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Theorem gt0srpr 6833
 Description: Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
Assertion
Ref Expression
gt0srpr (0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴)

Proof of Theorem gt0srpr
StepHypRef Expression
1 enrer 6820 . . . . 5 ~R Er (P × P)
2 erdm 6116 . . . . 5 ( ~R Er (P × P) → dom ~R = (P × P))
31, 2ax-mp 7 . . . 4 dom ~R = (P × P)
4 ltrelsr 6823 . . . . . . 7 <R ⊆ (R × R)
54brel 4392 . . . . . 6 (0R <R [⟨𝐴, 𝐵⟩] ~R → (0RR ∧ [⟨𝐴, 𝐵⟩] ~RR))
65simprd 107 . . . . 5 (0R <R [⟨𝐴, 𝐵⟩] ~R → [⟨𝐴, 𝐵⟩] ~RR)
7 df-nr 6812 . . . . 5 R = ((P × P) / ~R )
86, 7syl6eleq 2130 . . . 4 (0R <R [⟨𝐴, 𝐵⟩] ~R → [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R ))
9 ecelqsdm 6176 . . . 4 ((dom ~R = (P × P) ∧ [⟨𝐴, 𝐵⟩] ~R ∈ ((P × P) / ~R )) → ⟨𝐴, 𝐵⟩ ∈ (P × P))
103, 8, 9sylancr 393 . . 3 (0R <R [⟨𝐴, 𝐵⟩] ~R → ⟨𝐴, 𝐵⟩ ∈ (P × P))
11 opelxp 4374 . . 3 (⟨𝐴, 𝐵⟩ ∈ (P × P) ↔ (𝐴P𝐵P))
1210, 11sylib 127 . 2 (0R <R [⟨𝐴, 𝐵⟩] ~R → (𝐴P𝐵P))
13 ltrelpr 6603 . . . 4 <P ⊆ (P × P)
1413brel 4392 . . 3 (𝐵<P 𝐴 → (𝐵P𝐴P))
1514ancomd 254 . 2 (𝐵<P 𝐴 → (𝐴P𝐵P))
16 df-0r 6816 . . . . 5 0R = [⟨1P, 1P⟩] ~R
1716breq1i 3771 . . . 4 (0R <R [⟨𝐴, 𝐵⟩] ~R ↔ [⟨1P, 1P⟩] ~R <R [⟨𝐴, 𝐵⟩] ~R )
18 1pr 6652 . . . . 5 1PP
19 ltsrprg 6832 . . . . 5 (((1PP ∧ 1PP) ∧ (𝐴P𝐵P)) → ([⟨1P, 1P⟩] ~R <R [⟨𝐴, 𝐵⟩] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
2018, 18, 19mpanl12 412 . . . 4 ((𝐴P𝐵P) → ([⟨1P, 1P⟩] ~R <R [⟨𝐴, 𝐵⟩] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
2117, 20syl5bb 181 . . 3 ((𝐴P𝐵P) → (0R <R [⟨𝐴, 𝐵⟩] ~R ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
22 ltaprg 6717 . . . . 5 ((𝐵P𝐴P ∧ 1PP) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
2318, 22mp3an3 1221 . . . 4 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
2423ancoms 255 . . 3 ((𝐴P𝐵P) → (𝐵<P 𝐴 ↔ (1P +P 𝐵)<P (1P +P 𝐴)))
2521, 24bitr4d 180 . 2 ((𝐴P𝐵P) → (0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴))
2612, 15, 25pm5.21nii 620 1 (0R <R [⟨𝐴, 𝐵⟩] ~R𝐵<P 𝐴)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764   × cxp 4343  dom cdm 4345  (class class class)co 5512   Er wer 6103  [cec 6104   / cqs 6105  Pcnp 6389  1Pc1p 6390   +P cpp 6391
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