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Theorem prsrlt 6871
 Description: Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
Assertion
Ref Expression
prsrlt ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))

Proof of Theorem prsrlt
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pr 6652 . . . . 5 1PP
21a1i 9 . . . 4 ((𝐴P𝐵P) → 1PP)
3 simpr 103 . . . 4 ((𝐴P𝐵P) → 𝐵P)
4 addassprg 6677 . . . 4 ((1PP𝐵P ∧ 1PP) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
52, 3, 2, 4syl3anc 1135 . . 3 ((𝐴P𝐵P) → ((1P +P 𝐵) +P 1P) = (1P +P (𝐵 +P 1P)))
65breq2d 3776 . 2 ((𝐴P𝐵P) → (((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P) ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
7 simpl 102 . . . 4 ((𝐴P𝐵P) → 𝐴P)
8 ltaprg 6717 . . . 4 ((𝐴P𝐵P ∧ 1PP) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
97, 3, 2, 8syl3anc 1135 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
10 addcomprg 6676 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) = (1P +P 𝐴))
117, 2, 10syl2anc 391 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) = (1P +P 𝐴))
1211breq1d 3774 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ (1P +P 𝐴)<P (1P +P 𝐵)))
13 ltaprg 6717 . . . . 5 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1413adantl 262 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
15 addclpr 6635 . . . . 5 ((𝐴P ∧ 1PP) → (𝐴 +P 1P) ∈ P)
167, 2, 15syl2anc 391 . . . 4 ((𝐴P𝐵P) → (𝐴 +P 1P) ∈ P)
17 addclpr 6635 . . . . 5 ((1PP𝐵P) → (1P +P 𝐵) ∈ P)
182, 3, 17syl2anc 391 . . . 4 ((𝐴P𝐵P) → (1P +P 𝐵) ∈ P)
19 addcomprg 6676 . . . . 5 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2019adantl 262 . . . 4 (((𝐴P𝐵P) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2114, 16, 18, 2, 20caovord2d 5670 . . 3 ((𝐴P𝐵P) → ((𝐴 +P 1P)<P (1P +P 𝐵) ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
229, 12, 213bitr2d 205 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ((𝐴 +P 1P) +P 1P)<P ((1P +P 𝐵) +P 1P)))
23 addclpr 6635 . . . 4 ((𝐵P ∧ 1PP) → (𝐵 +P 1P) ∈ P)
243, 2, 23syl2anc 391 . . 3 ((𝐴P𝐵P) → (𝐵 +P 1P) ∈ P)
25 ltsrprg 6832 . . 3 ((((𝐴 +P 1P) ∈ P ∧ 1PP) ∧ ((𝐵 +P 1P) ∈ P ∧ 1PP)) → ([⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
2616, 2, 24, 2, 25syl22anc 1136 . 2 ((𝐴P𝐵P) → ([⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ↔ ((𝐴 +P 1P) +P 1P)<P (1P +P (𝐵 +P 1P))))
276, 22, 263bitr4d 209 1 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ [⟨(𝐴 +P 1P), 1P⟩] ~R <R [⟨(𝐵 +P 1P), 1P⟩] ~R ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  [cec 6104  Pcnp 6389  1Pc1p 6390   +P cpp 6391
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