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Theorem evth 22758

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 21198	. . . . . . . . . 10 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
7	1	toptopon 20722	. . . . . . . . 9 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	6, 7	sylib 208	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9		eqid 2622	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
10	9	cnfldtopon 22586	. . . . . . . . . 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 10056	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
13	8, 11, 12	cnmptc 21465	. . . . . . . 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 22566	. . . . . . . . . . . . . . . . . . 19 ⊢ ℝ = ∪ (topGen‘ran (,))
16	2	unieqi 4445	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ 𝐾 = ∪ (topGen‘ran (,))
17	15, 16	eqtr4i 2647	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ = ∪ 𝐾
18	1, 17	cnf 21050	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
19	14, 18	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
20		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
21	19, 20	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ⊆ ℝ)
22		fdm 6051	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
23	19, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom 𝐹 = 𝑋)
24		evth.5	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑋 ≠ ∅)
25	23, 24	eqnetrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 ≠ ∅)
26		dm0rn0 5342	. . . . . . . . . . . . . . . . 17 ⊢ (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
27	26	necon3bii 2846	. . . . . . . . . . . . . . . 16 ⊢ (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
28	25, 27	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran 𝐹 ≠ ∅)
29	1, 2, 3, 14	bndth 22757	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥)
30		ffn 6045	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋)
31	19, 30	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 Fn 𝑋)
32		breq1 4656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑧 ≤ 𝑥 ↔ (𝐹‘𝑦) ≤ 𝑥))
33	32	ralrn 6362	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
34	31, 33	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
35	34	rexbidv 3052	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ 𝑥))
36	29, 35	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥)
37	21, 28, 36	3jca 1242	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
38		suprcl 10983	. . . . . . . . . . . . . 14 ⊢ ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
39	37, 38	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
40	39	recnd 10068	. . . . . . . . . . . 12 ⊢ (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
41	40	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
42	8, 11, 41	cnmptc 21465	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
43	19	feqmptd 6249	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 = (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)))
44	9	cnfldtop 22587	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ Top
45		cnrest2r 21091	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂ_fld)))
46	44, 45	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂ_fld))
47	9	tgioo2 22606	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
48	2, 47	eqtri 2644	. . . . . . . . . . . . . . 15 ⊢ 𝐾 = ((TopOpen‘ℂ_fld) ↾_t ℝ)
49	48	oveq2i 6661	. . . . . . . . . . . . . 14 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ))
50	14, 49	syl6eleq 2711	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂ_fld) ↾_t ℝ)))
51	46, 50	sseldi 3601	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
52	43, 51	eqeltrrd 2702	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
53	52	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (𝐹‘𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
54	9	subcn 22669	. . . . . . . . . . 11 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
55	54	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
56	8, 42, 53, 55	cnmpt12f 21469	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
57	39	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
58		ffvelrn 6357	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
59	58	adantll 750	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
60		eldifsn 4317	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹‘𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
61	59, 60	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ ℝ ∧ (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < )))
62	61	simpld 475	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℝ)
63	57, 62	resubcld 10458	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℝ)
64	63	recnd 10068	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℂ)
65	57	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
66	62	recnd 10068	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ∈ ℂ)
67	61	simprd 479	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (𝐹‘𝑧) ≠ sup(ran 𝐹, ℝ, < ))
68	67	necomd 2849	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑧))
69	65, 66, 68	subne0d 10401	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ≠ 0)
70		eldifsn 4317	. . . . . . . . . . . . 13 ⊢ ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ (ℂ ∖ {0}) ↔ ((sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ ℂ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ≠ 0))
71	64, 69, 70	sylanbrc 698	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) ∈ (ℂ ∖ {0}))
72		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) = (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))
73	71, 72	fmptd 6385	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))):𝑋⟶(ℂ ∖ {0}))
74		frn 6053	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))):𝑋⟶(ℂ ∖ {0}) → ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ⊆ (ℂ ∖ {0}))
75	73, 74	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ⊆ (ℂ ∖ {0}))
76		difssd 3738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
77		cnrest2 21090	. . . . . . . . . 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})))))
78	11, 75, 76, 77	syl3anc 1326	. . . . . . . . 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})))))
79	56, 78	mpbid 222	. . . . . . . 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}))))
80		eqid 2622	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0})) = ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))
81	9, 80	divcn 22671	. . . . . . . . 9 ⊢ / ∈ (((TopOpen‘ℂ_fld) ×_t ((TopOpen‘ℂ_fld) ↾_t (ℂ ∖ {0}))) Cn (TopOpen‘ℂ_fld))
82	81	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)))
83	8, 13, 79, 82	cnmpt12f 21469	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂ_fld)))
84	63, 69	rereccld 10852	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) ∈ ℝ)
85		eqid 2622	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) = (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))
86	84, 85	fmptd 6385	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))):𝑋⟶ℝ)
87		frn 6053	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))):𝑋⟶ℝ → ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ⊆ ℝ)
88	86, 87	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ⊆ ℝ)
89		ax-resscn 9993	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
90	89	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
91		cnrest2 21090	. . . . . . . 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))))
92	11, 88, 90, 91	syl3anc 1326	. . . . . . 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))))
93	83, 92	mpbid 222	. . . . . 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)))
94	93, 49	syl6eleqr 2712	. . . . 5 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)))) ∈ (𝐽 Cn 𝐾))
95	1, 2, 4, 94	bndth 22757	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
96	39	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
97		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98		1re 10039	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
99		ifcl 4130	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
100	97, 98, 99	sylancl 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
101		0red 10041	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
102	98	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
103		0lt1 10550	. . . . . . . . . . . . 13 ⊢ 0 < 1
104	103	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
105		max1 12016	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
106	98, 97, 105	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
107	101, 102, 100, 104, 106	ltletrd 10197	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
