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Theorem caucvgprprlemnkltj 6785
Description: Lemma for caucvgprpr 6808. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprprlemnkj.k (𝜑𝐾N)
caucvgprprlemnkj.j (𝜑𝐽N)
caucvgprprlemnkj.s (𝜑𝑆Q)
Assertion
Ref Expression
caucvgprprlemnkltj ((𝜑𝐾 <N 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
Distinct variable groups:   𝑘,𝐹,𝑛   𝐽,𝑝,𝑞   𝐾,𝑝,𝑞   𝐾,𝑙,𝑝   𝑢,𝐾,𝑞   𝑆,𝑝,𝑞   𝑘,𝑙,𝑛   𝑢,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑆(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑞,𝑝,𝑙)   𝐽(𝑢,𝑘,𝑛,𝑙)   𝐾(𝑘,𝑛)

Proof of Theorem caucvgprprlemnkltj
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsopr 6692 . . . 4 <P Or P
2 ltrelpr 6601 . . . 4 <P ⊆ (P × P)
31, 2son2lpi 4721 . . 3 ¬ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽) ∧ (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
4 simprl 483 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾))
5 caucvgprpr.f . . . . . . . . . 10 (𝜑𝐹:NP)
6 caucvgprpr.cau . . . . . . . . . 10 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
75, 6caucvgprprlemval 6784 . . . . . . . . 9 ((𝜑𝐾 <N 𝐽) → ((𝐹𝐾)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐽)<P ((𝐹𝐾) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
87simpld 105 . . . . . . . 8 ((𝜑𝐾 <N 𝐽) → (𝐹𝐾)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
98adantr 261 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐾)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
101, 2sotri 4720 . . . . . . 7 (((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ (𝐹𝐾)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
114, 9, 10syl2anc 391 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
12 ltaprg 6715 . . . . . . . 8 ((𝑓P𝑔PP) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
1312adantl 262 . . . . . . 7 ((((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) ∧ (𝑓P𝑔PP)) → (𝑓<P 𝑔 ↔ ( +P 𝑓)<P ( +P 𝑔)))
14 caucvgprprlemnkj.s . . . . . . . . 9 (𝜑𝑆Q)
1514ad2antrr 457 . . . . . . . 8 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → 𝑆Q)
16 nqprlu 6643 . . . . . . . 8 (𝑆Q → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
1715, 16syl 14 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
18 caucvgprprlemnkj.j . . . . . . . . 9 (𝜑𝐽N)
195, 18ffvelrnd 5303 . . . . . . . 8 (𝜑 → (𝐹𝐽) ∈ P)
2019ad2antrr 457 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽) ∈ P)
21 caucvgprprlemnkj.k . . . . . . . . 9 (𝜑𝐾N)
22 recnnpr 6644 . . . . . . . . 9 (𝐾N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
2321, 22syl 14 . . . . . . . 8 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
2423ad2antrr 457 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
25 addcomprg 6674 . . . . . . . 8 ((𝑓P𝑔P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2625adantl 262 . . . . . . 7 ((((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) ∧ (𝑓P𝑔P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
2713, 17, 20, 24, 26caovord2d 5670 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽) ↔ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
2811, 27mpbird 156 . . . . 5 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽))
29 recnnpr 6644 . . . . . . . . 9 (𝐽N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
3018, 29syl 14 . . . . . . . 8 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
3130ad2antrr 457 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
32 ltaddpr 6693 . . . . . . 7 (((𝐹𝐽) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
3320, 31, 32syl2anc 391 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
34 simprr 484 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
351, 2sotri 4720 . . . . . 6 (((𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) → (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
3633, 34, 35syl2anc 391 . . . . 5 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
3728, 36jca 290 . . . 4 (((𝜑𝐾 <N 𝐽) ∧ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽) ∧ (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
3837ex 108 . . 3 ((𝜑𝐾 <N 𝐽) → (((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽) ∧ (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)))
393, 38mtoi 590 . 2 ((𝜑𝐾 <N 𝐽) → ¬ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
4014adantr 261 . . . . 5 ((𝜑𝐾 <N 𝐽) → 𝑆Q)
41 nnnq 6518 . . . . . . 7 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
42 recclnq 6488 . . . . . . 7 ([⟨𝐾, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
4321, 41, 423syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
4443adantr 261 . . . . 5 ((𝜑𝐾 <N 𝐽) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
45 addnqpr 6657 . . . . 5 ((𝑆Q ∧ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q) → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
4640, 44, 45syl2anc 391 . . . 4 ((𝜑𝐾 <N 𝐽) → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
4746breq1d 3774 . . 3 ((𝜑𝐾 <N 𝐽) → (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ↔ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾)))
4847anbi1d 438 . 2 ((𝜑𝐾 <N 𝐽) → ((⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) ↔ ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)))
4939, 48mtbird 598 1 ((𝜑𝐾 <N 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3a 885   = wceq 1243  wcel 1393  {cab 2026  wral 2306  cop 3378   class class class wbr 3764  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  Qcnq 6376   +Q cplq 6378  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
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-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemnkj  6788
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