Theorem ivthinclemlopn 13084

Description: Lemma for ivthinc 13091. 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 5470	. . . . . . . 8 ⊢ (𝑤 = 𝑄 → (𝐹‘𝑤) = (𝐹‘𝑄))
5	4	breq1d 3977	. . . . . . 7 ⊢ (𝑤 = 𝑄 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑄) < 𝑈))
6		ivthinclem.l	. . . . . . 7 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈}
7	5, 6	elrab2 2871	. . . . . 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 3129	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
11		ivth.3	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℝ)
12		fveq2 5470	. . . . . . 7 ⊢ (𝑥 = 𝑄 → (𝐹‘𝑥) = (𝐹‘𝑄))
13	12	eleq1d 2226	. . . . . 6 ⊢ (𝑥 = 𝑄 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑄) ∈ ℝ))
14		ivth.8	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15	14	ralrimiva 2530	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
16	13, 15, 9	rspcdva 2821	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) ∈ ℝ)
17	11, 16	resubcld 8260	. . . 4 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ)
18	8	simprd 113	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑄) < 𝑈)
19	16, 11	posdifd 8411	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑄) < 𝑈 ↔ 0 < (𝑈 − (𝐹‘𝑄))))
20	18, 19	mpbid 146	. . . 4 ⊢ (𝜑 → 0 < (𝑈 − (𝐹‘𝑄)))
21	17, 20	elrpd 9606	. . 3 ⊢ (𝜑 → (𝑈 − (𝐹‘𝑄)) ∈ ℝ+)
22		cncfi 13035	. . 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 1220	. 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 9848	. . . . . . . . . 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 994	. . . . . . 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 9622	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑑 / 2) ∈ ℝ+)
33	32	rpred 9609	. . . . . 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 7909	. . . . 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 995	. . . . . . 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 9643	. . . . . . 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 7998	. . . . . 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 8003	. . . . 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 9609	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑑 ∈ ℝ)
43	30, 42	readdcld 7909	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + 𝑑) ∈ ℝ)
44		rphalflt 9596	. . . . . . . . 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 8301	. . . . . . 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 9609	. . . . . . . . . . . . . . . . 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 8260	. . . . . . . . . . . . . . . 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 9642	. . . . . . . . . . . . . . . 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 996	. . . . . . . . . . . . . . . . 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 8302	. . . . . . . . . . . . . . 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 11089	. . . . . . . . . . . . . . 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 929	. . . . . . . . . . . . . 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 5849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝐵 → (abs‘(𝑧 − 𝑄)) = (abs‘(𝐵 − 𝑄)))
60	59	breq1d 3977	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝐵 → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘(𝐵 − 𝑄)) < 𝑑))
61	60	imbrov2fvoveq 5851	. . . . . . . . . . . . . . 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 7929	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
64	25	rexrd 7929	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ*)
65		ivth.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐴 < 𝐵)
66	24, 25, 65	ltled 7998	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
67		ubicc2 9895	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
68	63, 64, 66, 67	syl3anc 1220	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
69	2, 68	sseldd 3129	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . 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 5470	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
74	73	eleq1d 2226	. . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . 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 11089	. . . . . . . . . . . . 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 7908	. . . . . . . . . . . 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 7908	. . . . . . . . . . . . 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 8193	. . . . . . . . . . 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 3992	. . . . . . . . . 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 7970	. . . . . . . . . 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 8000	. . . . . . 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 8302	. . . . . 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 7998	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → (𝑄 + (𝑑 / 2)) ≤ 𝐵)
