ioodvbdlimc1lem1 - Mathbox for Glauco Siliprandi

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Theorem ioodvbdlimc1lem1 45922

Description: If 𝐹 has bounded derivative on (𝐴(,)𝐵) then a sequence of points in its image converges to its lim sup. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 3-Oct-2020.)

**Hypotheses**
Ref	Expression
ioodvbdlimc1lem1.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ioodvbdlimc1lem1.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ioodvbdlimc1lem1.altb	⊢ (𝜑 → 𝐴 < 𝐵)
ioodvbdlimc1lem1.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
ioodvbdlimc1lem1.dmdv	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
ioodvbdlimc1lem1.dvbd	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)
ioodvbdlimc1lem1.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ioodvbdlimc1lem1.r	⊢ (𝜑 → 𝑅:(ℤ_≥‘𝑀)⟶(𝐴(,)𝐵))
ioodvbdlimc1lem1.s	⊢ 𝑆 = (𝑗 ∈ (ℤ_≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))
ioodvbdlimc1lem1.rcnv	⊢ (𝜑 → 𝑅 ∈ dom ⇝ )
ioodvbdlimc1lem1.k	⊢ 𝐾 = inf({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )

**Assertion**
Ref	Expression
ioodvbdlimc1lem1	⊢ (𝜑 → 𝑆 ⇝ (lim sup‘𝑆))

Distinct variable groups: 𝐴,𝑖,𝑘,𝑥,𝑧 𝑦,𝐴,𝑖,𝑥,𝑧 𝐵,𝑖,𝑘,𝑥,𝑧 𝑦,𝐵 𝑖,𝐹,𝑗,𝑥 𝑘,𝐹,𝑧 𝑦,𝐹 𝑖,𝐾,𝑗 𝑘,𝐾 𝑦,𝐾 𝑖,𝑀,𝑗,𝑥 𝑘,𝑀 𝑅,𝑖,𝑗 𝑅,𝑘 𝑦,𝑅 𝑆,𝑖,𝑘,𝑥 𝜑,𝑖,𝑗,𝑥 𝜑,𝑘 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑧) 𝐴(𝑗) 𝐵(𝑗) 𝑅(𝑥,𝑧) 𝑆(𝑦,𝑧,𝑗) 𝐾(𝑥,𝑧) 𝑀(𝑦,𝑧)

Proof of Theorem ioodvbdlimc1lem1 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		ioodvbdlimc1lem1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
3		cncff 24819	. . . . . 6 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
5	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑀)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
6		ioodvbdlimc1lem1.r	. . . . 5 ⊢ (𝜑 → 𝑅:(ℤ_≥‘𝑀)⟶(𝐴(,)𝐵))
7	6	ffvelcdmda 7038	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑀)) → (𝑅‘𝑗) ∈ (𝐴(,)𝐵))
8	5, 7	ffvelcdmd 7039	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑅‘𝑗)) ∈ ℝ)
9		ioodvbdlimc1lem1.s	. . 3 ⊢ 𝑆 = (𝑗 ∈ (ℤ_≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))
10	8, 9	fmptd 7068	. 2 ⊢ (𝜑 → 𝑆:(ℤ_≥‘𝑀)⟶ℝ)
11		ssrab2 4039	. . . . 5 ⊢ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ⊆ (ℤ_≥‘𝑀)
12		ioodvbdlimc1lem1.k	. . . . . 6 ⊢ 𝐾 = inf({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )
13		rpre 12936	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
14	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
15		2fveq3 6845	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑧)) = (abs‘((ℝ D 𝐹)‘𝑥)))
16	15	cbvmptv 5206	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
17	16	rneqi 5890	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))) = ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
18	17	supeq1i 9374	. . . . . . . . . . . . . 14 ⊢ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )
19		ioodvbdlimc1lem1.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ)
20		ioodvbdlimc1lem1.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ)
21		ioodvbdlimc1lem1.altb	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 < 𝐵)
22		ioomidp 45505	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
23	19, 20, 21, 22	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
24	23	ne0d 4301	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
25		ioossre 13344	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 25888	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
28	4, 26, 27	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
29		ioodvbdlimc1lem1.dmdv	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
30	29	feq2d 6654	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
31	28, 30	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
32		ax-resscn 11101	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℂ
33	32	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ℝ ⊆ ℂ)
34	31, 33	fssd 6687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
35	34	ffvelcdmda 7038	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
36	35	abscld 15381	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ)
37		ioodvbdlimc1lem1.dvbd	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦)
38		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
39		eqid 2729	. . . . . . . . . . . . . . . 16 ⊢ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )
40	24, 36, 37, 38, 39	suprnmpt 45161	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) ∈ ℝ ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )))
41	40	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) ∈ ℝ)
42	18, 41	eqeltrid 2832	. . . . . . . . . . . . 13 ⊢ (𝜑 → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ)
43	42	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ)
44		peano2re 11323	. . . . . . . . . . . 12 ⊢ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
45	43, 44	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
46		0red 11153	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
47		1red 11151	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℝ)
48	46, 47	readdcld 11179	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0 + 1) ∈ ℝ)
49	42, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
50	46	ltp1d 12089	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (0 + 1))
