ivthlem2 - Metamath Proof Explorer

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Theorem ivthlem2 24056

Description: Lemma for ivth 24058. 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 484	. . . 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	. . . . . . . . . . 11 ⊢ 𝑆 = {𝑥 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑥) ≤ 𝑈}
6	5	ssrab3 4008	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (𝐴[,]𝐵)
7		ivth.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ivth.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		iccssre 12807	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
10	7, 8, 9	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
11	6, 10	sstrid 3926	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ⊆ ℝ)
12		ivth.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ ℝ)
13		ivth.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 < 𝐵)
14		ivth.8	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ)
15		ivth.9	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)))
16	7, 8, 12, 13, 3, 1, 14, 15, 5	ivthlem1 24055	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
17	16	simpld 498	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
18	17	ne0d 4251	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ≠ ∅)
19	16	simprd 499	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵)
20		brralrspcev 5090	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥)
21	8, 19, 20	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥)
22	11, 18, 21	suprcld 11591	. . . . . . . 8 ⊢ (𝜑 → sup(𝑆, ℝ, < ) ∈ ℝ)
23	4, 22	eqeltrid 2894	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ)
24	11, 18, 21, 17	suprubd 11590	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≤ sup(𝑆, ℝ, < ))
25	24, 4	breqtrrdi 5072	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
26	11, 18, 21	3jca 1125	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥))
27		suprleub 11594	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝐵 ∈ ℝ) → (sup(𝑆, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
28	26, 8, 27	syl2anc 587	. . . . . . . . 9 ⊢ (𝜑 → (sup(𝑆, ℝ, < ) ≤ 𝐵 ↔ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝐵))
29	19, 28	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → sup(𝑆, ℝ, < ) ≤ 𝐵)
30	4, 29	eqbrtrid 5065	. . . . . . 7 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
31		elicc2 12790	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
32	7, 8, 31	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
33	23, 25, 30, 32	mpbir3and 1339	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
34	3, 33	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
35	34	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → 𝐶 ∈ 𝐷)
36		fveq2 6645	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
37	36	eleq1d 2874	. . . . . . 7 ⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐶) ∈ ℝ))
38	14	ralrimiva 3149	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
39	37, 38, 33	rspcdva 3573	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ℝ)
40		difrp 12415	. . . . . 6 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((𝐹‘𝐶) < 𝑈 ↔ (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺))
41	39, 12, 40	syl2anc 587	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐶) < 𝑈 ↔ (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺))
42	41	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (𝑈 − (𝐹‘𝐶)) ∈ ℝ⁺)
43		cncfi 23499	. . . 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))))
44	2, 35, 42, 43	syl3anc 1368	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → ∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))))
45		ssralv 3981	. . . . . . 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)))))
46	3, 45	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)))))
47	46	ad2antrr 725	. . . . 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)))))
48	8	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
49	23	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 ∈ ℝ)
50		rphalfcl 12404	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) ∈ ℝ⁺)
51	50	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) ∈ ℝ⁺)
52	51	rpred 12419	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) ∈ ℝ)
53	49, 52	readdcld 10659	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + (𝑧 / 2)) ∈ ℝ)
54	48, 53	ifcld 4470	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ ℝ)
55	7	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
56	25	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ≤ 𝐶)
57	15	simprd 499	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵))
58		fveq2 6645	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
59	58	eleq1d 2874	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝐵) ∈ ℝ))
60	7	rexrd 10680	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
61	8	rexrd 10680	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
62	7, 8, 13	ltled 10777	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
63		ubicc2 12843	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
64	60, 61, 62, 63	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
