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Theorem gt0srpr 6656
 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 6643 . . . . 5 ~R Er (P × P)
2 erdm 6052 . . . . 5 ( ~R Er (P × P) → dom ~R = (P × P))
31, 2ax-mp 7 . . . 4 dom ~R = (P × P)
4 ltrelsr 6646 . . . . . . 7 <R ⊆ (R × R)
54brel 4335 . . . . . 6 (0R <R [⟨A, B⟩] ~R → (0R R [⟨A, B⟩] ~R R))
65simprd 107 . . . . 5 (0R <R [⟨A, B⟩] ~R → [⟨A, B⟩] ~R R)
7 df-nr 6635 . . . . 5 R = ((P × P) / ~R )
86, 7syl6eleq 2127 . . . 4 (0R <R [⟨A, B⟩] ~R → [⟨A, B⟩] ~R ((P × P) / ~R ))
9 ecelqsdm 6112 . . . 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 4317 . . 3 (⟨A, B (P × P) ↔ (A P B P))
1210, 11sylib 127 . 2 (0R <R [⟨A, B⟩] ~R → (A P B P))
13 ltrelpr 6487 . . . 4 <P ⊆ (P × P)
1413brel 4335 . . 3 (B<P A → (B P A P))
1514ancomd 254 . 2 (B<P A → (A P B P))
16 df-0r 6639 . . . . 5 0R = [⟨1P, 1P⟩] ~R
1716breq1i 3762 . . . 4 (0R <R [⟨A, B⟩] ~R ↔ [⟨1P, 1P⟩] ~R <R [⟨A, B⟩] ~R )
18 1pr 6534 . . . . 5 1P P
19 ltsrprg 6655 . . . . 5 (((1P P 1P P) (A P B P)) → ([⟨1P, 1P⟩] ~R <R [⟨A, B⟩] ~R ↔ (1P +P B)<P (1P +P A)))
2018, 18, 19mpanl12 412 . . . 4 ((A P B P) → ([⟨1P, 1P⟩] ~R <R [⟨A, B⟩] ~R ↔ (1P +P B)<P (1P +P A)))
2117, 20syl5bb 181 . . 3 ((A P B P) → (0R <R [⟨A, B⟩] ~R ↔ (1P +P B)<P (1P +P A)))
22 ltaprg 6591 . . . . 5 ((B P A P 1P P) → (B<P A ↔ (1P +P B)<P (1P +P A)))
2318, 22mp3an3 1220 . . . 4 ((B P A P) → (B<P A ↔ (1P +P B)<P (1P +P A)))
2423ancoms 255 . . 3 ((A P B P) → (B<P A ↔ (1P +P B)<P (1P +P A)))
2521, 24bitr4d 180 . 2 ((A P B P) → (0R <R [⟨A, B⟩] ~RB<P A))
2612, 15, 25pm5.21nii 619 1 (0R <R [⟨A, B⟩] ~RB<P A)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755   × cxp 4286  dom cdm 4288  (class class class)co 5455   Er wer 6039  [cec 6040   / cqs 6041  Pcnp 6275  1Pc1p 6276   +P cpp 6277
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