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Theorem evth 25010

Description: The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Hypotheses**
Ref	Expression
bndth.1	⊢ 𝑋 = ∪ 𝐽
bndth.2	⊢ 𝐾 = (topGen‘ran (,))
bndth.3	⊢ (𝜑 → 𝐽 ∈ Comp)
bndth.4	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
evth.5	⊢ (𝜑 → 𝑋 ≠ ∅)

**Assertion**
Ref	Expression
evth	⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑦,𝐾 𝜑,𝑥,𝑦 𝑥,𝑋,𝑦 𝑥,𝐽,𝑦 Allowed substitution hint: 𝐾(𝑥)

Proof of Theorem evth Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bndth.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2		bndth.2	. . . . 5 ⊢ 𝐾 = (topGen‘ran (,))
3		bndth.3	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Comp)
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Comp)
5		cmptop 23424	. . . . . . . . . 10 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
7	1	toptopon 22944	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	6, 7	sylib 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9		eqid 2740	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
10	9	cnfldtopon 24824	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
11	10	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
12		1cnd 11285	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
13	8, 11, 12	cnmptc 23691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
14		bndth.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
15		uniretop 24804	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ = ∪ (topGen‘ran (,))
16	2	unieqi 4943	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2771	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ = ∪ 𝐾
18	1, 17	cnf 23275	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
19	14, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
20	19	frnd 6755	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	19	fdmd 6757	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝑋)
22		evth.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
23	21, 22	eqnetrd 3014	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
24		dm0rn0 5949	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
25	24	necon3bii 2999	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
26	23, 25	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
27	1, 2, 3, 14	bndth 25009	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)
28	19	ffnd 6748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝑋)
29		breq1 5169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ 𝑥 ↔ (𝐹‘𝑦) ≤ 𝑥))
30	29	ralrn 7122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
31	28, 30	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
32	31	rexbidv 3185	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
33	27, 32	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥)
34	20, 26, 33	3jca 1128	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
35		suprcl 12255	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
36	34, 35	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
37	36	recnd 11318	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
38	37	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
39	8, 11, 38	cnmptc 23691	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
40	19	feqmptd 6990	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)))
41	9	cnfldtop 24825	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ Top
42		cnrest2r 23316	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂ_fld)))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂ_fld))
44	9	tgioo2 24844	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
45	2, 44	eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
46	45	oveq2i 7459	. . . . . . . . . . . . . 14 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
47	14, 46	eleqtrdi 2854	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
48	43, 47	sselid 4006	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
49	40, 48	eqeltrrd 2845	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
50	49	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
51	9	subcn 24907	. . . . . . . . . . 11 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
52	51	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
53	8, 39, 50, 52	cnmpt12f 23695	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
54	36	ad2antrr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
55		ffvelcdm 7115	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
56	55	adantll 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
57		eldifsn 4811	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
58	56, 57	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
59	58	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
60	54, 59	resubcld 11718	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℝ)
61	60	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℂ)
62	54	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
63	59	recnd 11318	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
64	58	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < ))
65	64	necomd 3002	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑧))
66	62, 63, 65	subne0d 11656	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ≠ 0)
67		eldifsn 4811	. . . . . . . . . . . . 13 ⊢ ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ (ℂ ∖ {0}) ↔ ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℂ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ≠ 0))
68	61, 66, 67	sylanbrc 582	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ (ℂ ∖ {0}))
69	68	fmpttd 7149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))):𝑋⟶(ℂ ∖ {0}))
70	69	frnd 6755	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ⊆ (ℂ ∖ {0}))
71		difssd 4160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
72		cnrest2 23315	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0})))))
73	11, 70, 71, 72	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ((𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0})))))
74	53, 73	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))))
75		eqid 2740	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0})) = ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))
76	9, 75	divcn 24911	. . . . . . . . 9 ⊢ / ∈ (((TopOpen‘ℂ_fld) ×_t ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))) Cn (TopOpen‘ℂ_fld))
77	76	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → / ∈ (((TopOpen‘ℂ_fld) ×_t ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))) Cn (TopOpen‘ℂ_fld)))
78	8, 13, 74, 77	cnmpt12f 23695	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
79	60, 66	rereccld 12121	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ ℝ)
80	79	fmpttd 7149	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))):𝑋⟶ℝ)
81	80	frnd 6755	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ⊆ ℝ)
82		ax-resscn 11241	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
84		cnrest2 23315	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
85	11, 81, 83, 84	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)) ↔ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))))
86	78, 85	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
87	86, 46	eleqtrrdi 2855	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn 𝐾))
88	1, 2, 4, 87	bndth 25009	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
89	36	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
90		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
91		1re 11290	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
92		ifcl 4593	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
93	90, 91, 92	sylancl 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
94		0red 11293	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
95	91	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
96		0lt1 11812	. . . . . . . . . . . . 13 ⊢ 0 < 1
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
98		max1 13247	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
99	91, 90, 98	sylancr 586	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
100	94, 95, 93, 97, 99	ltletrd 11450	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
101	100	gt0ne0d 11854	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
102	93, 101	rereccld 12121	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
103	93, 100	recgt0d 12229	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
104	102, 103	elrpd 13096	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ⁺)
