𝐴) ↔ 𝑥 ∈ (1st𝐴)))
3736anbi2d 460 . . . . . . . . . 10 (𝑥Q → (((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴))))
38 breq1 3939 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → (𝑦 <Q 𝑟 ↔ (*Q𝑥) <Q 𝑟))
3911eleq1d 2209 . . . . . . . . . . . . 13 (𝑦 = (*Q𝑥) → ((*Q𝑦) ∈ (1st𝐴) ↔ (*Q‘(*Q𝑥)) ∈ (1st𝐴)))
4038, 39anbi12d 465 . . . . . . . . . . . 12 (𝑦 = (*Q𝑥) → ((𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)) ↔ ((*Q𝑥) <Q 𝑟 ∧ (*Q‘(*Q𝑥)) ∈ (1st𝐴))))
4140spcegv 2777 . . . . . . . . . . 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 7454 . . . . . . . . 9 (𝑟 ∈ (2nd𝐵) ↔ ∃𝑦(𝑦 <Q 𝑟 ∧ (*Q𝑦) ∈ (1st𝐴)))
4543, 44syl6ibr 161 . . . . . . . 8 (𝑥Q → (((*Q𝑥) <Q 𝑟𝑥 ∈ (1st𝐴)) → 𝑟 ∈ (2nd𝐵)))
4645expcomd 1418 . . . . . . 7 (𝑥Q → (𝑥 ∈ (1st𝐴) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵))))
4746imp 123 . . . . . 6 ((𝑥Q𝑥 ∈ (1st𝐴)) → ((*Q𝑥) <Q 𝑟𝑟 ∈ (2nd𝐵)))
4847reximdv 2536 . . . . 5 ((𝑥Q𝑥 ∈ (1st𝐴)) → (∃𝑟Q (*Q𝑥) <Q 𝑟 → ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
4935, 48mpd 13 . . . 4 ((𝑥Q𝑥 ∈ (1st𝐴)) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5049rexlimiva 2547 . . 3 (∃𝑥Q 𝑥 ∈ (1st𝐴) → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
511, 26, 503syl 17 . 2 (𝐴P → ∃𝑟Q 𝑟 ∈ (2nd𝐵))
5225, 51jca 304 1 (𝐴P → (∃𝑞Q 𝑞 ∈ (1st𝐵) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  {cab 2126  wrex 2418  cop 3534   class class class wbr 3936  cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  Qcnq 7111  1Qc1q 7112   +Q cplq 7113  *Qcrq 7115   <Q cltq 7116  Pcnp 7122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-inp 7297
This theorem is referenced by:  recexprlempr  7463
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