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Theorem caucvgsrlemoffcau 7718
Description: Lemma for caucvgsr 7722. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
Hypotheses
Ref Expression
caucvgsr.f (𝜑𝐹:NR)
caucvgsr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
caucvgsrlembnd.bnd (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))
caucvgsrlembnd.offset 𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))
Assertion
Ref Expression
caucvgsrlemoffcau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Distinct variable groups:   𝐴,𝑎   𝐴,𝑚   𝐹,𝑎   𝑘,𝑎,𝑛,𝜑   𝑛,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑚,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑘,𝑚,𝑛,𝑙)   𝐺(𝑢,𝑘,𝑚,𝑛,𝑎,𝑙)

Proof of Theorem caucvgsrlemoffcau
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgsr.cau . 2 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
2 caucvgsr.f . . . . . . . . . . . 12 (𝜑𝐹:NR)
3 caucvgsrlembnd.bnd . . . . . . . . . . . 12 (𝜑 → ∀𝑚N 𝐴 <R (𝐹𝑚))
4 caucvgsrlembnd.offset . . . . . . . . . . . 12 𝐺 = (𝑎N ↦ (((𝐹𝑎) +R 1R) +R (𝐴 ·R -1R)))
52, 1, 3, 4caucvgsrlemoffval 7716 . . . . . . . . . . 11 ((𝜑𝑛N) → ((𝐺𝑛) +R 𝐴) = ((𝐹𝑛) +R 1R))
65adantr 274 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛) +R 𝐴) = ((𝐹𝑛) +R 1R))
76eqcomd 2163 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑛) +R 1R) = ((𝐺𝑛) +R 𝐴))
82ad2antrr 480 . . . . . . . . . . . 12 (((𝜑𝑛N) ∧ 𝑘N) → 𝐹:NR)
9 simpr 109 . . . . . . . . . . . 12 (((𝜑𝑛N) ∧ 𝑘N) → 𝑘N)
108, 9ffvelrnd 5603 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹𝑘) ∈ R)
11 simplr 520 . . . . . . . . . . . 12 (((𝜑𝑛N) ∧ 𝑘N) → 𝑛N)
12 recnnpr 7468 . . . . . . . . . . . 12 (𝑛N → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
13 prsrcl 7704 . . . . . . . . . . . 12 (⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
1411, 12, 133syl 17 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR)
15 1sr 7671 . . . . . . . . . . . 12 1RR
1615a1i 9 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → 1RR)
17 addcomsrg 7675 . . . . . . . . . . . 12 ((𝑓R𝑔R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
1817adantl 275 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ (𝑓R𝑔R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
19 addasssrg 7676 . . . . . . . . . . . 12 ((𝑓R𝑔RR) → ((𝑓 +R 𝑔) +R ) = (𝑓 +R (𝑔 +R )))
2019adantl 275 . . . . . . . . . . 11 ((((𝜑𝑛N) ∧ 𝑘N) ∧ (𝑓R𝑔RR)) → ((𝑓 +R 𝑔) +R ) = (𝑓 +R (𝑔 +R )))
2110, 14, 16, 18, 20caov32d 6001 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐹𝑘) +R 1R) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
222, 1, 3, 4caucvgsrlemoffval 7716 . . . . . . . . . . . 12 ((𝜑𝑘N) → ((𝐺𝑘) +R 𝐴) = ((𝐹𝑘) +R 1R))
2322adantlr 469 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘) +R 𝐴) = ((𝐹𝑘) +R 1R))
2423oveq1d 5839 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐺𝑘) +R 𝐴) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐹𝑘) +R 1R) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
252, 1, 3, 4caucvgsrlemofff 7717 . . . . . . . . . . . . 13 (𝜑𝐺:NR)
2625ad2antrr 480 . . . . . . . . . . . 12 (((𝜑𝑛N) ∧ 𝑘N) → 𝐺:NR)
2726, 9ffvelrnd 5603 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → (𝐺𝑘) ∈ R)
283caucvgsrlemasr 7710 . . . . . . . . . . . 12 (𝜑𝐴R)
2928ad2antrr 480 . . . . . . . . . . 11 (((𝜑𝑛N) ∧ 𝑘N) → 𝐴R)
3027, 29, 14, 18, 20caov32d 6001 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐺𝑘) +R 𝐴) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
3121, 24, 303eqtr2d 2196 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
327, 31breq12d 3978 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑛) +R 1R) <R (((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) ↔ ((𝐺𝑛) +R 𝐴) <R (((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
33 ltasrg 7690 . . . . . . . . . 10 ((𝑓R𝑔RR) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
3433adantl 275 . . . . . . . . 9 ((((𝜑𝑛N) ∧ 𝑘N) ∧ (𝑓R𝑔RR)) → (𝑓 <R 𝑔 ↔ ( +R 𝑓) <R ( +R 𝑔)))