65	59, 38, 64	rspcdva 3573	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
66		lttr 10706	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝐶) ∈ ℝ ∧ 𝑈 ∈ ℝ ∧ (𝐹‘𝐵) ∈ ℝ) → (((𝐹‘𝐶) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)) → (𝐹‘𝐶) < (𝐹‘𝐵)))
67	39, 12, 65, 66	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝐹‘𝐶) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵)) → (𝐹‘𝐶) < (𝐹‘𝐵)))
68	57, 67	mpan2d 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹‘𝐶) < 𝑈 → (𝐹‘𝐶) < (𝐹‘𝐵)))
69	68	imp 410	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (𝐹‘𝐶) < (𝐹‘𝐵))
70	69	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐹‘𝐶) < (𝐹‘𝐵))
71	39	ltnrd 10763	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ¬ (𝐹‘𝐶) < (𝐹‘𝐶))
72		fveq2 6645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 = 𝐶 → (𝐹‘𝐵) = (𝐹‘𝐶))
73	72	breq2d 5042	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 = 𝐶 → ((𝐹‘𝐶) < (𝐹‘𝐵) ↔ (𝐹‘𝐶) < (𝐹‘𝐶)))
74	73	notbid 321	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 = 𝐶 → (¬ (𝐹‘𝐶) < (𝐹‘𝐵) ↔ ¬ (𝐹‘𝐶) < (𝐹‘𝐶)))
75	71, 74	syl5ibrcom 250	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐵 = 𝐶 → ¬ (𝐹‘𝐶) < (𝐹‘𝐵)))
76	75	necon2ad 3002	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐵 ≠ 𝐶))
77	76, 30	jctild 529	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → (𝐶 ≤ 𝐵 ∧ 𝐵 ≠ 𝐶)))
78	23, 8	ltlend 10774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶 < 𝐵 ↔ (𝐶 ≤ 𝐵 ∧ 𝐵 ≠ 𝐶)))
79	77, 78	sylibrd 262	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐶 < 𝐵))
80	79	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ((𝐹‘𝐶) < (𝐹‘𝐵) → 𝐶 < 𝐵))
81	70, 80	mpd 15	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < 𝐵)
82	49, 51	ltaddrpd 12452	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < (𝐶 + (𝑧 / 2)))
83		breq2 5034	. . . . . . . . . . . 12 ⊢ (𝐵 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝐶 < 𝐵 ↔ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2)))))
84		breq2 5034	. . . . . . . . . . . 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)))))
85	83, 84	ifboth 4463	. . . . . . . . . . 11 ⊢ ((𝐶 < 𝐵 ∧ 𝐶 < (𝐶 + (𝑧 / 2))) → 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
86	81, 82, 85	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
87	49, 54, 86	ltled 10777	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐶 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
88	55, 49, 54, 56, 87	letrd 10786	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝐴 ≤ if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))))
89		min1 12570	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 + (𝑧 / 2)) ∈ ℝ) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)
90	48, 53, 89	syl2anc 587	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ 𝐵)
91		elicc2 12790	. . . . . . . . . 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)))
92	7, 8, 91	syl2anc 587	. . . . . . . . 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)))
93	92	ad2antrr 725	. . . . . . . 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)))
94	54, 88, 90, 93	mpbir3and 1339	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ∈ (𝐴[,]𝐵))
95	49, 54, 87	abssubge0d 14783	. . . . . . . 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))
96		rpre 12385	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ)
97	96	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → 𝑧 ∈ ℝ)
98	49, 97	readdcld 10659	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + 𝑧) ∈ ℝ)
99		min2 12571	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 + (𝑧 / 2)) ∈ ℝ) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ (𝐶 + (𝑧 / 2)))
100	48, 53, 99	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) ≤ (𝐶 + (𝑧 / 2)))
101		rphalflt 12406	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) < 𝑧)
102	101	adantl 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝑧 / 2) < 𝑧)
103	52, 97, 49, 102	ltadd2dd 10788	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (𝐶 + (𝑧 / 2)) < (𝐶 + 𝑧))
104	54, 53, 98, 100, 103	lelttrd 10787	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) < (𝐶 + 𝑧))
105	54, 49, 97	ltsubadd2d 11227	. . . . . . . . 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)))
106	104, 105	mpbird 260	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶) < 𝑧)
107	95, 106	eqbrtrd 5052	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (abs‘(if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) − 𝐶)) < 𝑧)
108		fvoveq1 7158	. . . . . . . . . 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)))
109	108	breq1d 5040	. . . . . . . . 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))
110		breq2 5034	. . . . . . . . 9 ⊢ (𝑦 = if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2))) → (𝐶 < 𝑦 ↔ 𝐶 < if(𝐵 ≤ (𝐶 + (𝑧 / 2)), 𝐵, (𝐶 + (𝑧 / 2)))))
111	109, 110	anbi12d 633	. . . . . . . 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))))))
112	111	rspcev 3571	. . . . . . 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))
113	94, 107, 86, 112	syl12anc 835	. . . . . 6 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → ∃𝑦 ∈ (𝐴[,]𝐵)((abs‘(𝑦 − 𝐶)) < 𝑧 ∧ 𝐶 < 𝑦))
