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Theorem recexpr 9752
 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 4586 . . . . . 6 (𝑧 = 𝑤 → (𝑧 <Q 𝑦𝑤 <Q 𝑦))
21anbi1d 737 . . . . 5 (𝑧 = 𝑤 → ((𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ (𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
32exbidv 1837 . . . 4 (𝑧 = 𝑤 → (∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) ↔ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)))
43cbvabv 2734 . . 3 {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} = {𝑤 ∣ ∃𝑦(𝑤 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
54reclem2pr 9749 . 2 (𝐴P → {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} ∈ P)
64reclem4pr 9751 . 2 (𝐴P → (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P)
7 oveq2 6557 . . . 4 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → (𝐴 ·P 𝑥) = (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}))
87eqeq1d 2612 . . 3 (𝑥 = {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)} → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P {𝑧 ∣ ∃𝑦(𝑧 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}) = 1P))
98rspcev 3282 . 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 691 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  *Qcrq 9558
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