Theorem caucvgsrlemoffcau 7606

Description: Lemma for caucvgsr 7610. 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 7604	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ N) → ((𝐺‘𝑛) +R 𝐴) = ((𝐹‘𝑛) +R 1R))
6	5	adantr 274	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑛) +R 𝐴) = ((𝐹‘𝑛) +R 1R))
7	6	eqcomd 2145	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑛) +R 1R) = ((𝐺‘𝑛) +R 𝐴))
8	2	ad2antrr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐹:N⟶R)
9		simpr 109	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝑘 ∈ N)
10	8, 9	ffvelrnd 5556	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐹‘𝑘) ∈ R)
11		simplr 519	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝑛 ∈ N)
12		recnnpr 7356	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ ∈ P)
13		prsrcl 7592	. . . . . . . . . . . 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 7559	. . . . . . . . . . . 12 ⊢ 1R ∈ R
16	15	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 1R ∈ R)
17		addcomsrg 7563	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
18	17	adantl 275	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) ∧ (𝑓 ∈ R ∧ 𝑔 ∈ R)) → (𝑓 +R 𝑔) = (𝑔 +R 𝑓))
19		addasssrg 7564	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → ((𝑓 +R 𝑔) +R ℎ) = (𝑓 +R (𝑔 +R ℎ)))
20	19	adantl 275	. . . . . . . . . . 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 5951	. . . . . . . . . 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 7604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) +R 𝐴) = ((𝐹‘𝑘) +R 1R))
23	22	adantlr 468	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐺‘𝑘) +R 𝐴) = ((𝐹‘𝑘) +R 1R))
24	23	oveq1d 5789	. . . . . . . . . 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 7605	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:N⟶R)
26	25	ad2antrr 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐺:N⟶R)
27	26, 9	ffvelrnd 5556	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐺‘𝑘) ∈ R)
28	3	caucvgsrlemasr 7598	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ R)
29	28	ad2antrr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → 𝐴 ∈ R)
30	27, 29, 14, 18, 20	caov32d 5951	. . . . . . . . . 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 2178	. . . . . . . . 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 3942	. . . . . . . 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 7578	. . . . . . . . . 10 ⊢ ((𝑓 ∈ R ∧ 𝑔 ∈ R ∧ ℎ ∈ R) → (𝑓 <R 𝑔 ↔ (ℎ +R 𝑓) <R (ℎ +R 𝑔)))
34	33	adantl 275	. . . . . . . . 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	ffvelrnd 5556	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐹‘𝑛) ∈ R)
36		addclsr 7561	. . . . . . . . . 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 408	. . . . . . . . 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 5940	. . . . . . . 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	ffvelrnd 5556	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → (𝐺‘𝑛) ∈ R)
40		addclsr 7561	. . . . . . . . . 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 408	. . . . . . . . 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 5940	. . . . . . . 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 219	. . . . . . 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 2145	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ N) ∧ 𝑘 ∈ N) → ((𝐹‘𝑘) +R 1R) = ((𝐺‘𝑘) +R 𝐴))
45	35, 14, 16, 18, 20	caov32d 5951	. . . . . . . . . 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 5789	. . . . . . . . . 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 5951	. . . . . . . . . 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 2178	. . . . . . . . 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 3942	. . . . . . . 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 7561	. . . . . . . . . 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 408	. . . . . . . . 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 5940	. . . . . . . 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 7561	. . . . . . . . . 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 408	. . . . . . . . 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 5940	. . . . . . . 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 219	. . . . . . 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 464	. . . . . 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 143	. . . . 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 2499	. . 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 2499	. 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 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ⟨cop 3530   class class class wbr 3929   ↦ cmpt 3989  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  1_oc1o 6306  [cec 6427  Ncnpi 7080   <_N clti 7083   ~_Q ceq 7087  *_Qcrq 7092   <_Q cltq 7093  Pcnp 7099  1_Pc1p 7100   +_P cpp 7101   ~_R cer 7104  Rcnr 7105  1_Rc1r 7107  -1_Rcm1r 7108   +_R cplr 7109   ·_R cmr 7110   <_R cltr 7111

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-imp 7277  df-iltp 7278  df-enr 7534  df-nr 7535  df-plr 7536  df-mr 7537  df-ltr 7538  df-0r 7539  df-1r 7540  df-m1r 7541

This theorem is referenced by:  caucvgsrlemoffres  7608