100		elicc2 9848	. . . . . 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 1165	. . . 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 5470	. . . . . . . . 9 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑥) = (𝐹‘(𝑄 + (𝑑 / 2))))
105	104	eleq1d 2226	. . . . . . . 8 ⊢ (𝑥 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘(𝑄 + (𝑑 / 2))) ∈ ℝ))
106	105, 75, 102	rspcdva 2821	. . . . . . 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 3971	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑦 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
108		fveq2 5470	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑦) = (𝐹‘(𝑄 + (𝑑 / 2))))
109	108	breq2d 3979	. . . . . . . . . . 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 3970	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑄 → (𝑥 < 𝑦 ↔ 𝑄 < 𝑦))
112	12	breq1d 3977	. . . . . . . . . . . . . 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 2457	. . . . . . . . . . . 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 2530	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → ∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
118	117	ralrimiva 2530	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)∀𝑦 ∈ (𝐴[,]𝐵)(𝑥 < 𝑦 → (𝐹‘𝑥) < (𝐹‘𝑦)))
119	114, 118, 9	rspcdva 2821	. . . . . . . . . . 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 2821	. . . . . . . . 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 7998	. . . . . . 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 11087	. . . . . 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 7908	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ ∀𝑧 ∈ 𝐷 ((abs‘(𝑧 − 𝑄)) < 𝑑 → (abs‘((𝐹‘𝑧) − (𝐹‘𝑄))) < (𝑈 − (𝐹‘𝑄))))) → 𝑄 ∈ ℂ)
126	33	recnd 7908	. . . . . . . . . . 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 8192	. . . . . . . . . 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 5474	. . . . . . . . 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 9613	. . . . . . . . . 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 11078	. . . . . . . . 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 2190	. . . . . . . 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 3988	. . . . . . 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 5849	. . . . . . . . . 10 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → (abs‘(𝑧 − 𝑄)) = (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)))
134	133	breq1d 3977	. . . . . . . . 9 ⊢ (𝑧 = (𝑄 + (𝑑 / 2)) → ((abs‘(𝑧 − 𝑄)) < 𝑑 ↔ (abs‘((𝑄 + (𝑑 / 2)) − 𝑄)) < 𝑑))
135	134	imbrov2fvoveq 5851	. . . . . . . 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 3129	. . . . . . . 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 2821	. . . . . . 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 3990	. . . . 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 8433	. . . . 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 5470	. . . . . 6 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → (𝐹‘𝑤) = (𝐹‘(𝑄 + (𝑑 / 2))))
146	145	breq1d 3977	. . . . 5 ⊢ (𝑤 = (𝑄 + (𝑑 / 2)) → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘(𝑄 + (𝑑 / 2))) < 𝑈))
147	146, 6	elrab2 2871	. . . 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 3971	. . . 4 ⊢ (𝑟 = (𝑄 + (𝑑 / 2)) → (𝑄 < 𝑟 ↔ 𝑄 < (𝑄 + (𝑑 / 2))))
150	149	rspcev 2816	. . 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 2579	1 ⊢ (𝜑 → ∃𝑟 ∈ 𝐿 𝑄 < 𝑟)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ∀wral 2435  ∃wrex 2436  {crab 2439   ⊆ wss 3102   class class class wbr 3967  ‘cfv 5172  (class class class)co 5826  ℂcc 7732  ℝcr 7733  0cc0 7734   + caddc 7737  ℝ^*cxr 7913   < clt 7914   ≤ cle 7915   − cmin 8050   / cdiv 8549  2c2 8889  ℝ⁺crp 9566  [,]cicc 9801  abscabs 10908  –cn→ccncf 13027

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-mulrcl 7833  ax-addcom 7834  ax-mulcom 7835  ax-addass 7836  ax-mulass 7837  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-1rid 7841  ax-0id 7842  ax-rnegex 7843  ax-precex 7844  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-apti 7849  ax-pre-ltadd 7850  ax-pre-mulgt0 7851  ax-pre-mulext 7852  ax-arch 7853  ax-caucvg 7854

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-ilim 4331  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-frec 6340  df-map 6597  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-reap 8454  df-ap 8461  df-div 8550  df-inn 8839  df-2 8897  df-3 8898  df-4 8899  df-n0 9096  df-z 9173  df-uz 9445  df-rp 9567  df-icc 9805  df-seqfrec 10354  df-exp 10428  df-cj 10753  df-re 10754  df-im 10755  df-rsqrt 10909  df-abs 10910  df-cncf 13028

This theorem is referenced by:  ivthinclemlr  13085