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Theorem ltpsrpr 9809
 Description: Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ltpsrpr.3 𝐶R
Assertion
Ref Expression
ltpsrpr ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵)

Proof of Theorem ltpsrpr
StepHypRef Expression
1 ltpsrpr.3 . . 3 𝐶R
2 ltasr 9800 . . 3 (𝐶R → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R )))
31, 2ax-mp 5 . 2 ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ))
4 addcompr 9722 . . . 4 (𝐴 +P 1P) = (1P +P 𝐴)
54breq1i 4590 . . 3 ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵))
6 ltsrpr 9777 . . 3 ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵))
7 1pr 9716 . . . 4 1PP
8 ltapr 9746 . . . 4 (1PP → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
97, 8ax-mp 5 . . 3 (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵))
105, 6, 93bitr4i 291 . 2 ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R𝐴<P 𝐵)
113, 10bitr3i 265 1 ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  ⟨cop 4131   class class class wbr 4583  (class class class)co 6549  [cec 7627  Pcnp 9560  1Pc1p 9561   +P cpp 9562
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