51	34, 23	ffvelcdmd 7039	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
52	51	abscld 15381	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
53	51	absge0d 15389	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
54	40	simprd 495	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ))
55		2fveq3 6845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑦)) = (abs‘((ℝ D 𝐹)‘𝑥)))
56	18	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ))
57	55, 56	breq12d 5115	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → ((abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ↔ (abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )))
58	57	cbvralvw 3213	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ↔ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ))
59	54, 58	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
60		2fveq3 6845	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑦)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
61	60	breq1d 5112	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ↔ (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < )))
62	61	rspcva 3583	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵) ∧ ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < )) → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
63	23, 59, 62	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
64	46, 52, 42, 53, 63	letrd 11307	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
65	46, 42, 47, 64	leadd1dd 11768	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0 + 1) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
66	46, 48, 49, 50, 65	ltletrd 11310	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
67	66	gt0ne0d 11718	. . . . . . . . . . . 12 ⊢ (𝜑 → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ≠ 0)
68	67	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ≠ 0)
69	14, 45, 68	redivcld 11986	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ)
70		rpgt0 12940	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 0 < 𝑥)
71	70	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 < 𝑥)
72	66	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 < (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
73	14, 45, 71, 72	divgt0d 12094	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
74	69, 73	elrpd 12968	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ⁺)
75		ioodvbdlimc1lem1.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
76	75	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑀 ∈ ℤ)
77		ioodvbdlimc1lem1.rcnv	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ dom ⇝ )
78	77	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅 ∈ dom ⇝ )
79	1	climcau 15613	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑅 ∈ dom ⇝ ) → ∀𝑤 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤)
80	76, 78, 79	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑤 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤)
81		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) → ((abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤 ↔ (abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
82	81	rexralbidv 3201	. . . . . . . . . 10 ⊢ (𝑤 = (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) → (∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤 ↔ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
83	82	rspcva 3583	. . . . . . . . 9 ⊢ (((𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ⁺ ∧ ∀𝑤 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤) → ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
84	74, 80, 83	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
85		rabn0 4348	. . . . . . . 8 ⊢ ({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ≠ ∅ ↔ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
86	84, 85	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ≠ ∅)
87		infssuzcl 12867	. . . . . . 7 ⊢ (({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ⊆ (ℤ_≥‘𝑀) ∧ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ≠ ∅) → inf({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < ) ∈ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))})
88	11, 86, 87	sylancr 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → inf({𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < ) ∈ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))})
89	12, 88	eqeltrid 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐾 ∈ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))})
90	11, 89	sselid 3941	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐾 ∈ (ℤ_≥‘𝑀))
91		2fveq3 6845	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝐹‘(𝑅‘𝑗)) = (𝐹‘(𝑅‘𝑖)))
92		uzss 12792	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑀))
93	90, 92	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑀))
94	93	sselda 3943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝑖 ∈ (ℤ_≥‘𝑀))
95	4	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
96	6	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝑅:(ℤ_≥‘𝑀)⟶(𝐴(,)𝐵))
97	96, 94	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) ∈ (𝐴(,)𝐵))
98	95, 97	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝑖)) ∈ ℝ)
99	9, 91, 94, 98	fvmptd3 6973	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑆‘𝑖) = (𝐹‘(𝑅‘𝑖)))
