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Theorem recexpr 11089
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 5151 . . . . . 6 (𝑧 = 𝑤 → (𝑧 <Q 𝑦𝑤 <Q 𝑦))
21anbi1d 631 . . . . 5 (𝑧 = 𝑤 → ((𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ (𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
32exbidv 1919 . . . 4 (𝑧 = 𝑤 → (∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
43cbvabv 2810 . . 3 {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} = {𝑤 ∣ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
54reclem2pr 11086 . 2 (𝐴P → {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P)
64reclem4pr 11088 . 2 (𝐴P → (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P)
7 oveq2 7439 . . . 4 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → (𝐴 ·P 𝑥) = (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}))
87eqeq1d 2737 . . 3 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P))
98rspcev 3622 . 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 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068   class class class wbr 5148  cfv 6563  (class class class)co 7431  *Qcrq 10895   <Q cltq 10896  Pcnp 10897  1Pc1p 10898   ·P cmp 10900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-ltpq 10948  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955  df-ltnq 10956  df-np 11019  df-1p 11020  df-mp 11022
This theorem is referenced by:  recexsrlem  11141
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