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Theorem ltexpri 10459
 Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexpri (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexpri
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 10414 . . 3 <P ⊆ (P × P)
21brel 5605 . 2 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltprord 10446 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
4 oveq2 7154 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤 +Q 𝑦) = (𝑤 +Q 𝑧))
54eleq1d 2900 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑤 +Q 𝑦) ∈ 𝐵 ↔ (𝑤 +Q 𝑧) ∈ 𝐵))
65anbi2d 631 . . . . . . . . 9 (𝑦 = 𝑧 → ((¬ 𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ (¬ 𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)))
76exbidv 1923 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)))
87cbvabv 2892 . . . . . . 7 {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} = {𝑧 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)}
98ltexprlem5 10456 . . . . . 6 ((𝐵P𝐴𝐵) → {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P)
109adantll 713 . . . . 5 (((𝐴P𝐵P) ∧ 𝐴𝐵) → {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P)
118ltexprlem6 10457 . . . . . 6 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) ⊆ 𝐵)
128ltexprlem7 10458 . . . . . 6 (((𝐴P𝐵P) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}))
1311, 12eqssd 3970 . . . . 5 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵)
14 oveq2 7154 . . . . . . 7 (𝑥 = {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → (𝐴 +P 𝑥) = (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}))
1514eqeq1d 2826 . . . . . 6 (𝑥 = {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵))
1615rspcev 3609 . . . . 5 (({𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P ∧ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
1710, 13, 16syl2anc 587 . . . 4 (((𝐴P𝐵P) ∧ 𝐴𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
1817ex 416 . . 3 ((𝐴P𝐵P) → (𝐴𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵))
193, 18sylbid 243 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵))
202, 19mpcom 38 1 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115  {cab 2802  ∃wrex 3134   ⊊ wpss 3920   class class class wbr 5053  (class class class)co 7146   +Q cplq 10271  Pcnp 10275   +P cpp 10277
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