100		2fveq3 6845	. . . . . . . . 9 ⊢ (𝑗 = 𝐾 → (𝐹‘(𝑅‘𝑗)) = (𝐹‘(𝑅‘𝐾)))
101	90	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐾 ∈ (ℤ_≥‘𝑀))
102	96, 101	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝐾) ∈ (𝐴(,)𝐵))
103	95, 102	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝐾)) ∈ ℝ)
104	9, 100, 101, 103	fvmptd3 6973	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑆‘𝐾) = (𝐹‘(𝑅‘𝐾)))
105	99, 104	oveq12d 7387	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑆‘𝑖) − (𝑆‘𝐾)) = ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))))
106	105	fveq2d 6844	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) = (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))))
107	98	recnd 11178	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝑖)) ∈ ℂ)
108	103	recnd 11178	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝐾)) ∈ ℂ)
109	107, 108	subcld 11509	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) ∈ ℂ)
110	109	abscld 15381	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ∈ ℝ)
111	110	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ∈ ℝ)
112	42	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ)
113	112	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ)
114	6	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑅:(ℤ_≥‘𝑀)⟶(𝐴(,)𝐵))
115	114, 90	ffvelcdmd 7039	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ (𝐴(,)𝐵))
116	25, 115	sselid 3941	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ ℝ)
117	116	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
118	25, 97	sselid 3941	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) ∈ ℝ)
119	118	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝑖) ∈ ℝ)
120	117, 119	resubcld 11582	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ∈ ℝ)
121	113, 120	remulcld 11180	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))) ∈ ℝ)
122	13	ad3antlr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝑥 ∈ ℝ)
123	107	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝐹‘(𝑅‘𝑖)) ∈ ℂ)
124	108	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝐹‘(𝑅‘𝐾)) ∈ ℂ)
125	123, 124	abssubd 15398	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) = (abs‘((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝑖)))))
126	19	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝐴 ∈ ℝ)
127	20	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝐵 ∈ ℝ)
128	95	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
129	29	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
130	59	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
131	97	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝑖) ∈ (𝐴(,)𝐵))
132	118	rexrd 11200	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) ∈ ℝ^*)
133	132	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝑖) ∈ ℝ^*)
134	20	rexrd 11200	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
135	134	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐵 ∈ ℝ^*)
136	135	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝐵 ∈ ℝ^*)
137		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝑖) < (𝑅‘𝐾))
138	19	rexrd 11200	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
139	138	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ∈ ℝ^*)
140	134	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐵 ∈ ℝ^*)
141		iooltub 45501	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑅‘𝐾) ∈ (𝐴(,)𝐵)) → (𝑅‘𝐾) < 𝐵)
142	139, 140, 115, 141	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) < 𝐵)
143	142	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) < 𝐵)
144	133, 136, 117, 137, 143	eliood 45489	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ((𝑅‘𝑖)(,)𝐵))
145	126, 127, 128, 129, 113, 130, 131, 144	dvbdfbdioolem1 45919	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((abs‘((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝑖)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))) ∧ (abs‘((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝑖)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · (𝐵 − 𝐴))))
146	145	simpld 494	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝑖)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))))
147	125, 146	eqbrtrd 5124	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))))
148	113, 44	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
149	148, 120	remulcld 11180	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝐾) − (𝑅‘𝑖))) ∈ ℝ)
150	119, 117	posdifd 11741	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝑖) < (𝑅‘𝐾) ↔ 0 < ((𝑅‘𝐾) − (𝑅‘𝑖))))
151	137, 150	mpbid 232	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 0 < ((𝑅‘𝐾) − (𝑅‘𝑖)))
152	120, 151	elrpd 12968	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ∈ ℝ⁺)
153	113	ltp1d 12089	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) < (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
154	113, 148, 152, 153	ltmul1dd 13026	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))) < ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝐾) − (𝑅‘𝑖))))
155	25, 102	sselid 3941	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
156	118, 155	resubcld 11582	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ)
157	156	recnd 11178	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℂ)
