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Theorem axprecex 6954
 Description: Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 6994. In treatments which assume excluded middle, the 0 <ℝ 𝐴 condition is generally replaced by 𝐴 ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)
Assertion
Ref Expression
axprecex ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Distinct variable group:   𝑥,𝐴

Proof of Theorem axprecex
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 6905 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2312 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 173 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 breq2 3768 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐴))
5 oveq1 5519 . . . . . . 7 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2048 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76anbi2d 437 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
87rexbidv 2327 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
94, 8imbi12d 223 . . 3 (⟨𝑦, 0R⟩ = 𝐴 → ((0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)) ↔ (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))))
10 df-0 6896 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 3771 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
12 ltresr 6915 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1311, 12bitri 173 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
14 recexgt0sr 6858 . . . . 5 (0R <R 𝑦 → ∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15 opelreal 6904 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1615anbi1i 431 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
1710breq1i 3771 . . . . . . . . . . . . 13 (0 <𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑧, 0R⟩)
18 ltresr 6915 . . . . . . . . . . . . 13 (⟨0R, 0R⟩ <𝑧, 0R⟩ ↔ 0R <R 𝑧)
1917, 18bitri 173 . . . . . . . . . . . 12 (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧)
2019a1i 9 . . . . . . . . . . 11 ((𝑦R𝑧R) → (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧))
21 mulresr 6914 . . . . . . . . . . . . 13 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2221eqeq1d 2048 . . . . . . . . . . . 12 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23 df-1 6897 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2050 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2040 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 6836 . . . . . . . . . . . . . . 15 1RR
27 0r 6835 . . . . . . . . . . . . . . 15 0RR
28 opthg2 3976 . . . . . . . . . . . . . . 15 ((1RR ∧ 0RR) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
2926, 27, 28mp2an 402 . . . . . . . . . . . . . 14 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
3025, 29mpbiran2 848 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
3124, 30bitri 173 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
3222, 31syl6bb 185 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
3320, 32anbi12d 442 . . . . . . . . . 10 ((𝑦R𝑧R) → ((0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)))
3433pm5.32da 425 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
3516, 34syl5bb 181 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
36 breq2 3768 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (0 < 𝑥 ↔ 0 <𝑧, 0R⟩))
37 oveq2 5520 . . . . . . . . . . 11 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3837eqeq1d 2048 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3936, 38anbi12d 442 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
4039rspcev 2656 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))
4135, 40syl6bir 153 . . . . . . 7 (𝑦R → ((𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4241expd 245 . . . . . 6 (𝑦R → (𝑧R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
4342rexlimdv 2432 . . . . 5 (𝑦R → (∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4414, 43syl5 28 . . . 4 (𝑦R → (0R <R 𝑦 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4513, 44syl5bi 141 . . 3 (𝑦R → (0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
463, 9, 45gencl 2586 . 2 (𝐴 ∈ ℝ → (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
4746imp 115 1 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  Rcnr 6395  0Rc0r 6396  1Rc1r 6397   ·R cmr 6400
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