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Theorem ltapr 9746
 Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltapr (𝐶P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))

Proof of Theorem ltapr
StepHypRef Expression
1 dmplp 9713 . 2 dom +P = (P × P)
2 ltrelpr 9699 . 2 <P ⊆ (P × P)
3 0npr 9693 . 2 ¬ ∅ ∈ P
4 ltaprlem 9745 . . . . . 6 (𝐶P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
54adantr 480 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
6 olc 398 . . . . . . . . 9 ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
7 ltaprlem 9745 . . . . . . . . . . . 12 (𝐶P → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
87adantr 480 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 → (𝐶 +P 𝐵)<P (𝐶 +P 𝐴)))
9 ltsopr 9733 . . . . . . . . . . . . 13 <P Or P
10 sotric 4985 . . . . . . . . . . . . 13 ((<P Or P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
119, 10mpan 702 . . . . . . . . . . . 12 ((𝐵P𝐴P) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
1211adantl 481 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐵<P 𝐴 ↔ ¬ (𝐵 = 𝐴𝐴<P 𝐵)))
13 addclpr 9719 . . . . . . . . . . . . 13 ((𝐶P𝐵P) → (𝐶 +P 𝐵) ∈ P)
14 addclpr 9719 . . . . . . . . . . . . 13 ((𝐶P𝐴P) → (𝐶 +P 𝐴) ∈ P)
1513, 14anim12dan 878 . . . . . . . . . . . 12 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P))
16 sotric 4985 . . . . . . . . . . . 12 ((<P Or P ∧ ((𝐶 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐴) ∈ P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
179, 15, 16sylancr 694 . . . . . . . . . . 11 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐵)<P (𝐶 +P 𝐴) ↔ ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
188, 12, 173imtr3d 281 . . . . . . . . . 10 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ (𝐵 = 𝐴𝐴<P 𝐵) → ¬ ((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))))
1918con4d 113 . . . . . . . . 9 ((𝐶P ∧ (𝐵P𝐴P)) → (((𝐶 +P 𝐵) = (𝐶 +P 𝐴) ∨ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)) → (𝐵 = 𝐴𝐴<P 𝐵)))
206, 19syl5 33 . . . . . . . 8 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (𝐵 = 𝐴𝐴<P 𝐵)))
21 df-or 384 . . . . . . . 8 ((𝐵 = 𝐴𝐴<P 𝐵) ↔ (¬ 𝐵 = 𝐴𝐴<P 𝐵))
2220, 21syl6ib 240 . . . . . . 7 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → (¬ 𝐵 = 𝐴𝐴<P 𝐵)))
2322com23 84 . . . . . 6 ((𝐶P ∧ (𝐵P𝐴P)) → (¬ 𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵)))
249, 2soirri 5441 . . . . . . . 8 ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)
25 oveq2 6557 . . . . . . . . 9 (𝐵 = 𝐴 → (𝐶 +P 𝐵) = (𝐶 +P 𝐴))
2625breq2d 4595 . . . . . . . 8 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐴)))
2724, 26mtbiri 316 . . . . . . 7 (𝐵 = 𝐴 → ¬ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))
2827pm2.21d 117 . . . . . 6 (𝐵 = 𝐴 → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
2923, 28pm2.61d2 171 . . . . 5 ((𝐶P ∧ (𝐵P𝐴P)) → ((𝐶 +P 𝐴)<P (𝐶 +P 𝐵) → 𝐴<P 𝐵))
305, 29impbid 201 . . . 4 ((𝐶P ∧ (𝐵P𝐴P)) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
31303impb 1252 . . 3 ((𝐶P𝐵P𝐴P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
32313com13 1262 . 2 ((𝐴P𝐵P𝐶P) → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
331, 2, 3, 32ndmovord 6722 1 (𝐶P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583   Or wor 4958  (class class class)co 6549  Pcnp 9560   +P cpp 9562
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