105	89, 104	ltsubrpd 13131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) < sup(ran 𝐹, ℝ, < ))
106	89, 102	resubcld 11718	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
107	106, 89	ltnled 11437	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ((sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) < sup(ran 𝐹, ℝ, < ) ↔ ¬ sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
108	105, 107	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))))
109		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ ℝ)
110		max2 13249	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
111	91, 109, 110	sylancr 586	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
112	36	ad2antrr 725	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113		ffvelcdm 7115	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
114	113	ad2ant2l 745	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
115		eldifsn 4811	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < )))
116	114, 115	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦) ∈ ℝ ∧ (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < )))
117	116	simpld 494	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ ℝ)
118	112, 117	resubcld 11718	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ∈ ℝ)
119		fnfvelrn 7114	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
120	28, 119	sylan 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
121		suprub 12256	. . . . . . . . . . . . . . . . . 18 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥) ∧ (𝐹‘𝑦) ∈ ran 𝐹) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
122	34, 120, 121	syl2an2r 684	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
123	122	ad2ant2rl 748	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
124	116	simprd 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < ))
125	124	necomd 3002	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑦))
126	117, 112, 123, 125	leneltd 11444	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) < sup(ran 𝐹, ℝ, < ))
127	117, 112	posdifd 11877	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
128	126, 127	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
129	128	gt0ne0d 11854	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ≠ 0)
130	118, 129	rereccld 12121	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ ℝ)
131	109, 91, 92	sylancl 585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
132		letr 11384	. . . . . . . . . . . 12 ⊢ (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ) → (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥 ∧ 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
133	130, 109, 131, 132	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥 ∧ 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
134	111, 133	mpan2d 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
135		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
136	135	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
137	136	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
138		eqid 2740	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) = (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))
139		ovex 7481	. . . . . . . . . . . . 13 ⊢ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ V
140	137, 138, 139	fvmpt 7029	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
141	140	breq1d 5176	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑋 → (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥))
142	141	ad2antll 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥))
143	102	adantrr 716	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
144	100	adantrr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
145	131, 144	recgt0d 12229	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
146		lerec 12178	. . . . . . . . . . . 12 ⊢ ((((1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ ∧ 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∧ ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ∈ ℝ ∧ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
147	143, 145, 118, 128, 146	syl22anc 838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
148		lesub 11769	. . . . . . . . . . . 12 ⊢ (((1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ (𝐹‘𝑦) ∈ ℝ) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ↔ (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
149	143, 112, 117, 148	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ↔ (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
150	131	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
151	101	adantrr 716	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
152	150, 151	recrecd 12067	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) = if(1 ≤ 𝑥, 𝑥, 1))
153	152	breq2d 5178	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
154	147, 149, 153	3bitr3d 309	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
155	134, 142, 154	3imtr4d 294	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 → (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
156	155	anassrs 467	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝑋) → (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 → (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
157	156	ralimdva 3173	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
158	34	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
159		suprleub 12261	. . . . . . . . 9 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥) ∧ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
160	158, 106, 159	syl2anc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
161	28	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
162		breq1 5169	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
163	162	ralrn 7122	. . . . . . . . 9 ⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
164	161, 163	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
165	160, 164	bitrd 279	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
166	157, 165	sylibrd 259	. . . . . 6 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 → sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
167	108, 166	mtod 198	. . . . 5 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
168	167	nrexdv 3155	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
169	88, 168	pm2.65da 816	. . 3 ⊢ (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
170	122	ralrimiva 3152	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
171		breq2 5170	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < )))
172	171	ralbidv 3184	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < )))
173	170, 172	syl5ibrcom 247	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
174	173	necon3bd 2960	. . . . . . 7 ⊢ (𝜑 → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
175	174	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
176	19	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ)
177		eldifsn 4811	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
178	177	baib 535	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ ℝ → ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
179	176, 178	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
180	175, 179	sylibrd 259	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
181	180	ralimdva 3173	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
182		ffnfv 7153	. . . . . 6 ⊢ (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
183	182	baib 535	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
184	28, 183	syl 17	. . . 4 ⊢ (𝜑 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
185	181, 184	sylibrd 259	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
186	169, 185	mtod 198	. 2 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
187		dfrex2 3079	. 2 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
188	186, 187	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ∖ cdif 3973 ⊆ wss 3976 ∅c0 4352 ifcif 4548 {csn 4648 ∪ cuni 4931 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 Fn wfn 6568 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 supcsup 9509 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 (,)cioo 13407 ↾_t crest 17480 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 Topctop 22920 TopOnctopon 22937 Cn ccn 23253 Compccmp 23415 ×_t ctx 23589

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-cnp 23257 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353

This theorem is referenced by: evth2 25011 evthicc 25513 evthf 44927 cncmpmax 44932

W3C validator