108	107	gt0ne0d 10592	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
109	100, 108	rereccld 10852	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
110	100, 107	recgt0d 10958	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
111	109, 110	elrpd 11869	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ⁺)
112	96, 111	ltsubrpd 11904	. . . . . . 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, < ))
113	96, 109	resubcld 10458	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
114	113, 96	ltnled 10184	. . . . . . 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)))))
115	112, 114	mpbid 222	. . . . . 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))))
116		simprl 794	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ∈ ℝ)
117		max2 12018	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
118	98, 116, 117	sylancr 695	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
119	39	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
120		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
121	120	ad2ant2l 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
122		eldifsn 4317	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹‘𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑦) ∈ ℝ ∧ (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < )))
123	121, 122	sylib 208	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦) ∈ ℝ ∧ (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < )))
124	123	simpld 475	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ ℝ)
125	119, 124	resubcld 10458	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ∈ ℝ)
126	37	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
127		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
128	31, 127	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ ran 𝐹)
129		suprub 10984	. . . . . . . . . . . . . . . . . 18 ⊢ (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥) ∧ (𝐹‘𝑦) ∈ ran 𝐹) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
130	126, 128, 129	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
131	130	ad2ant2rl 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
132	123	simprd 479	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ≠ sup(ran 𝐹, ℝ, < ))
133	132	necomd 2849	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑦))
134	124, 119	ltlend 10182	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ((𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ) ∧ sup(ran 𝐹, ℝ, < ) ≠ (𝐹‘𝑦))))
135	131, 133, 134	mpbir2and 957	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) < sup(ran 𝐹, ℝ, < ))
136	124, 119	posdifd 10614	. . . . . . . . . . . . . . 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))))
137	135, 136	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
138	137	gt0ne0d 10592	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)) ≠ 0)
139	125, 138	rereccld 10852	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ ℝ)
140	116, 98, 99	sylancl 694	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
141		letr 10131	. . . . . . . . . . . 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)))
142	139, 116, 140, 141	syl3anc 1326	. . . . . . . . . . 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)))
143	118, 142	mpan2d 710	. . . . . . . . . 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)))
144		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
145	144	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦)))
146	145	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
147		ovex 6678	. . . . . . . . . . . . 13 ⊢ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ∈ V
148	146, 85, 147	fvmpt 6282	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑋 → ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))))
149	148	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑋 → (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑦))) ≤ 𝑥))
150	149	ad2antll 765	. . . . . . . . . 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))
151	109	adantrr 753	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
152	107	adantrr 753	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
153	140, 152	recgt0d 10958	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
154		lerec 10906	. . . . . . . . . . . 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)))))
155	151, 153, 125, 137, 154	syl22anc 1327	. . . . . . . . . . 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)))))
156		lesub 10507	. . . . . . . . . . . 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)))))
157	151, 119, 124, 156	syl3anc 1326	. . . . . . . . . . 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)))))
158	140	recnd 10068	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
159	108	adantrr 753	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ 𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
160	158, 159	recrecd 10798	. . . . . . . . . . . 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))
161	160	breq2d 4665	. . . . . . . . . . 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)))
162	155, 157, 161	3bitr3d 298	. . . . . . . . . 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)))
163	143, 150, 162	3imtr4d 283	. . . . . . . . 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)))))
164	163	anassrs 680	. . . . . . . 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)))))
165	164	ralimdva 2962	. . . . . . 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)))))
166	37	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧 ≤ 𝑥))
167		suprleub 10989	. . . . . . . . 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)))))
168	166, 113, 167	syl2anc 693	. . . . . . . 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)))))
169	31	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
170		breq1 4656	. . . . . . . . . 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)))))
171	170	ralrn 6362	. . . . . . . . 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)))))
172	169, 171	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)))))
173	168, 172	bitrd 268	. . . . . . 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)))))
174	165, 173	sylibrd 249	. . . . . 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)))))
175	115, 174	mtod 189	. . . . 5 ⊢ (((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
176	175	nrexdv 3001	. . . 4 ⊢ ((𝜑 ∧ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹‘𝑧))))‘𝑦) ≤ 𝑥)
177	95, 176	pm2.65da 600	. . 3 ⊢ (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
178	130	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < ))
179		breq2 4657	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < )))
180	179	ralbidv 2986	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ sup(ran 𝐹, ℝ, < )))
181	178, 180	syl5ibrcom 237	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)))
182	181	necon3bd 2808	. . . . . . 7 ⊢ (𝜑 → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
183	182	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
184	19	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) ∈ ℝ)
185		eldifsn 4317	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
186	185	baib 944	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ ℝ → ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
187	184, 186	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹‘𝑥) ≠ sup(ran 𝐹, ℝ, < )))
188	183, 187	sylibrd 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
189	188	ralimdva 2962	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
190		ffnfv 6388	. . . . . 6 ⊢ (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
191	190	baib 944	. . . . 5 ⊢ (𝐹 Fn 𝑋 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
192	31, 191	syl 17	. . . 4 ⊢ (𝜑 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥 ∈ 𝑋 (𝐹‘𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
193	189, 192	sylibrd 249	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
194	177, 193	mtod 189	. 2 ⊢ (𝜑 → ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
195		dfrex2 2996	. 2 ⊢ (∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ 𝑋 ¬ ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
196	194, 195	sylibr 224	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 {csn 4177 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 (,)cioo 12175 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 Compccmp 21189 ×_t ctx 21363

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127

This theorem is referenced by: evth2 22759 evthicc 23228 evthf 39186 cncmpmax 39191

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