Theorem ivthinclemlopn 15613

Description: Lemma for ivthinc 15620. The lower cut is open. (Contributed by Jim Kingdon, 6-Feb-2024.)

Hypotheses

Ref	Expression
ivth.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ivth.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ivth.3	⊢ (𝜑 → 𝑈 ∈ ℝ)
ivth.4	⊢ (𝜑 → 𝐴 < 𝐵)
ivth.5	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
ivth.7	⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
ivth.8	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
ivth.9	⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
ivthinc.i	⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
ivthinclem.l	⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
ivthinclem.r	⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)}
ivthinclemlopn.q	⊢ (𝜑 → 𝑄 ∈ 𝐿)

Assertion

Ref	Expression
ivthinclemlopn	⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)

Distinct variable groups:   𝑤,𝐴   𝑥,𝐴,𝑦   𝑤,𝐵   𝑥,𝐵,𝑦   𝑤,𝐹   𝑥,𝐹,𝑦   𝐿,𝑟   𝑄,𝑟   𝑤,𝑄   𝑥,𝑄,𝑦   𝑤,𝑈   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑤,𝑟)   𝐴(𝑟)   𝐵(𝑟)   𝐷(𝑥,𝑦,𝑤,𝑟)   𝑅(𝑥,𝑦,𝑤,𝑟)   𝑈(𝑥,𝑦,𝑟)   𝐹(𝑟)   𝐿(𝑥,𝑦,𝑤)