114		r19.29 3216	. . . . . . 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)))
115		pm3.45 624	. . . . . . . . . 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)))
116	115	imp 410	. . . . . . . . 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))
117		simprr 772	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝐶 < 𝑦)
118		fveq2 6645	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
119	118	eleq1d 2874	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑦) ∈ ℝ))
120		simplll 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝜑)
121	120, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ)
122		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑦 ∈ (𝐴[,]𝐵))
123	119, 121, 122	rspcdva 3573	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝑦) ∈ ℝ)
124	120, 39	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝐶) ∈ ℝ)
125	120, 12	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑈 ∈ ℝ)
126	125, 124	resubcld 11057	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑈 − (𝐹‘𝐶)) ∈ ℝ)
127	123, 124, 126	absdifltd 14785	. . . . . . . . . . . . . 14 ⊢ ((((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))))))
128		ltle 10718	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹‘𝑦) ∈ ℝ ∧ 𝑈 ∈ ℝ) → ((𝐹‘𝑦) < 𝑈 → (𝐹‘𝑦) ≤ 𝑈))
129	123, 125, 128	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < 𝑈 → (𝐹‘𝑦) ≤ 𝑈))
130	124	recnd 10658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐹‘𝐶) ∈ ℂ)
131	125	recnd 10658	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑈 ∈ ℂ)
132	130, 131	pncan3d 10989	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) = 𝑈)
133	132	breq2d 5042	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) ↔ (𝐹‘𝑦) < 𝑈))
134	118	breq1d 5040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≤ 𝑈 ↔ (𝐹‘𝑦) ≤ 𝑈))
135	134, 5	elrab2 3631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑦) ≤ 𝑈))
136	135	baib 539	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (𝐴[,]𝐵) → (𝑦 ∈ 𝑆 ↔ (𝐹‘𝑦) ≤ 𝑈))
137	136	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 ↔ (𝐹‘𝑦) ≤ 𝑈))
138	129, 133, 137	3imtr4d 297	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) → 𝑦 ∈ 𝑆))
139		suprub 11589	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ sup(𝑆, ℝ, < ))
140	139, 4	breqtrrdi 5072	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) ∧ 𝑦 ∈ 𝑆) → 𝑦 ≤ 𝐶)
141	140	ex 416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑥) → (𝑦 ∈ 𝑆 → 𝑦 ≤ 𝐶))
142	120, 26, 141	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 → 𝑦 ≤ 𝐶))
143	120, 10	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝐴[,]𝐵) ⊆ ℝ)
144	143, 122	sseldd 3916	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝑦 ∈ ℝ)
145	120, 23	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → 𝐶 ∈ ℝ)
146	144, 145	lenltd 10775	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ≤ 𝐶 ↔ ¬ 𝐶 < 𝑦))
147	142, 146	sylibd 242	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → (𝑦 ∈ 𝑆 → ¬ 𝐶 < 𝑦))
148	138, 147	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((𝐹‘𝑦) < ((𝐹‘𝐶) + (𝑈 − (𝐹‘𝐶))) → ¬ 𝐶 < 𝑦))
149	148	adantld 494	. . . . . . . . . . . . . 14 ⊢ ((((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))
150	127, 149	sylbid 243	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ 𝐶 < 𝑦))
151	117, 150	mt2d 138	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ¬ (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)))
152	151	pm2.21d 121	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝐶 < 𝑦)) → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ (𝐹‘𝐶) < 𝑈))
153	152	expr 460	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (𝐶 < 𝑦 → ((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) → ¬ (𝐹‘𝐶) < 𝑈)))
154	153	impcomd 415	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ (𝐴[,]𝐵)) → (((abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶)) ∧ 𝐶 < 𝑦) → ¬ (𝐹‘𝐶) < 𝑈))
155	116, 154	syl5 34	. . . . . . . 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))
156	155	rexlimdva 3243	. . . . . . 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))
157	114, 156	syl5 34	. . . . . 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))
158	113, 157	mpan2d 693	. . . . 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))
159	47, 158	syld 47	. . . 4 ⊢ (((𝜑 ∧ (𝐹‘𝐶) < 𝑈) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ¬ (𝐹‘𝐶) < 𝑈))
160	159	rexlimdva 3243	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → (∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝐷 ((abs‘(𝑦 − 𝐶)) < 𝑧 → (abs‘((𝐹‘𝑦) − (𝐹‘𝐶))) < (𝑈 − (𝐹‘𝐶))) → ¬ (𝐹‘𝐶) < 𝑈))
161	44, 160	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐶) < 𝑈) → ¬ (𝐹‘𝐶) < 𝑈)
162	161	pm2.01da 798	1 ⊢ (𝜑 → ¬ (𝐹‘𝐶) < 𝑈)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 ⊆ wss 3881 ∅c0 4243 ifcif 4425 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℂcc 10524 ℝcr 10525 + caddc 10529 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 ℝ⁺crp 12377 [,]cicc 12729 abscabs 14585 –cn→ccncf 23481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-icc 12733 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-cncf 23483

This theorem is referenced by: ivthlem3 24057

W3C validator