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Theorem caucvgsrlemasr 6902
 Description: Lemma for caucvgsr 6914. 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 6851 . . . . . 6 <R ⊆ (R × R)
32brel 4417 . . . . 5 (𝐴 <R (𝐹𝑚) → (𝐴R ∧ (𝐹𝑚) ∈ R))
43simpld 109 . . . 4 (𝐴 <R (𝐹𝑚) → 𝐴R)
54ralimi 2399 . . 3 (∀𝑚N 𝐴 <R (𝐹𝑚) → ∀𝑚N 𝐴R)
61, 5syl 14 . 2 (𝜑 → ∀𝑚N 𝐴R)
7 1pi 6441 . . 3 1𝑜N
8 elex2 2585 . . 3 (1𝑜N → ∃𝑥 𝑥N)
9 r19.3rmv 3337 . . 3 (∃𝑥 𝑥N → (𝐴R ↔ ∀𝑚N 𝐴R))
107, 8, 9mp2b 8 . 2 (𝐴R ↔ ∀𝑚N 𝐴R)
116, 10sylibr 141 1 (𝜑𝐴R)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102  ∃wex 1395   ∈ wcel 1407  ∀wral 2321   class class class wbr 3789  ‘cfv 4927  1𝑜c1o 6022  Ncnpi 6398  Rcnr 6423
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