Theorem ivthinclemlopn 14117

Description: Lemma for ivthinc 14124. 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 5516	. . . . . . . 8 ⊢ (𝑤 = 𝑄 → (𝐹‘𝑤) = (𝐹‘𝑄))
5	4	breq1d 4014	. . . . . . 7 ⊢ (𝑤 = 𝑄 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑄) < 𝑈))
6		ivthinclem.l	. . . . . . 7 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
7	5, 6	elrab2 2897	. . . . . 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 3157	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
11		ivth.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
12		fveq2 5516	. . . . . . 7 ⊢ (𝑥 = 𝑄 → (𝐹‘𝑥) = (𝐹‘𝑄))
13	12	eleq1d 2246	. . . . . 6 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑄) ∈ ℝ))
14		ivth.8	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15	14	ralrimiva 2550	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
16	13, 15, 9	rspcdva 2847	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) ∈ ℝ)
17	11, 16	resubcld 8338	. . . 4 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ)
18	8	simprd 114	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) < 𝑈)
19	16, 11	posdifd 8489	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑄) < 𝑈 ↔ 0 < (𝑈 − (𝐹‘𝑄))))
20	18, 19	mpbid 147	. . . 4 ⊢ (𝜑 → 0 < (𝑈 − (𝐹‘𝑄)))
21	17, 20	elrpd 9693	. . 3 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ+)
22		cncfi 14068	. . 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 1238	. 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 9938	. . . . . . . . . 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 1009	. . . . . . 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 529	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ+)
32	31	rphalfcld 9709	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℝ+)
33	32	rpred 9696	. . . . . 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 7987	. . . . 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 1010	. . . . . . 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 9730	. . . . . . 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 8076	. . . . . 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 8081	. . . . 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 9696	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ)
43	30, 42	readdcld 7987	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + 𝑑) ∈ ℝ)
44		rphalflt 9683	. . . . . . . . 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 8379	. . . . . . 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 9696	. . . . . . . . . . . . . . . . 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 8338	. . . . . . . . . . . . . . . 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 9729	. . . . . . . . . . . . . . . 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 1011	. . . . . . . . . . . . . . . . 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 8380	. . . . . . . . . . . . . . 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 11187	. . . . . . . . . . . . . . 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 944	. . . . . . . . . . . . . 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 5898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐵 → (abs‘(𝑧 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
60	59	breq1d 4014	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝐵 → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘(𝐵 − 𝑄)) < 𝑑))
61	60	imbrov2fvoveq 5900	. . . . . . . . . . . . . . 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 536	. . . . . . . . . . . . . . 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 8007	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
64	25	rexrd 8007	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
65		ivth.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 < 𝐵)
66	24, 25, 65	ltled 8076	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 9985	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	63, 64, 66, 67	syl3anc 1238	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	2, 68	sseldd 3157	. . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . 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 5516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
74	73	eleq1d 2246	. . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . 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 11187	. . . . . . . . . . . . 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 7986	. . . . . . . . . . . 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 7986	. . . . . . . . . . . . 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 8271	. . . . . . . . . . 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 4030	. . . . . . . . . 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 8048	. . . . . . . . . 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 661	. . . . . . . 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 8078	. . . . . . 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 8380	. . . . . 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 8076	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ≤ 𝐵)
100		elicc2 9938	. . . . . 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 1180	. . . 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 5516	. . . . . . . . 9 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑥) = (𝐹‘(𝑄 + (𝑑 / 2))))
105	104	eleq1d 2246	. . . . . . . 8 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘(𝑄 + (𝑑 / 2))) ∈ ℝ))
106	105, 75, 102	rspcdva 2847	. . . . . . 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 4008	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑦 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
108		fveq2 5516	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑦) = (𝐹‘(𝑄 + (𝑑 / 2))))
109	108	breq2d 4016	. . . . . . . . . . 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 4007	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → (𝑥 < 𝑦 ↔ 𝑄 < 𝑦))
112	12	breq1d 4014	. . . . . . . . . . . . . 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 2477	. . . . . . . . . . . 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 2550	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
118	117	ralrimiva 2550	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
119	114, 118, 9	rspcdva 2847	. . . . . . . . . . 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 2847	. . . . . . . . 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 8076	. . . . . . 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 11185	. . . . . 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 7986	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℂ)
126	33	recnd 7986	. . . . . . . . . . 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 8270	. . . . . . . . . 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 5520	. . . . . . . . 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 9700	. . . . . . . . . 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 11176	. . . . . . . . 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 2210	. . . . . . . 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 4026	. . . . . . 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 5898	. . . . . . . . . 10 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → (abs‘(𝑧 − 𝑄)) = (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)))
134	133	breq1d 4014	. . . . . . . . 9 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑))
135	134	imbrov2fvoveq 5900	. . . . . . . 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 531	. . . . . . . 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 3157	. . . . . . . 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 2847	. . . . . . 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 4028	. . . . 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 8511	. . . . 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 5516	. . . . . 6 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑤) = (𝐹‘(𝑄 + (𝑑 / 2))))
146	145	breq1d 4014	. . . . 5 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
147	146, 6	elrab2 2897	. . . 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 4008	. . . 4 ⊢ (𝑟 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑟 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
150	149	rspcev 2842	. . 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 2599	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   ⊆ wss 3130   class class class wbr 4004  ‘cfv 5217  (class class class)co 5875  ℂcc 7809  ℝcr 7810  0cc0 7811   + caddc 7814  ℝ^*cxr 7991   < clt 7992   ≤ cle 7993   − cmin 8128   / cdiv 8629  2c2 8970  ℝ⁺crp 9653  [,]cicc 9891  abscabs 11006  –cn→ccncf 14060

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-map 6650  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-rp 9654  df-icc 9895  df-seqfrec 10446  df-exp 10520  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-cncf 14061

This theorem is referenced by:  ivthinclemlr  14118