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Theorem caucvgsrlemasr 7610
 Description: Lemma for caucvgsr 7622. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
Hypothesis
Ref Expression
caucvgsrlemasr.bnd (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))
Assertion
Ref Expression
caucvgsrlemasr (𝜑𝐴R)
Distinct variable group:   𝐴,𝑚
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑚)

Proof of Theorem caucvgsrlemasr
StepHypRef Expression
1 caucvgsrlemasr.bnd . . 3 (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))
2 ltrelsr 7558 . . . . . 6 <R ⊆ (R × R)
32brel 4591 . . . . 5 (𝐴 <R (𝐹𝑚) → (𝐴R ∧ (𝐹𝑚) ∈ R))
43simpld 111 . . . 4 (𝐴 <R (𝐹𝑚) → 𝐴R)
54ralimi 2495 . . 3 (∀𝑚N 𝐴 <R (𝐹𝑚) → ∀𝑚N 𝐴R)
61, 5syl 14 . 2 (𝜑 → ∀𝑚N 𝐴R)
7 1pi 7135 . . 3 1oN
8 elex2 2702 . . 3 (1oN → ∃𝑥 𝑥N)
9 r19.3rmv 3453 . . 3 (∃𝑥 𝑥N → (𝐴R ↔ ∀𝑚N 𝐴R))
107, 8, 9mp2b 8 . 2 (𝐴R ↔ ∀𝑚N 𝐴R)
116, 10sylibr 133 1 (𝜑𝐴R)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∃wex 1468   ∈ wcel 1480  ∀wral 2416   class class class wbr 3929  ‘cfv 5123  1oc1o 6306  Ncnpi 7092  Rcnr 7117
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