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Theorem recexpr 10807
Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
recexpr (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexpr
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5077 . . . . . 6 (𝑧 = 𝑤 → (𝑧 <Q 𝑦𝑤 <Q 𝑦))
21anbi1d 630 . . . . 5 (𝑧 = 𝑤 → ((𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ (𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
32exbidv 1924 . . . 4 (𝑧 = 𝑤 → (∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
43cbvabv 2811 . . 3 {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} = {𝑤 ∣ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
54reclem2pr 10804 . 2 (𝐴P → {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P)
64reclem4pr 10806 . 2 (𝐴P → (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P)
7 oveq2 7283 . . . 4 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → (𝐴 ·P 𝑥) = (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}))
87eqeq1d 2740 . . 3 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P))
98rspcev 3561 . 2 (({𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P ∧ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P) → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
105, 6, 9syl2anc 584 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wrex 3065   class class class wbr 5074  cfv 6433  (class class class)co 7275  *Qcrq 10613   <Q cltq 10614  Pcnp 10615  1Pc1p 10616   ·P cmp 10618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-ni 10628  df-pli 10629  df-mi 10630  df-lti 10631  df-plpq 10664  df-mpq 10665  df-ltpq 10666  df-enq 10667  df-nq 10668  df-erq 10669  df-plq 10670  df-mq 10671  df-1nq 10672  df-rq 10673  df-ltnq 10674  df-np 10737  df-1p 10738  df-mp 10740
This theorem is referenced by:  recexsrlem  10859
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