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Theorem caucvgprprlemloccalc 6840
 Description: Lemma for caucvgprpr 6868. Rearranging some expressions for caucvgprprlemloc 6859. (Contributed by Jim Kingdon, 8-Feb-2021.)
Hypotheses
Ref Expression
caucvgprprlemloccalc.st (𝜑𝑆 <Q 𝑇)
caucvgprprlemloccalc.y (𝜑𝑌Q)
caucvgprprlemloccalc.syt (𝜑 → (𝑆 +Q 𝑌) = 𝑇)
caucvgprprlemloccalc.x (𝜑𝑋Q)
caucvgprprlemloccalc.xxy (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)
caucvgprprlemloccalc.m (𝜑𝑀N)
caucvgprprlemloccalc.mx (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)
Assertion
Ref Expression
caucvgprprlemloccalc (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
Distinct variable groups:   𝑀,𝑙,𝑢   𝑆,𝑙,𝑢   𝑇,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝑋(𝑢,𝑙)   𝑌(𝑢,𝑙)

Proof of Theorem caucvgprprlemloccalc
StepHypRef Expression
1 caucvgprprlemloccalc.st . . . . . 6 (𝜑𝑆 <Q 𝑇)
2 ltrelnq 6521 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4420 . . . . . 6 (𝑆 <Q 𝑇 → (𝑆Q𝑇Q))
41, 3syl 14 . . . . 5 (𝜑 → (𝑆Q𝑇Q))
54simpld 109 . . . 4 (𝜑𝑆Q)
6 caucvgprprlemloccalc.m . . . . 5 (𝜑𝑀N)
7 nnnq 6578 . . . . 5 (𝑀N → [⟨𝑀, 1𝑜⟩] ~QQ)
8 recclnq 6548 . . . . 5 ([⟨𝑀, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
96, 7, 83syl 17 . . . 4 (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
10 addclnq 6531 . . . 4 ((𝑆Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
115, 9, 10syl2anc 397 . . 3 (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
12 addnqpr 6717 . . 3 (((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → ⟨{𝑙𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
1311, 9, 12syl2anc 397 . 2 (𝜑 → ⟨{𝑙𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
14 addassnqg 6538 . . . . 5 ((𝑆Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) = (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))))
155, 9, 9, 14syl3anc 1146 . . . 4 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) = (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))))
16 caucvgprprlemloccalc.mx . . . . . . . 8 (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)
17 caucvgprprlemloccalc.x . . . . . . . . 9 (𝜑𝑋Q)
18 lt2addnq 6560 . . . . . . . . 9 ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q𝑋Q) ∧ ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q𝑋Q)) → (((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋 ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋)))
199, 17, 9, 17, 18syl22anc 1147 . . . . . . . 8 (𝜑 → (((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋 ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋)))
2016, 16, 19mp2and 417 . . . . . . 7 (𝜑 → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋))
21 caucvgprprlemloccalc.xxy . . . . . . 7 (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)
22 ltsonq 6554 . . . . . . . 8 <Q Or Q
2322, 2sotri 4748 . . . . . . 7 ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋) ∧ (𝑋 +Q 𝑋) <Q 𝑌) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌)
2420, 21, 23syl2anc 397 . . . . . 6 (𝜑 → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌)
25 ltanqi 6558 . . . . . 6 ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌𝑆Q) → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q (𝑆 +Q 𝑌))
2624, 5, 25syl2anc 397 . . . . 5 (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q (𝑆 +Q 𝑌))
27 caucvgprprlemloccalc.syt . . . . 5 (𝜑 → (𝑆 +Q 𝑌) = 𝑇)
2826, 27breqtrd 3816 . . . 4 (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q 𝑇)
2915, 28eqbrtrd 3812 . . 3 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑇)
30 ltnqpri 6750 . . 3 (((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑇 → ⟨{𝑙𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
3129, 30syl 14 . 2 (𝜑 → ⟨{𝑙𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
3213, 31eqbrtrrd 3814 1 (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259   ∈ wcel 1409  {cab 2042  ⟨cop 3406   class class class wbr 3792  ‘cfv 4930  (class class class)co 5540  1𝑜c1o 6025  [cec 6135  Ncnpi 6428   ~Q ceq 6435  Qcnq 6436   +Q cplq 6438  *Qcrq 6440
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