ivthlem2 - Metamath Proof Explorer

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Theorem ivthlem2 23028

Description: Lemma for ivth 23030. Show that the supremum of 𝑆 cannot be less than 𝑈. If it was, continuity of 𝐹 implies that there are points just above the supremum that are also less than 𝑈, a contradiction. (Contributed by Mario Carneiro, 17-Jun-2014.)

**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)))
ivth.10	⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}
ivth.11	⊢ 𝐶 = sup(𝑆, ℝ, < )

**Assertion**
Ref	Expression
ivthlem2	⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈)

Distinct variable groups: 𝑥,𝐵 𝑥,𝐷 𝑥,𝐹 𝜑,𝑥 𝑥,𝐴 𝑥,𝐶 𝑥,𝑆 𝑥,𝑈

Proof of Theorem ivthlem2 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ivth.7	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ))
2	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → 𝐹 ∈ (𝐷–cn→ℂ))
3		ivth.5	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷)
4		ivth.11	. . . . . . . 8 ⊢ 𝐶 = sup(𝑆, ℝ, < )
5		ivth.10	. . . . . . . . . . . 12 ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}
6		ssrab2 3650	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈} ⊆ (𝐴[,]𝐵)
7	5, 6	eqsstri 3598	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ (𝐴[,]𝐵)
8		ivth.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		ivth.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
10		iccssre 12126	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 691	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12	7, 11	syl5ss 3579	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
13		ivth.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈 ∈ ℝ)
14		ivth.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 < 𝐵)
15		ivth.8	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
16		ivth.9	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
17	8, 9, 13, 14, 3, 1, 15, 16, 5	ivthlem1 23027	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
18	17	simpld 474	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
19		ne0i 3880	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅)
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≠ ∅)
21	17	simprd 478	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)
22		breq2 4587	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝐵 → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ 𝐵))
23	22	ralbidv 2969	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝐵 → (∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
24	23	rspcev 3282	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥)
25	9, 21, 24	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥)
26	12, 20, 25	3jca 1235	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥))
27		suprcl 10862	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) → sup(𝑆, ℝ, < ) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . 8 ⊢ (𝜑 → sup(𝑆, ℝ, < ) ∈ ℝ)
29	4, 28	syl5eqel 2692	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
30		suprub 10863	. . . . . . . . 9 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝐴 ∈ 𝑆) → 𝐴 ≤ sup(𝑆, ℝ, < ))
31	26, 18, 30	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ sup(𝑆, ℝ, < ))
32	31, 4	syl6breqr 4625	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
33		suprleub 10866	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝑆, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
34	26, 9, 33	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (sup(𝑆, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
35	21, 34	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → sup(𝑆, ℝ, < ) ≤ 𝐵)
36	4, 35	syl5eqbr 4618	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
37		elicc2 12109	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
38	8, 9, 37	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
39	29, 32, 36, 38	mpbir3and 1238	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
40	3, 39	sseldd 3569	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
41	40	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → 𝐶 ∈ 𝐷)
42	15	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
43		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
44	43	eleq1d 2672	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐶) ∈ ℝ))
45	44	rspcv 3278	. . . . . . 7 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ → (𝐹‘𝐶) ∈ ℝ))
46	39, 42, 45	sylc 63	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
47		difrp 11744	. . . . . 6 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((𝐹‘𝐶) < 𝑈 ↔ (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺))
48	46, 13, 47	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶) < 𝑈 ↔ (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺))
49	48	biimpa 500	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺)
50		cncfi 22505	. . . 4 ⊢ ((𝐹 ∈ (𝐷–cn→ℂ) ∧ 𝐶 ∈ 𝐷 ∧ (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺) → ∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))))
51	2, 41, 49, 50	syl3anc 1318	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → ∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))))
52		ssralv 3629	. . . . . . 7 ⊢ ((𝐴[,]𝐵) ⊆ 𝐷 → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)))))
53	3, 52	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)))))
54	53	ad2antrr 758	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)))))
55	9	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
56	29	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
57		rphalfcl 11734	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) ∈ ℝ⁺)
58	57	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) ∈ ℝ⁺)
59	58	rpred 11748	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) ∈ ℝ)
60	56, 59	readdcld 9948	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + (𝑧 / 2)) ∈ ℝ)
61	55, 60	ifcld 4081	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ ℝ)
62	8	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
63	32	ad2antrr 758	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ≤ 𝐶)
64	16	simprd 478	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵))
65	8	rexrd 9968	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
