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Theorem recexpr 7469
 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.)
Assertion
Ref Expression
recexpr (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexpr
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3941 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑧 <Q 𝑤𝑢 <Q 𝑣))
2 simpr 109 . . . . . . . . 9 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑤 = 𝑣)
32fveq2d 5432 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → (*Q𝑤) = (*Q𝑣))
43eleq1d 2209 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (2nd𝐴) ↔ (*Q𝑣) ∈ (2nd𝐴)))
51, 4anbi12d 465 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
65cbvexdva 1902 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
76cbvabv 2265 . . . 4 {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}
8 simpl 108 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑧 = 𝑢)
92, 8breq12d 3949 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑤 <Q 𝑧𝑣 <Q 𝑢))
103eleq1d 2209 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (1st𝐴) ↔ (*Q𝑣) ∈ (1st𝐴)))
119, 10anbi12d 465 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1211cbvexdva 1902 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1312cbvabv 2265 . . . 4 {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}
147, 13opeq12i 3717 . . 3 ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ = ⟨{𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}, {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}⟩
1514recexprlempr 7463 . 2 (𝐴P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P)
1614recexprlemex 7468 . 2 (𝐴P → (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P)
17 oveq2 5789 . . . 4 (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ → (𝐴 ·P 𝑥) = (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩))
1817eqeq1d 2149 . . 3 (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P))
1918rspcev 2792 . 2 ((⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P ∧ (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P) → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
2015, 16, 19syl2anc 409 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
 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  *Qcrq 7115
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