158	157	abscld 15381	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
159	158	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
160	69	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ)
161	120	leabsd 15357	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ≤ (abs‘((𝑅‘𝐾) − (𝑅‘𝑖))))
162	117	recnd 11178	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℂ)
163	118	recnd 11178	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) ∈ ℂ)
164	163	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝑖) ∈ ℂ)
165	162, 164	abssubd 15398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝑅‘𝐾) − (𝑅‘𝑖))) = (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
166	161, 165	breqtrd 5128	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ≤ (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
167		fveq2 6840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐾 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝐾))
168		fveq2 6840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝐾 → (𝑅‘𝑘) = (𝑅‘𝐾))
169	168	oveq2d 7385	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝐾 → ((𝑅‘𝑖) − (𝑅‘𝑘)) = ((𝑅‘𝑖) − (𝑅‘𝐾)))
170	169	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝐾 → (abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) = (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
171	170	breq1d 5112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐾 → ((abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔ (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
172	167, 171	raleqbidv 3316	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝐾 → (∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
173	172	elrab 3656	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ {𝑘 ∈ (ℤ_≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} ↔ (𝐾 ∈ (ℤ_≥‘𝑀) ∧ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
174	89, 173	sylib 218	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝐾 ∈ (ℤ_≥‘𝑀) ∧ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
175	174	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
176	175	r19.21bi 3227	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
177	176	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
178	120, 159, 160, 166, 177	lelttrd 11308	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
179	49, 66	elrpd 12968	. . . . . . . . . . . 12 ⊢ (𝜑 → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ⁺)
180	179	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ⁺)
181	120, 122, 180	ltmuldiv2d 13019	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝐾) − (𝑅‘𝑖))) < 𝑥 ↔ ((𝑅‘𝐾) − (𝑅‘𝑖)) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
182	178, 181	mpbird 257	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝐾) − (𝑅‘𝑖))) < 𝑥)
183	121, 149, 122, 154, 182	lttrd 11311	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝐾) − (𝑅‘𝑖))) < 𝑥)
184	111, 121, 122, 147, 183	lelttrd 11308	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
185		fveq2 6840	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝑖) = (𝑅‘𝐾) → (𝐹‘(𝑅‘𝑖)) = (𝐹‘(𝑅‘𝐾)))
186	185	oveq1d 7384	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝑖) = (𝑅‘𝐾) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) = ((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝐾))))
187	108	subidd 11497	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝐾))) = 0)
188	186, 187	sylan9eqr 2786	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) = 0)
189	188	abs00bd 15233	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) = 0)
190	70	ad3antlr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → 0 < 𝑥)
191	189, 190	eqbrtrd 5124	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
192	191	adantlr 715	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
193		simpll 766	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → ((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)))
194	155	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
195	118	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (𝑅‘𝑖) ∈ ℝ)
196		id 22	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝐾) = (𝑅‘𝑖) → (𝑅‘𝐾) = (𝑅‘𝑖))
197	196	eqcomd 2735	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝐾) = (𝑅‘𝑖) → (𝑅‘𝑖) = (𝑅‘𝐾))
198	197	necon3bi 2951	. . . . . . . . . . 11 ⊢ (¬ (𝑅‘𝑖) = (𝑅‘𝐾) → (𝑅‘𝐾) ≠ (𝑅‘𝑖))
199	198	adantl 481	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (𝑅‘𝐾) ≠ (𝑅‘𝑖))
200		simplr 768	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → ¬ (𝑅‘𝑖) < (𝑅‘𝐾))
201	194, 195, 199, 200	lttri5d 45290	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (𝑅‘𝐾) < (𝑅‘𝑖))
202	110	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ∈ ℝ)
203	112, 156	remulcld 11180	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
204	203	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
205	13	ad3antlr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝑥 ∈ ℝ)
206	19	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝐴 ∈ ℝ)
207	20	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝐵 ∈ ℝ)
208	95	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
209	29	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
210	42	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) ∈ ℝ)
