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Theorem recexsr 10112
 Description: The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
recexsr ((𝐴R𝐴 ≠ 0R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexsr
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sqgt0sr 10111 . 2 ((𝐴R𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴))
2 recexsrlem 10108 . . . 4 (0R <R (𝐴 ·R 𝐴) → ∃𝑦R ((𝐴 ·R 𝐴) ·R 𝑦) = 1R)
3 mulclsr 10089 . . . . . . 7 ((𝐴R𝑦R) → (𝐴 ·R 𝑦) ∈ R)
4 mulasssr 10095 . . . . . . . . 9 ((𝐴 ·R 𝐴) ·R 𝑦) = (𝐴 ·R (𝐴 ·R 𝑦))
54eqeq1i 2757 . . . . . . . 8 (((𝐴 ·R 𝐴) ·R 𝑦) = 1R ↔ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R)
6 oveq2 6813 . . . . . . . . . 10 (𝑥 = (𝐴 ·R 𝑦) → (𝐴 ·R 𝑥) = (𝐴 ·R (𝐴 ·R 𝑦)))
76eqeq1d 2754 . . . . . . . . 9 (𝑥 = (𝐴 ·R 𝑦) → ((𝐴 ·R 𝑥) = 1R ↔ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R))
87rspcev 3441 . . . . . . . 8 (((𝐴 ·R 𝑦) ∈ R ∧ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
95, 8sylan2b 493 . . . . . . 7 (((𝐴 ·R 𝑦) ∈ R ∧ ((𝐴 ·R 𝐴) ·R 𝑦) = 1R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
103, 9sylan 489 . . . . . 6 (((𝐴R𝑦R) ∧ ((𝐴 ·R 𝐴) ·R 𝑦) = 1R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
1110exp31 631 . . . . 5 (𝐴R → (𝑦R → (((𝐴 ·R 𝐴) ·R 𝑦) = 1R → ∃𝑥R (𝐴 ·R 𝑥) = 1R)))
1211rexlimdv 3160 . . . 4 (𝐴R → (∃𝑦R ((𝐴 ·R 𝐴) ·R 𝑦) = 1R → ∃𝑥R (𝐴 ·R 𝑥) = 1R))
132, 12syl5 34 . . 3 (𝐴R → (0R <R (𝐴 ·R 𝐴) → ∃𝑥R (𝐴 ·R 𝑥) = 1R))
1413imp 444 . 2 ((𝐴R ∧ 0R <R (𝐴 ·R 𝐴)) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
151, 14syldan 488 1 ((𝐴R𝐴 ≠ 0R) → ∃𝑥R (𝐴 ·R 𝑥) = 1R)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∃wrex 3043   class class class wbr 4796  (class class class)co 6805  Rcnr 9871  0Rc0r 9872  1Rc1r 9873   ·R cmr 9876
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