66	9	rexrd 9968	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
67	8, 9, 14	ltled 10064	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
68		ubicc2 12160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
69	65, 66, 67, 68	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
70		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
71	70	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
72	71	rspcv 3278	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (𝐴[,]𝐵) → (∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ → (𝐹‘𝐵) ∈ ℝ))
73	69, 42, 72	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
74		lttr 9993	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → (((𝐹‘𝐶) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)) → (𝐹‘𝐶) < (𝐹‘𝐵)))
75	46, 13, 73, 74	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝐹‘𝐶) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)) → (𝐹‘𝐶) < (𝐹‘𝐵)))
76	64, 75	mpan2d 706	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹‘𝐶) < 𝑈 → (𝐹‘𝐶) < (𝐹‘𝐵)))
77	76	imp 444	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (𝐹‘𝐶) < (𝐹‘𝐵))
78	77	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐹‘𝐶) < (𝐹‘𝐵))
79	46	ltnrd 10050	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐹‘𝐶) < (𝐹‘𝐶))
80		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 → (𝐹‘𝐵) = (𝐹‘𝐶))
81	80	breq2d 4595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = 𝐶 → ((𝐹‘𝐶) < (𝐹‘𝐵) ↔ (𝐹‘𝐶) < (𝐹‘𝐶)))
82	81	notbid 307	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐶 → (¬ (𝐹‘𝐶) < (𝐹‘𝐵) ↔ ¬ (𝐹‘𝐶) < (𝐹‘𝐶)))
83	79, 82	syl5ibrcom 236	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 = 𝐶 → ¬ (𝐹‘𝐶) < (𝐹‘𝐵)))
84	83	necon2ad 2797	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐵 ≠ 𝐶))
85	84, 36	jctild 564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → (𝐶 ≤ 𝐵 ∧ 𝐵 ≠ 𝐶)))
86	29, 9	ltlend 10061	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ (𝐶 ≤ 𝐵 ∧ 𝐵 ≠ 𝐶)))
87	85, 86	sylibrd 248	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐶 < 𝐵))
88	87	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐶 < 𝐵))
89	78, 88	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < 𝐵)
90	56, 58	ltaddrpd 11781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < (𝐶 + (𝑧 / 2)))
91		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝐵 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝐶 < 𝐵 ↔ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2)))))
92		breq2 4587	. . . . . . . . . . . 12 ⊢ ((𝐶 + (𝑧 / 2)) = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝐶 < (𝐶 + (𝑧 / 2)) ↔ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2)))))
93	91, 92	ifboth 4074	. . . . . . . . . . 11 ⊢ ((𝐶 < 𝐵 ∧ 𝐶 < (𝐶 + (𝑧 / 2))) → 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
94	89, 90, 93	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
95	56, 61, 94	ltled 10064	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
96	62, 56, 61, 63, 95	letrd 10073	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
97		min1 11894	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 + (𝑧 / 2)) ∈ ℝ) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)
98	55, 60, 97	syl2anc 691	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)
99		elicc2 12109	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵) ↔ (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ ℝ ∧ 𝐴 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∧ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)))
100	8, 9, 99	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵) ↔ (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ ℝ ∧ 𝐴 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∧ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)))
101	100	ad2antrr 758	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵) ↔ (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ ℝ ∧ 𝐴 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∧ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)))
102	61, 96, 98, 101	mpbir3and 1238	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵))
103	56, 61, 95	abssubge0d 14018	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) = (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶))
104		rpre 11715	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
105	104	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝑧 ∈ ℝ)
106	56, 105	readdcld 9948	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + 𝑧) ∈ ℝ)
107		min2 11895	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 + (𝑧 / 2)) ∈ ℝ) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ (𝐶 + (𝑧 / 2)))
108	55, 60, 107	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ (𝐶 + (𝑧 / 2)))
109		rphalflt 11736	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) < 𝑧)
110	109	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) < 𝑧)
111	59, 105, 56, 110	ltadd2dd 10075	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + (𝑧 / 2)) < (𝐶 + 𝑧))
112	61, 60, 106, 108, 111	lelttrd 10074	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) < (𝐶 + 𝑧))
113	61, 56, 105	ltsubadd2d 10504	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ((if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶) < 𝑧 ↔ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) < (𝐶 + 𝑧)))
114	112, 113	mpbird 246	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶) < 𝑧)
115	103, 114	eqbrtrd 4605	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) < 𝑧)
116		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝑦 − 𝐶) = (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶))
117	116	fveq2d 6107	. . . . . . . . . 10 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (abs‘(𝑦 − 𝐶)) = (abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)))
118	117	breq1d 4593	. . . . . . . . 9 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → ((abs‘(𝑦 − 𝐶)) < 𝑧 ↔ (abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) < 𝑧))
119		breq2 4587	. . . . . . . . 9 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝐶 < 𝑦 ↔ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2)))))