211	59	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ∀𝑦 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑦)) ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
212	102	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝐾) ∈ (𝐴(,)𝐵))
213	116	rexrd 11200	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ ℝ^*)
214	213	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝐾) ∈ ℝ^*)
215	207	rexrd 11200	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝐵 ∈ ℝ^*)
216	118	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝑖) ∈ ℝ)
217		simpr 484	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝐾) < (𝑅‘𝑖))
218	138	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐴 ∈ ℝ^*)
219		iooltub 45501	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑅‘𝑖) ∈ (𝐴(,)𝐵)) → (𝑅‘𝑖) < 𝐵)
220	218, 135, 97, 219	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) < 𝐵)
221	220	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝑖) < 𝐵)
222	214, 215, 216, 217, 221	eliood 45489	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝑖) ∈ ((𝑅‘𝐾)(,)𝐵))
223	206, 207, 208, 209, 210, 211, 212, 222	dvbdfbdioolem1 45919	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))) ∧ (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · (𝐵 − 𝐴))))
224	223	simpld 494	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))))
225		1red 11151	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 1 ∈ ℝ)
226	210, 225	readdcld 11179	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
227	155	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝐾) ∈ ℝ)
228	216, 227	resubcld 11582	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ)
229	226, 228	remulcld 11180	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
230	210, 44	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ)
231	227, 216	posdifd 11741	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝐾) < (𝑅‘𝑖) ↔ 0 < ((𝑅‘𝑖) − (𝑅‘𝐾))))
232	217, 231	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 0 < ((𝑅‘𝑖) − (𝑅‘𝐾)))
233	228, 232	elrpd 12968	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ⁺)
234	210	ltp1d 12089	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) < (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
235	210, 230, 233, 234	ltmul1dd 13026	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))) < ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝑖) − (𝑅‘𝐾))))
236	158	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) ∈ ℝ)
237	69	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ)
238	228	leabsd 15357	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ≤ (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
239	176	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
240	228, 236, 237, 238, 239	lelttrd 11308	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
241	179	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ⁺)
242	228, 205, 241	ltmuldiv2d 13019	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝑖) − (𝑅‘𝐾))) < 𝑥 ↔ ((𝑅‘𝑖) − (𝑅‘𝐾)) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))))
243	240, 242	mpbird 257	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) · ((𝑅‘𝑖) − (𝑅‘𝐾))) < 𝑥)
244	204, 229, 205, 235, 243	lttrd 11311	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) · ((𝑅‘𝑖) − (𝑅‘𝐾))) < 𝑥)
245	202, 204, 205, 224, 244	lelttrd 11308	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
246	193, 201, 245	syl2anc 584	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
247	192, 246	pm2.61dan 812	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
248	184, 247	pm2.61dan 812	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
249	106, 248	eqbrtrd 5124	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥)
250	249	ralrimiva 3125	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥)
251		fveq2 6840	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑆‘𝑘) = (𝑆‘𝐾))
252	251	oveq2d 7385	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑆‘𝑖) − (𝑆‘𝑘)) = ((𝑆‘𝑖) − (𝑆‘𝐾)))
253	252	fveq2d 6844	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) = (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))))
254	253	breq1d 5112	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥 ↔ (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥))
255	167, 254	raleqbidv 3316	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥 ↔ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥))
256	255	rspcev 3585	. . . 4 ⊢ ((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥) → ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥)
257	90, 250, 256	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥)
258	257	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥)
259	1, 10, 258	caurcvg 15619	1 ⊢ (𝜑 → 𝑆 ⇝ (lim sup‘𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3402 ⊆ wss 3911 ∅c0 4292 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5631 ran crn 5632 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 supcsup 9367 infcinf 9368 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 ℝ^*cxr 11183 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 (,)cioo 13282 abscabs 15176 lim supclsp 15412 ⇝ cli 15426 –cn→ccncf 24802 D cdv 25797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: ioodvbdlimc1lem2 45923 ioodvbdlimc2lem 45925

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