ioodvbdlimc1lem1 - Mathbox for Glauco Siliprandi

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Theorem ioodvbdlimc1lem1 45852

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 2740	. 2 ⊢ (ℤ_≥‘𝑀) = (ℤ_≥‘𝑀)
2		ioodvbdlimc1lem1.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
3		cncff 24938	. . . . . 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 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑀)) → (𝑅‘𝑗) ∈ (𝐴(,)𝐵))
8	5, 7	ffvelcdmd 7119	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑀)) → (𝐹‘(𝑅‘𝑗)) ∈ ℝ)
9		ioodvbdlimc1lem1.s	. . 3 ⊢ 𝑆 = (𝑗 ∈ (ℤ_≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))
10	8, 9	fmptd 7148	. 2 ⊢ (𝜑 → 𝑆:(ℤ_≥‘𝑀)⟶ℝ)
11		ssrab2 4103	. . . . 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 13065	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
14	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
15		2fveq3 6925	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑧)) = (abs‘((ℝ D 𝐹)‘𝑥)))
16	15	cbvmptv 5279	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
17	16	rneqi 5962	. . . . . . . . . . . . . . 15 ⊢ ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))) = ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
18	17	supeq1i 9516	. . . . . . . . . . . . . 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 45432	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
23	19, 20, 21, 22	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵))
24	23	ne0d 4365	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅)
25		ioossre 13468	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴(,)𝐵) ⊆ ℝ
26	25	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
27		dvfre 26009	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
28	4, 26, 27	syl2anc 583	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
29		ioodvbdlimc1lem1.dmdv	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
30	29	feq2d 6733	. . . . . . . . . . . . . . . . . . . 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 11241	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℝ ⊆ ℂ
33	32	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ℝ ⊆ ℂ)
34	31, 33	fssd 6764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
35	34	ffvelcdmda 7118	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
36	35	abscld 15485	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥)))
39		eqid 2740	. . . . . . . . . . . . . . . 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 45081	. . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . 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 11463	. . . . . . . . . . . 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 11293	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
47		1red 11291	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℝ)
48	46, 47	readdcld 11319	. . . . . . . . . . . . . 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 12225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (0 + 1))
51	34, 23	ffvelcdmd 7119	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ)
52	51	abscld 15485	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ)
53	51	absge0d 15493	. . . . . . . . . . . . . . . 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 6925	. . . . . . . . . . . . . . . . . . . 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 5179	. . . . . . . . . . . . . . . . . . 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 3243	. . . . . . . . . . . . . . . . . 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 6925	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑦)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))))
61	60	breq1d 5176	. . . . . . . . . . . . . . . . . 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 3633	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 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 11447	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ≤ sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ))
65	46, 42, 47, 64	leadd1dd 11904	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0 + 1) ≤ (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
66	46, 48, 49, 50, 65	ltletrd 11450	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))
67	66	gt0ne0d 11854	. . . . . . . . . . . 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 12122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ∈ ℝ)
70		rpgt0 13069	. . . . . . . . . . . 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 12230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 0 < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))
74	69, 73	elrpd 13096	. . . . . . . . 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 15719	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑅 ∈ dom ⇝ ) → ∀𝑤 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤)
80	76, 78, 79	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑤 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < 𝑤)
81		breq2 5170	. . . . . . . . . . 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 3229	. . . . . . . . . 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 3633	. . . . . . . . 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 583	. . . . . . . 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 4412	. . . . . . . 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 12997	. . . . . . 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 586	. . . . . 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 2848	. . . . 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 4006	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝐾 ∈ (ℤ_≥‘𝑀))
91		2fveq3 6925	. . . . . . . . 9 ⊢ (𝑗 = 𝑖 → (𝐹‘(𝑅‘𝑗)) = (𝐹‘(𝑅‘𝑖)))