120	118, 119	anbi12d 743	. . . . . . . 8 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦) ↔ ((abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) < 𝑧 ∧ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))))
121	120	rspcev 3282	. . . . . . 7 ⊢ ((if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵) ∧ ((abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) < 𝑧 ∧ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))) → ∃𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦))
122	102, 115, 94, 121	syl12anc 1316	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ∃𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦))
123		r19.29 3054	. . . . . . 7 ⊢ ((∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ∃𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)) → ∃𝑦 ∈ (𝐴[,]𝐵)(((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)))
124		pm3.45 875	. . . . . . . . . 10 ⊢ (((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → (((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) ∧ 𝐶 < 𝑦)))
125	124	imp 444	. . . . . . . . 9 ⊢ ((((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) ∧ 𝐶 < 𝑦))
126		simprr 792	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝐶 < 𝑦)
127		simprl 790	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑦 ∈ (𝐴[,]𝐵))
128		simplll 794	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝜑)
129	128, 42	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
130		fveq2 6103	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
131	130	eleq1d 2672	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑦) ∈ ℝ))
132	131	rspcv 3278	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ → (𝐹‘𝑦) ∈ ℝ))
133	127, 129, 132	sylc 63	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝑦) ∈ ℝ)
134	128, 46	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝐶) ∈ ℝ)
135	128, 13	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑈 ∈ ℝ)
136	135, 134	resubcld 10337	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑈 − (𝐹‘𝐶)) ∈ ℝ)
137	133, 134, 136	absdifltd 14020	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) ↔ (((𝐹‘𝐶) − (𝑈 − (𝐹‘𝐶))) < (𝐹‘𝑦) ∧ (𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))))))
138		ltle 10005	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((𝐹‘𝑦) < 𝑈 → (𝐹‘𝑦) ≤ 𝑈))
139	133, 135, 138	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < 𝑈 → (𝐹‘𝑦) ≤ 𝑈))
140	134	recnd 9947	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝐶) ∈ ℂ)
141	135	recnd 9947	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑈 ∈ ℂ)
142	140, 141	pncan3d 10274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) = 𝑈)
143	142	breq2d 4595	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) ↔ (𝐹‘𝑦) < 𝑈))
144	130	breq1d 4593	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≤ 𝑈 ↔ (𝐹‘𝑦) ≤ 𝑈))
145	144, 5	elrab2 3333	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑦) ≤ 𝑈))
146	145	baib 942	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ 𝑆 ↔ (𝐹‘𝑦) ≤ 𝑈))
147	146	ad2antrl 760	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 ↔ (𝐹‘𝑦) ≤ 𝑈))
148	139, 143, 147	3imtr4d 282	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) → 𝑦 ∈ 𝑆))
149		suprub 10863	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ sup(𝑆, ℝ, < ))
150	149, 4	syl6breqr 4625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ 𝐶)
151	150	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) → (𝑦 ∈ 𝑆 → 𝑦 ≤ 𝐶))
152	128, 26, 151	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 → 𝑦 ≤ 𝐶))
153	128, 11	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐴[,]𝐵) ⊆ ℝ)
154	153, 127	sseldd 3569	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑦 ∈ ℝ)
155	128, 29	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝐶 ∈ ℝ)
156	154, 155	lenltd 10062	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ≤ 𝐶 ↔ ¬ 𝐶 < 𝑦))
157	152, 156	sylibd 228	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 → ¬ 𝐶 < 𝑦))
158	148, 157	syld 46	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) → ¬ 𝐶 < 𝑦))
159	158	adantld 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((((𝐹‘𝐶) − (𝑈 − (𝐹‘𝐶))) < (𝐹‘𝑦) ∧ (𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶)))) → ¬ 𝐶 < 𝑦))
160	137, 159	sylbid 229	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ 𝐶 < 𝑦))
161	126, 160	mt2d 130	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ¬ (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)))
162	161	pm2.21d 117	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ (𝐹‘𝐶) < 𝑈))
163	162	expr 641	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐶 < 𝑦 → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ (𝐹‘𝐶) < 𝑈)))
164	163	com23 84	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → (𝐶 < 𝑦 → ¬ (𝐹‘𝐶) < 𝑈)))
165	164	impd 446	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) ∧ 𝐶 < 𝑦) → ¬ (𝐹‘𝐶) < 𝑈))
166	125, 165	syl5 33	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → ((((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)) → ¬ (𝐹‘𝐶) < 𝑈))
167	166	rexlimdva 3013	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (∃𝑦 ∈ (𝐴[,]𝐵)(((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)) → ¬ (𝐹‘𝐶) < 𝑈))
168	123, 167	syl5 33	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ((∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) ∧ ∃𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦)) → ¬ (𝐹‘𝐶) < 𝑈))
169	122, 168	mpan2d 706	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ¬ (𝐹‘𝐶) < 𝑈))
170	54, 169	syld 46	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ¬ (𝐹‘𝐶) < 𝑈))
171	170	rexlimdva 3013	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ¬ (𝐹‘𝐶) < 𝑈))
172	51, 171	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → ¬ (𝐹‘𝐶) < 𝑈)
173	172	pm2.01da 457	1 ⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 ifcif 4036 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℂcc 9813 ℝcr 9814 + caddc 9818 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 [,]cicc 12049 abscabs 13822 –cn→ccncf 22487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-icc 12053 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-cncf 22489

This theorem is referenced by: ivthlem3 23029

W3C validator