Proof of Theorem ivthinclemlopn

Dummy variables 𝑧 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivth.7	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
2		ivth.5	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
3		ivthinclemlopn.q	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐿)
4		fveq2 5675	. . . . . . . 8 ⊢ (𝑤 = 𝑄 → (𝐹‘𝑤) = (𝐹‘𝑄))
5	4	breq1d 4124	. . . . . . 7 ⊢ (𝑤 = 𝑄 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑄) < 𝑈))
6		ivthinclem.l	. . . . . . 7 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
7	5, 6	elrab2 2979	. . . . . 6 ⊢ (𝑄 ∈ 𝐿 ↔ (𝑄 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑄) < 𝑈))
8	3, 7	sylib 122	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑄) < 𝑈))
9	8	simpld 112	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝐴[,]𝐵))
10	2, 9	sseldd 3243	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
11		ivth.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
12		fveq2 5675	. . . . . . 7 ⊢ (𝑥 = 𝑄 → (𝐹‘𝑥) = (𝐹‘𝑄))
13	12	eleq1d 2303	. . . . . 6 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑄) ∈ ℝ))
14		ivth.8	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15	14	ralrimiva 2617	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
16	13, 15, 9	rspcdva 2928	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) ∈ ℝ)
17	11, 16	resubcld 8671	. . . 4 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ)
18	8	simprd 114	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) < 𝑈)
19	16, 11	posdifd 8823	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑄) < 𝑈 ↔ 0 < (𝑈 − (𝐹‘𝑄))))
20	18, 19	mpbid 147	. . . 4 ⊢ (𝜑 → 0 < (𝑈 − (𝐹‘𝑄)))
21	17, 20	elrpd 10044	. . 3 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ+)
22		cncfi 15555	. . 3 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ 𝑄 ∈ 𝐷 ∧ (𝑈 − (𝐹‘𝑄)) ∈ ℝ+) → ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
23	1, 10, 21, 22	syl3anc 1274	. 2 ⊢ (𝜑 → ∃𝑑 ∈ ℝ+ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
24		ivth.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
25		ivth.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ ℝ)
26		elicc2 10290	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑄 ∈ (𝐴[,]𝐵) ↔ (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵)))
27	24, 25, 26	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ (𝐴[,]𝐵) ↔ (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵)))
28	9, 27	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵))
29	28	simp1d 1036	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℝ)
30	29	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℝ)
31		simprl 531	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ+)
32	31	rphalfcld 10060	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℝ+)
33	32	rpred 10047	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℝ)
34	30, 33	readdcld 8319	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ∈ ℝ)
35	24	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐴 ∈ ℝ)
36	28	simp2d 1037	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝑄)
37	36	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐴 ≤ 𝑄)
38	30, 32	ltaddrpd 10081	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 < (𝑄 + (𝑑 / 2)))
39	30, 34, 38	ltled 8408	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ≤ (𝑄 + (𝑑 / 2)))
40	35, 30, 34, 37, 39	letrd 8413	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐴 ≤ (𝑄 + (𝑑 / 2)))
41	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐵 ∈ ℝ)
42	31	rpred 10047	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ)
43	30, 42	readdcld 8319	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + 𝑑) ∈ ℝ)
44		rphalflt 10034	. . . . . . . . 9 ⊢ (𝑑 ∈ ℝ+ → (𝑑 / 2) < 𝑑)
45	31, 44	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) < 𝑑)
46	33, 42, 30, 45	ltadd2dd 8713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) < (𝑄 + 𝑑))
47	29	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑄 ∈ ℝ)
48	31	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑑 ∈ ℝ+)
49	48	rpred 10047	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑑 ∈ ℝ)
50	47, 49	resubcld 8671	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝑄 − 𝑑) ∈ ℝ)
51	25	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝐵 ∈ ℝ)
52	47, 48	ltsubrpd 10080	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝑄 − 𝑑) < 𝑄)
53	28	simp3d 1038	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ≤ 𝐵)
54	53	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑄 ≤ 𝐵)
55	50, 47, 51, 52, 54	ltletrd 8714	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝑄 − 𝑑) < 𝐵)
56		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝐵 < (𝑄 + 𝑑))
57	51, 47, 49	absdifltd 11888	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ((abs‘(𝐵 − 𝑄)) < 𝑑 ↔ ((𝑄 − 𝑑) < 𝐵 ∧ 𝐵 < (𝑄 + 𝑑))))
58	55, 56, 57	mpbir2and 953	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (abs‘(𝐵 − 𝑄)) < 𝑑)
59		fvoveq1 6081	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐵 → (abs‘(𝑧 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
60	59	breq1d 4124	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝐵 → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘(𝐵 − 𝑄)) < 𝑑))
61	60	imbrov2fvoveq 6083	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐵 → (((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))) ↔ ((abs‘(𝐵 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝐵) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄)))))
62		simplrr 538	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
63	24	rexrd 8339	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
64	25	rexrd 8339	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
65		ivth.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 < 𝐵)
66	24, 25, 65	ltled 8408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 10337	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	63, 64, 66, 67	syl3anc 1274	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	2, 68	sseldd 3243	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
70	69	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝐵 ∈ 𝐷)
71	61, 62, 70	rspcdva 2928	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ((abs‘(𝐵 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝐵) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
72	58, 71	mpd 13	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (abs‘((𝐹‘𝐵) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄)))
73		fveq2 5675	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
74	73	eleq1d 2303	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
75	15	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
76	68	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐵 ∈ (𝐴[,]𝐵))
77	74, 75, 76	rspcdva 2928	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝐵) ∈ ℝ)
78	77	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝐹‘𝐵) ∈ ℝ)
79	16	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝐹‘𝑄) ∈ ℝ)
80	17	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝑈 − (𝐹‘𝑄)) ∈ ℝ)
81	78, 79, 80	absdifltd 11888	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ((abs‘((𝐹‘𝐵) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄)) ↔ (((𝐹‘𝑄) − (𝑈 − (𝐹‘𝑄))) < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < ((𝐹‘𝑄) + (𝑈 − (𝐹‘𝑄))))))
82	72, 81	mpbid 147	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (((𝐹‘𝑄) − (𝑈 − (𝐹‘𝑄))) < (𝐹‘𝐵) ∧ (𝐹‘𝐵) < ((𝐹‘𝑄) + (𝑈 − (𝐹‘𝑄)))))
83	82	simprd 114	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝐹‘𝐵) < ((𝐹‘𝑄) + (𝑈 − (𝐹‘𝑄))))
84	79	recnd 8318	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝐹‘𝑄) ∈ ℂ)
85	11	recnd 8318	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ ℂ)