92		uzss 12926	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑀))
93	90, 92	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (ℤ_≥‘𝐾) ⊆ (ℤ_≥‘𝑀))
94	93	sselda 4008	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝑖 ∈ (ℤ_≥‘𝑀))
95	4	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
96	6	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝑅:(ℤ_≥‘𝑀)⟶(𝐴(,)𝐵))
97	96, 94	ffvelcdmd 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝑖) ∈ (𝐴(,)𝐵))
98	95, 97	ffvelcdmd 7119	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝑖)) ∈ ℝ)
99	9, 91, 94, 98	fvmptd3 7052	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑆‘𝑖) = (𝐹‘(𝑅‘𝑖)))
100		2fveq3 6925	. . . . . . . . 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 7119	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝐾) ∈ (𝐴(,)𝐵))
103	95, 102	ffvelcdmd 7119	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝐾)) ∈ ℝ)
104	9, 100, 101, 103	fvmptd3 7052	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑆‘𝐾) = (𝐹‘(𝑅‘𝐾)))
105	99, 104	oveq12d 7466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑆‘𝑖) − (𝑆‘𝐾)) = ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))))
106	105	fveq2d 6924	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) = (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))))
107	98	recnd 11318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝑖)) ∈ ℂ)
108	103	recnd 11318	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝐹‘(𝑅‘𝐾)) ∈ ℂ)
109	107, 108	subcld 11647	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) ∈ ℂ)
110	109	abscld 15485	. . . . . . . . 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 725	. . . . . . . . . 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 7119	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ (𝐴(,)𝐵))
116	25, 115	sselid 4006	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ ℝ)
117	116	ad2antrr 725	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
118	25, 97	sselid 4006	. . . . . . . . . . 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 11718	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ∈ ℝ)
121	113, 120	remulcld 11320	. . . . . . . 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 730	. . . . . . . 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 15502	. . . . . . . . 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 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → 𝐴 ∈ ℝ)
127	20	ad3antrrr 729	. . . . . . . . . . 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 729	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
130	59	ad3antrrr 729	. . . . . . . . . . 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 11340	. . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
135	134	ad2antrr 725	. . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . 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 45428	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑅‘𝐾) ∈ (𝐴(,)𝐵)) → (𝑅‘𝐾) < 𝐵)
142	139, 140, 115, 141	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) < 𝐵)
143	142	ad2antrr 725	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) < 𝐵)
144	133, 136, 117, 137, 143	eliood 45416	. . . . . . . . . . 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 45849	. . . . . . . . . 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 5188	. . . . . . . 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 11320	. . . . . . . . 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 11877	. . . . . . . . . . . 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 13096	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ∈ ℝ⁺)
153	113	ltp1d 12225	. . . . . . . . . 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 13154	. . . . . . . . 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 4006	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
156	118, 155	resubcld 11718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ)
157	156	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℂ)
158	157	abscld 15485	. . . . . . . . . . . 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 725	. . . . . . . . . . 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 15463	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ≤ (abs‘((𝑅‘𝐾) − (𝑅‘𝑖))))
162	117	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℂ)
163	118	recnd 11318	. . . . . . . . . . . . . 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 15502	. . . . . . . . . . . 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 5192	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → ((𝑅‘𝐾) − (𝑅‘𝑖)) ≤ (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
167		fveq2 6920	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝐾 → (ℤ_≥‘𝑘) = (ℤ_≥‘𝐾))
168		fveq2 6920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝐾 → (𝑅‘𝑘) = (𝑅‘𝐾))
169	168	oveq2d 7464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝐾 → ((𝑅‘𝑖) − (𝑅‘𝑘)) = ((𝑅‘𝑖) − (𝑅‘𝐾)))
170	169	fveq2d 6924	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝐾 → (abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) = (abs‘((𝑅‘𝑖) − (𝑅‘𝐾))))
171	170	breq1d 5176	. . . . . . . . . . . . . . . . 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 3354	. . . . . . . . . . . . . . . 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 3708	. . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . 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 11448	. . . . . . . . . 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 13096	. . . . . . . . . . . 12 ⊢ (𝜑 → (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1) ∈ ℝ⁺)
