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Theorem evth 24944

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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Comp)
5		cmptop 23378	. . . . . . . . . 10 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
7	1	toptopon 22900	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	6, 7	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9		eqid 2739	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
10	9	cnfldtopon 24765	. . . . . . . . . 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 11130	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
13	8, 11, 12	cnmptc 23645	. . . . . . . 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 24745	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ = ∪ (topGen‘ran (,))
16	2	unieqi 4850	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2765	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ = ∪ 𝐾
18	1, 17	cnf 23229	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
19	14, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
20	19	frnd 6663	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
21	19	fdmd 6665	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝑋)
22		evth.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
23	21, 22	eqnetrd 3001	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
24		dm0rn0 5866	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
25	24	necon3bii 2986	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
26	23, 25	sylib 219	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
27	1, 2, 3, 14	bndth 24943	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)
28	19	ffnd 6656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝑋)
29		breq1 5075	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ 𝑥 ↔ (𝐹‘𝑦) ≤ 𝑥))
30	29	ralrn 7029	. . . . . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
33	27, 32	mpbird 258	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥)
34	20, 26, 33	3jca 1134	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
35		suprcl 12107	. . . . . . . . . . . . . 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 11164	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
38	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
39	8, 11, 38	cnmptc 23645	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
40	19	feqmptd 6895	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)))
41	9	cnfldtop 24766	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ Top
42		cnrest2r 23270	. . . . . . . . . . . . . 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		tgioo4 24788	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
45	2, 44	eqtri 2762	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
46	45	oveq2i 7367	. . . . . . . . . . . . . 14 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
47	14, 46	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
48	43, 47	sselid 3913	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
49	40, 48	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
50	49	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
51	9	subcn 24850	. . . . . . . . . . 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 23649	. . . . . . . . 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 732	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
55		ffvelcdm 7022	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
56	55	adantll 720	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
57		eldifsn 4719	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
58	56, 57	sylib 219	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
59	58	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
60	54, 59	resubcld 11569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℝ)
61	60	recnd 11164	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℂ)
62	54	recnd 11164	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
63	59	recnd 11164	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
64	58	simprd 496	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < ))
65	64	necomd 2989	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑧))
66	62, 63, 65	subne0d 11505	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ≠ 0)
67		eldifsn 4719	. . . . . . . . . . . . 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 589	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ (ℂ ∖ {0}))
69	68	fmpttd 7056	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))):𝑋⟶(ℂ ∖ {0}))
70	69	frnd 6663	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ⊆ (ℂ ∖ {0}))
71		difssd 4067	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
72		cnrest2 23269	. . . . . . . . . 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 1379	. . . . . . . . 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 233	. . . . . . . 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 2739	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0})) = ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))
76	9, 75	divcn 24853	. . . . . . . . 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 23649	. . . . . . 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 11973	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ ℝ)
80	79	fmpttd 7056	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))):𝑋⟶ℝ)
81	80	frnd 6663	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ⊆ ℝ)
82		ax-resscn 11086	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
84		cnrest2 23269	. . . . . . . 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 1379	. . . . . . 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 233	. . . . . 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 2850	. . . . 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 24943	. . . 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 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
90		simpr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
91		1re 11135	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
92		ifcl 4500	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
93	90, 91, 92	sylancl 592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
94		0red 11138	. . . . . . . . . . . 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 11663	. . . . . . . . . . . . 13 ⊢ 0 < 1
97	96	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
98		max1 13128	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
99	91, 90, 98	sylancr 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
100	94, 95, 93, 97, 99	ltletrd 11297	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
101	100	gt0ne0d 11705	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
102	93, 101	rereccld 11973	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
103	93, 100	recgt0d 12081	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
104	102, 103	elrpd 12974	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ⁺)
105	89, 104	ltsubrpd 13009	. . . . . . 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 11569	. . . . . . . 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 11284	. . . . . . 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 233	. . . . . 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 776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ ℝ)
110		max2 13130	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
111	91, 109, 110	sylancr 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
112	36	ad2antrr 732	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113		ffvelcdm 7022	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
114	113	ad2ant2l 752	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
115		eldifsn 4719	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < )))
116	114, 115	sylib 219	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ ℝ)
118	112, 117	resubcld 11569	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ∈ ℝ)
119		fnfvelrn 7021	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
120	28, 119	sylan 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
121		suprub 12108	. . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
123	122	ad2ant2rl 755	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
124	116	simprd 496	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < ))
125	124	necomd 2989	. . . . . . . . . . . . . . . 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 11291	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) < sup(ran 𝐹, ℝ, < ))
127	117, 112	posdifd 11728	. . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
129	128	gt0ne0d 11705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ≠ 0)
130	118, 129	rereccld 11973	. . . . . . . . . . . 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 592	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
132		letr 11231	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 700	. . . . . . . . . 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 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
136	135	oveq2d 7372	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
137	136	oveq2d 7372	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
138		eqid 2739	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) = (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))
139		ovex 7389	. . . . . . . . . . . . 13 ⊢ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ V
140	137, 138, 139	fvmpt 6935	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
141	140	breq1d 5082	. . . . . . . . . . 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 735	. . . . . . . . . 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 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
144	100	adantrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
145	131, 144	recgt0d 12081	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
146		lerec 12030	. . . . . . . . . . . 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 844	. . . . . . . . . . 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 11620	. . . . . . . . . . . 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 1379	. . . . . . . . . . 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 11164	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
151	101	adantrr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
152	150, 151	recrecd 11919	. . . . . . . . . . . 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 5084	. . . . . . . . . . 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 310	. . . . . . . . . 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 295	. . . . . . . . 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 468	. . . . . . . 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 3151	. . . . . . 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 732	. . . . . . . . 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 12113	. . . . . . . . 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 590	. . . . . . . 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 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
162		breq1 5075	. . . . . . . . . 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 7029	. . . . . . . . 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 280	. . . . . . 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 260	. . . . . 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 199	. . . . 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 3134	. . . 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 822	. . 3 ⊢ (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
170	122	ralrimiva 3131	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
171		breq2 5076	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < )))
172	171	ralbidv 3162	. . . . . . . . 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 248	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
174	173	necon3bd 2948	. . . . . . 7 ⊢ (𝜑 → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
175	174	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
176	19	ffvelcdmda 7025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ)
177		eldifsn 4719	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
178	177	baib 540	. . . . . . 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 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
181	180	ralimdva 3151	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
182		ffnfv 7060	. . . . . 6 ⊢ (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
183	182	baib 540	. . . . 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 260	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
186	169, 185	mtod 199	. 2 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
187		dfrex2 3066	. 2 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
188	186, 187	sylibr 235	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4261 ifcif 4454 {csn 4555 ∪ cuni 4838 class class class wbr 5072 ↦ cmpt 5153 dom cdm 5618 ran crn 5619 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 supcsup 9343 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 (,)cioo 13289 ↾_t crest 17374 TopOpenctopn 17375 topGenctg 17391 ℂ_fldccnfld 21347 Topctop 22876 TopOnctopon 22893 Cn ccn 23207 Compccmp 23369 ×_t ctx 23543

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-icc 13296 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cn 23210 df-cnp 23211 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-xms 24303 df-ms 24304 df-tms 24305

This theorem is referenced by: evth2 24945 evthicc 25444 evthf 45475 cncmpmax 45480

W3C validator