Theorem ivthinclemlopn 13254

Description: Lemma for ivthinc 13261. 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 5486	. . . . . . . 8 ⊢ (𝑤 = 𝑄 → (𝐹‘𝑤) = (𝐹‘𝑄))
5	4	breq1d 3992	. . . . . . 7 ⊢ (𝑤 = 𝑄 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑄) < 𝑈))
6		ivthinclem.l	. . . . . . 7 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
7	5, 6	elrab2 2885	. . . . . 6 ⊢ (𝑄 ∈ 𝐿 ↔ (𝑄 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑄) < 𝑈))
8	3, 7	sylib 121	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑄) < 𝑈))
9	8	simpld 111	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (𝐴[,]𝐵))
10	2, 9	sseldd 3143	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
11		ivth.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
12		fveq2 5486	. . . . . . 7 ⊢ (𝑥 = 𝑄 → (𝐹‘𝑥) = (𝐹‘𝑄))
13	12	eleq1d 2235	. . . . . 6 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑄) ∈ ℝ))
14		ivth.8	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15	14	ralrimiva 2539	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
16	13, 15, 9	rspcdva 2835	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) ∈ ℝ)
17	11, 16	resubcld 8279	. . . 4 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ)
18	8	simprd 113	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) < 𝑈)
19	16, 11	posdifd 8430	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑄) < 𝑈 ↔ 0 < (𝑈 − (𝐹‘𝑄))))
20	18, 19	mpbid 146	. . . 4 ⊢ (𝜑 → 0 < (𝑈 − (𝐹‘𝑄)))
21	17, 20	elrpd 9629	. . 3 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ+)
22		cncfi 13205	. . 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 1228	. 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 9874	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑄 ∈ (𝐴[,]𝐵) ↔ (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵)))
27	24, 25, 26	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ∈ (𝐴[,]𝐵) ↔ (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵)))
28	9, 27	mpbid 146	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ ℝ ∧ 𝐴 ≤ 𝑄 ∧ 𝑄 ≤ 𝐵))
29	28	simp1d 999	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℝ)
30	29	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℝ)
31		simprl 521	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ+)
32	31	rphalfcld 9645	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℝ+)
33	32	rpred 9632	. . . . . 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 7928	. . . . 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 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐴 ∈ ℝ)
36	28	simp2d 1000	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝑄)
37	36	adantr 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐴 ≤ 𝑄)
38	30, 32	ltaddrpd 9666	. . . . . . 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 8017	. . . . . 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 8022	. . . . 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 274	. . . . . 6 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝐵 ∈ ℝ)
42	31	rpred 9632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ)
43	30, 42	readdcld 7928	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + 𝑑) ∈ ℝ)
44		rphalflt 9619	. . . . . . . . 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 8320	. . . . . . 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 480	. . . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . . . 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 9632	. . . . . . . . . . . . . . . . 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 8279	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 9665	. . . . . . . . . . . . . . . 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 1001	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ≤ 𝐵)
54	53	ad2antrr 480	. . . . . . . . . . . . . . . 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 8321	. . . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . . 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 11120	. . . . . . . . . . . . . . 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 934	. . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐵 → (abs‘(𝑧 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
60	59	breq1d 3992	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝐵 → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘(𝐵 − 𝑄)) < 𝑑))
61	60	imbrov2fvoveq 5867	. . . . . . . . . . . . . . 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 526	. . . . . . . . . . . . . . 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 7948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
64	25	rexrd 7948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
65		ivth.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 < 𝐵)
66	24, 25, 65	ltled 8017	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 9921	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	63, 64, 66, 67	syl3anc 1228	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	2, 68	sseldd 3143	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
70	69	ad2antrr 480	. . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
74	73	eleq1d 2235	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
75	15	adantr 274	. . . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝐵) ∈ ℝ)
78	77	adantr 274	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 11120	. . . . . . . . . . . . 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 146	. . . . . . . . . . . 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 113	. . . . . . . . . . 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 7927	. . . . . . . . . . . 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 7927	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ ℂ)
86	85	ad2antrr 480	. . . . . . . . . . . 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 8212	. . . . . . . . . . 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 4008	. . . . . . . . . 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 113	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵))