180	179	ad3antrrr 729	. . . . . . . . . . 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 13147	. . . . . . . . . 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 11451	. . . . . . . 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 11448	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) < (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
185		fveq2 6920	. . . . . . . . . . . . 13 ⊢ ((𝑅‘𝑖) = (𝑅‘𝐾) → (𝐹‘(𝑅‘𝑖)) = (𝐹‘(𝑅‘𝐾)))
186	185	oveq1d 7463	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝑖) = (𝑅‘𝐾) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) = ((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝐾))))
187	108	subidd 11635	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → ((𝐹‘(𝑅‘𝐾)) − (𝐹‘(𝑅‘𝐾))) = 0)
188	186, 187	sylan9eqr 2802	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → ((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾))) = 0)
189	188	abs00bd 15340	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) = 0)
190	70	ad3antlr 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → 0 < 𝑥)
191	189, 190	eqbrtrd 5188	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝑖) = (𝑅‘𝐾)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
192	191	adantlr 714	. . . . . . . 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 725	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ ¬ (𝑅‘𝑖) < (𝑅‘𝐾)) ∧ ¬ (𝑅‘𝑖) = (𝑅‘𝐾)) → (𝑅‘𝐾) ∈ ℝ)
195	118	ad2antrr 725	. . . . . . . . . 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 2746	. . . . . . . . . . . 12 ⊢ ((𝑅‘𝐾) = (𝑅‘𝑖) → (𝑅‘𝑖) = (𝑅‘𝐾))
198	197	necon3bi 2973	. . . . . . . . . . 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 45214	. . . . . . . . 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 11320	. . . . . . . . . . 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 730	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝑥 ∈ ℝ)
206	19	ad3antrrr 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 𝐴 ∈ ℝ)
207	20	ad3antrrr 729	. . . . . . . . . . . 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 729	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
210	42	ad3antrrr 729	. . . . . . . . . . . 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 729	. . . . . . . . . . . 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 11340	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (𝑅‘𝐾) ∈ ℝ^*)
214	213	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (𝑅‘𝐾) ∈ ℝ^*)
215	207	rexrd 11340	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → 𝐴 ∈ ℝ^*)
219		iooltub 45428	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑅‘𝑖) ∈ (𝐴(,)𝐵)) → (𝑅‘𝑖) < 𝐵)
220	218, 135, 97, 219	syl3anc 1371	. . . . . . . . . . . . . 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 45416	. . . . . . . . . . . 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 45849	. . . . . . . . . . 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 11291	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → 1 ∈ ℝ)
226	210, 225	readdcld 11319	. . . . . . . . . . . 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 11718	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ)
229	226, 228	remulcld 11320	. . . . . . . . . . 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 11877	. . . . . . . . . . . . . 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 13096	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → ((𝑅‘𝑖) − (𝑅‘𝐾)) ∈ ℝ⁺)
234	210	ltp1d 12225	. . . . . . . . . . . 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 13154	. . . . . . . . . . 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 725	. . . . . . . . . . . . 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 15463	. . . . . . . . . . . . 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 11448	. . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 13147	. . . . . . . . . . . 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 11451	. . . . . . . . . 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 11448	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) ∧ (𝑅‘𝐾) < (𝑅‘𝑖)) → (abs‘((𝐹‘(𝑅‘𝑖)) − (𝐹‘(𝑅‘𝐾)))) < 𝑥)
246	193, 201, 245	syl2anc 583	. . . . . . . 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 5188	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑖 ∈ (ℤ_≥‘𝐾)) → (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥)
250	249	ralrimiva 3152	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥)
251		fveq2 6920	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑆‘𝑘) = (𝑆‘𝐾))
252	251	oveq2d 7464	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((𝑆‘𝑖) − (𝑆‘𝑘)) = ((𝑆‘𝑖) − (𝑆‘𝐾)))
253	252	fveq2d 6924	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) = (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))))
254	253	breq1d 5176	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥 ↔ (abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥))
255	167, 254	raleqbidv 3354	. . . . 5 ⊢ (𝑘 = 𝐾 → (∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥 ↔ ∀𝑖 ∈ (ℤ_≥‘𝐾)(abs‘((𝑆‘𝑖) − (𝑆‘𝐾))) < 𝑥))
256	255	rspcev 3635	. . . 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 583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥)
258	257	ralrimiva 3152	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ⁺ ∃𝑘 ∈ (ℤ_≥‘𝑀)∀𝑖 ∈ (ℤ_≥‘𝑘)(abs‘((𝑆‘𝑖) − (𝑆‘𝑘))) < 𝑥)
259	1, 10, 258	caurcvg 15725	1 ⊢ (𝜑 → 𝑆 ⇝ (lim sup‘𝑆))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 supcsup 9509 infcinf 9510 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 2c2 12348 ℤcz 12639 ℤ_≥cuz 12903 ℝ⁺crp 13057 (,)cioo 13407 abscabs 15283 lim supclsp 15516 ⇝ cli 15530 –cn→ccncf 24921 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: ioodvbdlimc1lem2 45853 ioodvbdlimc2lem 45855

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