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Theorem prarloclemup 6593
Description: Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6601. (Contributed by Jim Kingdon, 10-Nov-2019.)
Assertion
Ref Expression
prarloclemup (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 → (((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))

Proof of Theorem prarloclemup
StepHypRef Expression
1 simpllr 486 . . 3 (((((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) ∧ ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → 𝑦 ∈ ω)
2 simprl 483 . . 3 (((((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) ∧ ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → (𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿)
3 simplr 482 . . 3 (((((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) ∧ ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)
4 rspe 2370 . . 3 ((𝑦 ∈ ω ∧ ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
51, 2, 3, 4syl12anc 1133 . 2 (((((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) ∧ ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))
65exp31 346 1 (((𝑋 ∈ ω ∧ (⟨𝐿, 𝑈⟩ ∈ P𝐴𝐿𝑃Q)) ∧ 𝑦 ∈ ω) → ((𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈 → (((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 suc 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑦 ∈ ω ((𝐴 +Q0 ([⟨𝑦, 1𝑜⟩] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([⟨((𝑦 +𝑜 2𝑜) +𝑜 𝑋), 1𝑜⟩] ~Q ·Q 𝑃)) ∈ 𝑈))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885  wcel 1393  wrex 2307  cop 3378  suc csuc 4102  ωcom 4313  (class class class)co 5512  1𝑜c1o 5994  2𝑜c2o 5995   +𝑜 coa 5998  [cec 6104   ~Q ceq 6377  Qcnq 6378   +Q cplq 6380   ·Q cmq 6381   ~Q0 ceq0 6384   +Q0 cplq0 6387   ·Q0 cmq0 6388  Pcnp 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400
This theorem depends on definitions:  df-bi 110  df-rex 2312
This theorem is referenced by:  prarloclem3step  6594
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