358, 11ffvelrnd 5603 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → (𝐹𝑛) ∈ R)
36 addclsr 7673 . . . . . . . . . 10 (((𝐹𝑘) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
3710, 14, 36syl2anc 409 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
3834, 35, 37, 16, 18caovord2d 5990 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐹𝑛) +R 1R) <R (((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R)))
3926, 11ffvelrnd 5603 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → (𝐺𝑛) ∈ R)
40 addclsr 7673 . . . . . . . . . 10 (((𝐺𝑘) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
4127, 14, 40syl2anc 409 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
4234, 39, 41, 29, 18caovord2d 5990 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐺𝑛) +R 𝐴) <R (((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
4332, 38, 423bitr4d 219 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
4423eqcomd 2163 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑘) +R 1R) = ((𝐺𝑘) +R 𝐴))
4535, 14, 16, 18, 20caov32d 6001 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐹𝑛) +R 1R) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
466oveq1d 5839 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐺𝑛) +R 𝐴) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐹𝑛) +R 1R) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
4739, 29, 14, 18, 20caov32d 6001 . . . . . . . . . 10 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐺𝑛) +R 𝐴) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
4845, 46, 473eqtr2d 2196 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
4944, 48breq12d 3978 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑘) +R 1R) <R (((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) ↔ ((𝐺𝑘) +R 𝐴) <R (((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
50 addclsr 7673 . . . . . . . . . 10 (((𝐹𝑛) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
5135, 14, 50syl2anc 409 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
5234, 10, 51, 16, 18caovord2d 5990 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐹𝑘) +R 1R) <R (((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R)))
53 addclsr 7673 . . . . . . . . . 10 (((𝐺𝑛) ∈ R ∧ [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~RR) → ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
5439, 14, 53syl2anc 409 . . . . . . . . 9 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
5534, 27, 54, 29, 18caovord2d 5990 . . . . . . . 8 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐺𝑘) +R 𝐴) <R (((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
5649, 52, 553bitr4d 219 . . . . . . 7 (((𝜑𝑛N) ∧ 𝑘N) → ((𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
5743, 56anbi12d 465 . . . . . 6 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
5857biimpd 143 . . . . 5 (((𝜑𝑛N) ∧ 𝑘N) → (((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
5958imim2d 54 . . . 4 (((𝜑𝑛N) ∧ 𝑘N) → ((𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
6059ralimdva 2524 . . 3 ((𝜑𝑛N) → (∀𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
6160ralimdva 2524 . 2 (𝜑 → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
621, 61mpd 13 1 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐺𝑛) <R ((𝐺𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺𝑘) <R ((𝐺𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1335  wcel 2128  {cab 2143  wral 2435  cop 3563   class class class wbr 3965  cmpt 4025  wf 5166  cfv 5170  (class class class)co 5824  1oc1o 6356  [cec 6478  Ncnpi 7192   <N clti 7195   ~Q ceq 7199  *Qcrq 7204   <Q cltq 7205  Pcnp 7211  1Pc1p 7212   +P cpp 7213   ~R cer 7216  Rcnr 7217  1Rc1r 7219  -1Rcm1r 7220   +R cplr 7221   ·R cmr 7222   <R cltr 7223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-imp 7389  df-iltp 7390  df-enr 7646  df-nr 7647  df-plr 7648  df-mr 7649  df-ltr 7650  df-0r 7651  df-1r 7652  df-m1r 7653
This theorem is referenced by:  caucvgsrlemoffres  7720
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