91	90	ad2antrr 480	. . . . . . . . . 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 304	. . . . . . . . 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 480	. . . . . . . . . 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 7989	. . . . . . . . . 10 ⊢ (((𝐹‘𝐵) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ¬ ((𝐹‘𝐵) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
95	78, 93, 94	syl2anc 409	. . . . . . . . 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 651	. . . . . . . 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 8019	. . . . . . 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 8321	. . . . . 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 8017	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ≤ 𝐵)
100		elicc2 9874	. . . . . 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 409	. . . . 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 1170	. . . 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 274	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝐹‘𝑄) ∈ ℝ)
104		fveq2 5486	. . . . . . . . 9 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑥) = (𝐹‘(𝑄 + (𝑑 / 2))))
105	104	eleq1d 2235	. . . . . . . 8 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘(𝑄 + (𝑑 / 2))) ∈ ℝ))
106	105, 75, 102	rspcdva 2835	. . . . . . 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 3986	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑦 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
108		fveq2 5486	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑦) = (𝐹‘(𝑄 + (𝑑 / 2))))
109	108	breq2d 3994	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑄) < (𝐹‘𝑦) ↔ (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2)))))
110	107, 109	imbi12d 233	. . . . . . . . . 10 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → ((𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦)) ↔ (𝑄 < (𝑄 + (𝑑 / 2)) → (𝐹‘𝑄) < (𝐹‘(𝑄 + (𝑑 / 2))))))
111		breq1 3985	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → (𝑥 < 𝑦 ↔ 𝑄 < 𝑦))
112	12	breq1d 3992	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) < (𝐹‘𝑦) ↔ (𝐹‘𝑄) < (𝐹‘𝑦)))
113	111, 112	imbi12d 233	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑄 → ((𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)) ↔ (𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦))))
114	113	ralbidv 2466	. . . . . . . . . . . 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 373	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
117	116	ralrimiva 2539	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
118	117	ralrimiva 2539	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
119	114, 118, 9	rspcdva 2835	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑄 < 𝑦 → (𝐹‘𝑄) < (𝐹‘𝑦)))
120	119	adantr 274	. . . . . . . . . 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 2835	. . . . . . . . 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 8017	. . . . . . 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 11118	. . . . . 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 7927	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℂ)
126	33	recnd 7927	. . . . . . . . . . 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 8211	. . . . . . . . . 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 5490	. . . . . . . . 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 9636	. . . . . . . . . 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 11109	. . . . . . . . 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 2198	. . . . . . . 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 4004	. . . . . . 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 5865	. . . . . . . . . 10 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → (abs‘(𝑧 − 𝑄)) = (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)))
134	133	breq1d 3992	. . . . . . . . 9 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑))
135	134	imbrov2fvoveq 5867	. . . . . . . 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 522	. . . . . . . 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 274	. . . . . . . . 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 3143	. . . . . . . 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 2835	. . . . . . 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 4006	. . . . 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 274	. . . . . 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 8452	. . . . 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 166	. . . 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 5486	. . . . . 6 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑤) = (𝐹‘(𝑄 + (𝑑 / 2))))
146	145	breq1d 3992	. . . . 5 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
147	146, 6	elrab2 2885	. . . 4 ⊢ ((𝑄 + (𝑑 / 2)) ∈ 𝐿 ↔ ((𝑄 + (𝑑 / 2)) ∈ (𝐴[,]𝐵) ∧ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
148	102, 144, 147	sylanbrc 414	. . 3 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ∈ 𝐿)
149		breq2 3986	. . . 4 ⊢ (𝑟 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑟 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
150	149	rspcev 2830	. . 3 ⊢ (((𝑄 + (𝑑 / 2)) ∈ 𝐿 ∧ 𝑄 < (𝑄 + (𝑑 / 2))) → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)
151	148, 38, 150	syl2anc 409	. 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 2588	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  {crab 2448   ⊆ wss 3116   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  ℂcc 7751  ℝcr 7752  0cc0 7753   + caddc 7756  ℝ^*cxr 7932   < clt 7933   ≤ cle 7934   − cmin 8069   / cdiv 8568  2c2 8908  ℝ⁺crp 9589  [,]cicc 9827  abscabs 10939  –cn→ccncf 13197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-map 6616  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-rp 9590  df-icc 9831  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-cncf 13198

This theorem is referenced by:  ivthinclemlr  13255