86	85	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑈 ∈ ℂ)
87	84, 86	pncan3d 8603	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ((𝐹‘𝑄) + (𝑈 − (𝐹‘𝑄))) = 𝑈)
88	83, 87	breqtrd 4140	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → (𝐹‘𝐵) < 𝑈)
89		ivth.9	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
90	89	simprd 114	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵))
91	90	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑈 < (𝐹‘𝐵))
92	88, 91	jca 306	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
93	11	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → 𝑈 ∈ ℝ)
94		ltnsym2 8380	. . . . . . . . . 10 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ¬ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
95	78, 93, 94	syl2anc 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) ∧ 𝐵 < (𝑄 + 𝑑)) → ¬ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
96	92, 95	pm2.65da 667	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ¬ 𝐵 < (𝑄 + 𝑑))
97	43, 41, 96	nltled 8410	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + 𝑑) ≤ 𝐵)
98	34, 43, 41, 46, 97	ltletrd 8714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) < 𝐵)
99	34, 41, 98	ltled 8408	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ≤ 𝐵)
100		elicc2 10290	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝑄 + (𝑑 / 2)) ∈ (𝐴[,]𝐵) ↔ ((𝑄 + (𝑑 / 2)) ∈ ℝ ∧ 𝐴 ≤ (𝑄 + (𝑑 / 2)) ∧ (𝑄 + (𝑑 / 2)) ≤ 𝐵)))
101	35, 41, 100	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ((𝑄 + (𝑑 / 2)) ∈ (𝐴[,]𝐵) ↔ ((𝑄 + (𝑑 / 2)) ∈ ℝ ∧ 𝐴 ≤ (𝑄 + (𝑑 / 2)) ∧ (𝑄 + (𝑑 / 2)) ≤ 𝐵)))
102	34, 40, 99, 101	mpbir3and 1207	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ∈ (𝐴[,]𝐵))
103	16	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝑄) ∈ ℝ)
104		fveq2 5675	. . . . . . . . 9 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑥) = (𝐹‘(𝑄 + (𝑑 / 2))))
105	104	eleq1d 2303	. . . . . . . 8 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘(𝑄 + (𝑑 / 2))) ∈ ℝ))
106	105, 75, 102	rspcdva 2928	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘(𝑄 + (𝑑 / 2))) ∈ ℝ)
107		breq2 4118	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑦 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
108		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑦) = (𝐹‘(𝑄 + (𝑑 / 2))))
109	108	breq2d 4126	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑄) < (𝐹‘𝑦) ↔ (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2)))))
110	107, 109	imbi12d 234	. . . . . . . . . 10 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → ((𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦)) ↔ (𝑄 < (𝑄 + (𝑑 / 2)) → (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2))))))
111		breq1 4117	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → (𝑥 < 𝑦 ↔ 𝑄 < 𝑦))
112	12	breq1d 4124	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘𝑄) < (𝐹‘𝑦)))
113	111, 112	imbi12d 234	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑄 → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦))))
114	113	ralbidv 2544	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑄 → (∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ ∀𝑦 ∈ (𝐴[,]𝐵)(𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦))))
115		ivthinc.i	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦))
116	115	expr 375	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
117	116	ralrimiva 2617	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
118	117	ralrimiva 2617	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
119	114, 118, 9	rspcdva 2928	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦)))
120	119	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦)))
121	110, 120, 102	rspcdva 2928	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 < (𝑄 + (𝑑 / 2)) → (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2)))))
122	38, 121	mpd 13	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2))))
123	103, 106, 122	ltled 8408	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝑄) ≤ (𝐹‘(𝑄 + (𝑑 / 2))))
124	103, 106, 123	abssubge0d 11886	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄))) = ((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄)))
125	30	recnd 8318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℂ)
126	33	recnd 8318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℂ)
127	125, 126	pncan2d 8602	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ((𝑄 + (𝑑 / 2)) − 𝑄) = (𝑑 / 2))
128	127	fveq2d 5679	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) = (abs‘(𝑑 / 2)))
129	32	rpge0d 10051	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 0 ≤ (𝑑 / 2))
130	33, 129	absidd 11877	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘(𝑑 / 2)) = (𝑑 / 2))
131	128, 130	eqtrd 2267	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) = (𝑑 / 2))
132	131, 45	eqbrtrd 4136	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑)
133		fvoveq1 6081	. . . . . . . . . 10 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → (abs‘(𝑧 − 𝑄)) = (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)))
134	133	breq1d 4124	. . . . . . . . 9 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑))
135	134	imbrov2fvoveq 6083	. . . . . . . 8 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → (((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))) ↔ ((abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑 → (abs‘((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄)))))
136		simprr 533	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
137	2	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐴[,]𝐵) ⊆ 𝐷)
138	137, 102	sseldd 3243	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ∈ 𝐷)
139	135, 136, 138	rspcdva 2928	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ((abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑 → (abs‘((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))
140	132, 139	mpd 13	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (abs‘((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄)))
141	124, 140	eqbrtrrd 4138	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄)) < (𝑈 − (𝐹‘𝑄)))
142	11	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑈 ∈ ℝ)
143	106, 142, 103	ltsub1d 8845	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ((𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈 ↔ ((𝐹‘(𝑄 + (𝑑 / 2))) − (𝐹‘𝑄)) < (𝑈 − (𝐹‘𝑄))))
144	141, 143	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈)
145		fveq2 5675	. . . . . 6 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑤) = (𝐹‘(𝑄 + (𝑑 / 2))))
146	145	breq1d 4124	. . . . 5 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
147	146, 6	elrab2 2979	. . . 4 ⊢ ((𝑄 + (𝑑 / 2)) ∈ 𝐿 ↔ ((𝑄 + (𝑑 / 2)) ∈ (𝐴[,]𝐵) ∧ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
148	102, 144, 147	sylanbrc 417	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ∈ 𝐿)
149		breq2 4118	. . . 4 ⊢ (𝑟 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑟 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
150	149	rspcev 2923	. . 3 ⊢ (((𝑄 + (𝑑 / 2)) ∈ 𝐿 ∧ 𝑄 < (𝑄 + (𝑑 / 2))) → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)
151	148, 38, 150	syl2anc 411	. 2 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)
152	23, 151	rexlimddv 2667	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526   ⊆ wss 3214   class class class wbr 4114  ‘cfv 5357  (class class class)co 6058  ℂcc 8141  ℝcr 8142  0cc0 8143   + caddc 8146  ℝ^*cxr 8323   < clt 8324   ≤ cle 8325   − cmin 8460   / cdiv 8963  2c2 9305  ℝ⁺crp 10004  [,]cicc 10243  abscabs 11707  –cn→ccncf 15547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-icc 10247  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-cncf 15548

This theorem is referenced by:  ivthinclemlr  15614