Theorem caucvgsrlemoffcau 8159

Description: Lemma for caucvgsr 8163. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	⊢ (𝜑 → 𝐹:N⟶R)
caucvgsr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
caucvgsrlembnd.bnd	⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))
caucvgsrlembnd.offset	⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))

Assertion

Ref	Expression
caucvgsrlemoffcau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgsr.cau	. 2 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
2		caucvgsr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:N⟶R)
3		caucvgsrlembnd.bnd	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑚 ∈ N 𝐴 <R (𝐹‘𝑚))
4		caucvgsrlembnd.offset	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑎 ∈ N ↦ (((𝐹‘𝑎) +R 1R) +R (𝐴 ·R -1R)))
5	2, 1, 3, 4	caucvgsrlemoffval 8157	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ N) → ((𝐺‘𝑛) +R 𝐴) = ((𝐹‘𝑛) +R 1R))
6	5	adantr 276	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑛) +R 𝐴) = ((𝐹‘𝑛) +R 1R))
7	6	eqcomd 2244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑛) +R 1R) = ((𝐺‘𝑛) +R 𝐴))
8	2	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐹:N⟶R)
9		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝑘 ∈ N)
10	8, 9	ffvelcdmd 5838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐹‘𝑘) ∈ R)
11		simplr 533	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝑛 ∈ N)
12		recnnpr 7909	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
13		prsrcl 8145	. . . . . . . . . . . 12 ⊢ (⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
14	11, 12, 13	3syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
15		1sr 8112	. . . . . . . . . . . 12 ⊢ 1R ∈ R
16	15	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 1R ∈ R)
17		addcomsrg 8116	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
18	17	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
19		addasssrg 8117	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 +R 𝑔) +R ℎ) = (𝑓 +R (𝑔 +R ℎ)))
20	19	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → ((𝑓 +R 𝑔) +R ℎ) = (𝑓 +R (𝑔 +R ℎ)))
21	10, 14, 16, 18, 20	caov32d 6264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐹‘𝑘) +R 1R) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
22	2, 1, 3, 4	caucvgsrlemoffval 8157	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) +R 𝐴) = ((𝐹‘𝑘) +R 1R))
23	22	adantlr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) +R 𝐴) = ((𝐹‘𝑘) +R 1R))
24	23	oveq1d 6094	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐺‘𝑘) +R 𝐴) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐹‘𝑘) +R 1R) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
25	2, 1, 3, 4	caucvgsrlemofff 8158	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:N⟶R)
26	25	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐺:N⟶R)
27	26, 9	ffvelcdmd 5838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐺‘𝑘) ∈ R)
28	3	caucvgsrlemasr 8151	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ R)
29	28	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐴 ∈ R)
30	27, 29, 14, 18, 20	caov32d 6264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐺‘𝑘) +R 𝐴) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
31	21, 24, 30	3eqtr2d 2277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
32	7, 31	breq12d 4141	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑛) +R 1R) <R (((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) ↔ ((𝐺‘𝑛) +R 𝐴) <R (((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
33		ltasrg 8131	. . . . . . . . . 10 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
34	33	adantl 277	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R)) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
35	8, 11	ffvelcdmd 5838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐹‘𝑛) ∈ R)
36		addclsr 8114	. . . . . . . . . 10 ⊢ (((𝐹‘𝑘) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
37	10, 14, 36	syl2anc 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
38	34, 35, 37, 16, 18	caovord2d 6253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐹‘𝑛) +R 1R) <R (((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R)))
39	26, 11	ffvelcdmd 5838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐺‘𝑛) ∈ R)
40		addclsr 8114	. . . . . . . . . 10 ⊢ (((𝐺‘𝑘) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
41	27, 14, 40	syl2anc 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
42	34, 39, 41, 29, 18	caovord2d 6253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐺‘𝑛) +R 𝐴) <R (((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
43	32, 38, 42	3bitr4d 220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
44	23	eqcomd 2244	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑘) +R 1R) = ((𝐺‘𝑘) +R 𝐴))
45	35, 14, 16, 18, 20	caov32d 6264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐹‘𝑛) +R 1R) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
46	6	oveq1d 6094	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐺‘𝑛) +R 𝐴) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐹‘𝑛) +R 1R) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))
47	39, 29, 14, 18, 20	caov32d 6264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐺‘𝑛) +R 𝐴) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) = (((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
48	45, 46, 47	3eqtr2d 2277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) = (((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴))
49	44, 48	breq12d 4141	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑘) +R 1R) <R (((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R) ↔ ((𝐺‘𝑘) +R 𝐴) <R (((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
50		addclsr 8114	. . . . . . . . . 10 ⊢ (((𝐹‘𝑛) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
51	35, 14, 50	syl2anc 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
52	34, 10, 51, 16, 18	caovord2d 6253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐹‘𝑘) +R 1R) <R (((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 1R)))
53		addclsr 8114	. . . . . . . . . 10 ⊢ (((𝐺‘𝑛) ∈ R ∧ [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R) → ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
54	39, 14, 53	syl2anc 415	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∈ R)
55	34, 27, 54, 29, 18	caovord2d 6253	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ ((𝐺‘𝑘) +R 𝐴) <R (((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) +R 𝐴)))
56	49, 52, 55	3bitr4d 220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ↔ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))
57	43, 56	anbi12d 477	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ↔ ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
58	57	biimpd 144	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
59	58	imim2d 54	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
60	59	ralimdva 2617	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ N) → (∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
61	60	ralimdva 2617	. 2 ⊢ (𝜑 → (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )))))
62	1, 61	mpd 13	1 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐺‘𝑛) <R ((𝐺‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐺‘𝑘) <R ((𝐺‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ⟨cop 3711   class class class wbr 4128   ↦ cmpt 4190  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079  1_oc1o 6674  [cec 6799  Ncnpi 7633   <_N clti 7636   ~_Q ceq 7640  *_Qcrq 7645   <_Q cltq 7646  Pcnp 7652  1_Pc1p 7653   +_P cpp 7654   ~_R cer 7657  Rcnr 7658  1_Rc1r 7660  -1_Rcm1r 7661   +_R cplr 7662   ·_R cmr 7663   <_R cltr 7664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-i1p 7828  df-iplp 7829  df-imp 7830  df-iltp 7831  df-enr 8087  df-nr 8088  df-plr 8089  df-mr 8090  df-ltr 8091  df-0r 8092  df-1r 8093  df-m1r 8094

This theorem is referenced by:  caucvgsrlemoffres  8161