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Theorem ltpsrprg 7618
 Description: Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.)
Assertion
Ref Expression
ltpsrprg ((𝐴P𝐵P𝐶R) → ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵))

Proof of Theorem ltpsrprg
StepHypRef Expression
1 simp1 981 . . . 4 ((𝐴P𝐵P𝐶R) → 𝐴P)
2 1pr 7369 . . . 4 1PP
3 enrex 7552 . . . . 5 ~R ∈ V
4 df-nr 7542 . . . . 5 R = ((P × P) / ~R )
53, 4ecopqsi 6484 . . . 4 ((𝐴P ∧ 1PP) → [⟨𝐴, 1P⟩] ~RR)
61, 2, 5sylancl 409 . . 3 ((𝐴P𝐵P𝐶R) → [⟨𝐴, 1P⟩] ~RR)
7 simp2 982 . . . 4 ((𝐴P𝐵P𝐶R) → 𝐵P)
83, 4ecopqsi 6484 . . . 4 ((𝐵P ∧ 1PP) → [⟨𝐵, 1P⟩] ~RR)
97, 2, 8sylancl 409 . . 3 ((𝐴P𝐵P𝐶R) → [⟨𝐵, 1P⟩] ~RR)
10 simp3 983 . . 3 ((𝐴P𝐵P𝐶R) → 𝐶R)
11 ltasrg 7585 . . 3 (([⟨𝐴, 1P⟩] ~RR ∧ [⟨𝐵, 1P⟩] ~RR𝐶R) → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R )))
126, 9, 10, 11syl3anc 1216 . 2 ((𝐴P𝐵P𝐶R) → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R )))
13 addcomprg 7393 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) = (1P +P 𝐴))
141, 2, 13sylancl 409 . . . 4 ((𝐴P𝐵P𝐶R) → (𝐴 +P 1P) = (1P +P 𝐴))
1514breq1d 3939 . . 3 ((𝐴P𝐵P𝐶R) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
162a1i 9 . . . 4 ((𝐴P𝐵P𝐶R) → 1PP)
17 ltsrprg 7562 . . . 4 (((𝐴P ∧ 1PP) ∧ (𝐵P ∧ 1PP)) → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵)))
181, 16, 7, 16, 17syl22anc 1217 . . 3 ((𝐴P𝐵P𝐶R) → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R ↔ (𝐴 +P 1P)<P (1P +P 𝐵)))
19 ltaprg 7434 . . . 4 ((𝐴P𝐵P ∧ 1PP) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
201, 7, 16, 19syl3anc 1216 . . 3 ((𝐴P𝐵P𝐶R) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
2115, 18, 203bitr4d 219 . 2 ((𝐴P𝐵P𝐶R) → ([⟨𝐴, 1P⟩] ~R <R [⟨𝐵, 1P⟩] ~R𝐴<P 𝐵))
2212, 21bitr3d 189 1 ((𝐴P𝐵P𝐶R) → ((𝐶 +R [⟨𝐴, 1P⟩] ~R ) <R (𝐶 +R [⟨𝐵, 1P⟩] ~R ) ↔ 𝐴<P 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929  (class class class)co 5774  [cec 6427  Pcnp 7106  1Pc1p 7107   +P cpp 7108
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