Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  evth Structured version   Visualization version   GIF version

Theorem evth 22566
 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 (𝜑𝐽 ∈ Comp)
bndth.4 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
evth.5 (𝜑𝑋 ≠ ∅)
Assertion
Ref Expression
evth (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
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.
StepHypRef Expression
1 bndth.1 . . . . 5 𝑋 = 𝐽
2 bndth.2 . . . . 5 𝐾 = (topGen‘ran (,))
3 bndth.3 . . . . . 6 (𝜑𝐽 ∈ Comp)
43adantr 480 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Comp)
5 cmptop 21008 . . . . . . . . . 10 (𝐽 ∈ Comp → 𝐽 ∈ Top)
64, 5syl 17 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
71toptopon 20548 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
86, 7sylib 207 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9 eqid 2610 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
109cnfldtopon 22396 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
1110a1i 11 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
12 1cnd 9935 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
138, 11, 12cnmptc 21275 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
14 bndth.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
15 uniretop 22376 . . . . . . . . . . . . . . . . . . 19 ℝ = (topGen‘ran (,))
162unieqi 4381 . . . . . . . . . . . . . . . . . . 19 𝐾 = (topGen‘ran (,))
1715, 16eqtr4i 2635 . . . . . . . . . . . . . . . . . 18 ℝ = 𝐾
181, 17cnf 20860 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
1914, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℝ)
20 frn 5966 . . . . . . . . . . . . . . . 16 (𝐹:𝑋⟶ℝ → ran 𝐹 ⊆ ℝ)
2119, 20syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ ℝ)
22 fdm 5964 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑋⟶ℝ → dom 𝐹 = 𝑋)
2319, 22syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑋)
24 evth.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
2523, 24eqnetrd 2849 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 ≠ ∅)
26 dm0rn0 5263 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2726necon3bii 2834 . . . . . . . . . . . . . . . 16 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2825, 27sylib 207 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ≠ ∅)
291, 2, 3, 14bndth 22565 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
30 ffn 5958 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝑋⟶ℝ → 𝐹 Fn 𝑋)
3119, 30syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝑋)
32 breq1 4586 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐹𝑦) → (𝑧𝑥 ↔ (𝐹𝑦) ≤ 𝑥))
3332ralrn 6270 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3431, 33syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3534rexbidv 3034 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3629, 35mpbird 246 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥)
3721, 28, 363jca 1235 . . . . . . . . . . . . . 14 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
38 suprcl 10862 . . . . . . . . . . . . . 14 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3937, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
4039recnd 9947 . . . . . . . . . . . 12 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
4140adantr 480 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
428, 11, 41cnmptc 21275 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4319feqmptd 6159 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑧𝑋 ↦ (𝐹𝑧)))
449cnfldtop 22397 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ Top
45 cnrest2r 20901 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ∈ Top → (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂfld)))
4644, 45ax-mp 5 . . . . . . . . . . . . 13 (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂfld))
479tgioo2 22414 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
482, 47eqtri 2632 . . . . . . . . . . . . . . 15 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ)
4948oveq2i 6560 . . . . . . . . . . . . . 14 (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))
5014, 49syl6eleq 2698 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
5146, 50sseldi 3566 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5243, 51eqeltrrd 2689 . . . . . . . . . . 11 (𝜑 → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5352adantr 480 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
549subcn 22477 . . . . . . . . . . 11 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
5554a1i 11 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
568, 42, 53, 55cnmpt12f 21279 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5739ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
58 ffvelrn 6265 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
5958adantll 746 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
60 eldifsn 4260 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
6159, 60sylib 207 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
6261simpld 474 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
6357, 62resubcld 10337 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℝ)
6463recnd 9947 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ)
6557recnd 9947 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
6662recnd 9947 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
6761simprd 478 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < ))
6867necomd 2837 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑧))
6965, 66, 68subne0d 10280 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0)
70 eldifsn 4260 . . . . . . . . . . . . 13 ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}) ↔ ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0))
7164, 69, 70sylanbrc 695 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}))
72 eqid 2610 . . . . . . . . . . . 12 (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) = (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))
7371, 72fmptd 6292 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))):𝑋⟶(ℂ ∖ {0}))
74 frn 5966 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))):𝑋⟶(ℂ ∖ {0}) → ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}))
7573, 74syl 17 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}))
76 difssd 3700 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
77 cnrest2 20900 . . . . . . . . . 10 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}) ∧ (ℂ ∖ {0}) ⊆ ℂ) → ((𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))))
7811, 75, 76, 77syl3anc 1318 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ((𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})))))
7956, 78mpbid 221 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))))
80 eqid 2610 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
819, 80divcn 22479 . . . . . . . . 9 / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld))
8281a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld)))
838, 13, 79, 82cnmpt12f 21279 . . . . . . 7 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
8463, 69rereccld 10731 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ ℝ)
85 eqid 2610 . . . . . . . . . 10 (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) = (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))
8684, 85fmptd 6292 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))):𝑋⟶ℝ)
87 frn 5966 . . . . . . . . 9 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))):𝑋⟶ℝ → ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ)
8886, 87syl 17 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ)
89 ax-resscn 9872 . . . . . . . . 9 ℝ ⊆ ℂ
9089a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
91 cnrest2 20900 . . . . . . . 8 (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ ∧ ℝ ⊆ ℂ) → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))))
9211, 88, 90, 91syl3anc 1318 . . . . . . 7 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)) ↔ (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))))
9383, 92mpbid 221 . . . . . 6 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
9493, 49syl6eleqr 2699 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn 𝐾))
951, 2, 4, 94bndth 22565 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
9639ad2antrr 758 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
97 simpr 476 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
98 1re 9918 . . . . . . . . . . 11 1 ∈ ℝ
99 ifcl 4080 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
10097, 98, 99sylancl 693 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
101 0red 9920 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
10298a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
103 0lt1 10429 . . . . . . . . . . . . 13 0 < 1
104103a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
105 max1 11890 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
10698, 97, 105sylancr 694 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
107101, 102, 100, 104, 106ltletrd 10076 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
108107gt0ne0d 10471 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
109100, 108rereccld 10731 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
110100, 107recgt0d 10837 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
111109, 110elrpd 11745 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ+)
11296, 111ltsubrpd 11780 . . . . . . 7 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) < sup(ran 𝐹, ℝ, < ))
11396, 109resubcld 10337 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
114113, 96ltnled 10063 . . . . . . 7 (((𝜑𝐹: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)))))
115112, 114mpbid 221 . . . . . 6 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))))
116 simprl 790 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ∈ ℝ)
117 max2 11892 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11898, 116, 117sylancr 694 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11939ad2antrr 758 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
120 ffvelrn 6265 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
121120ad2ant2l 778 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
122 eldifsn 4260 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
123121, 122sylib 207 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
124123simpld 474 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ ℝ)
125119, 124resubcld 10337 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ∈ ℝ)
12637adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
127 fnfvelrn 6264 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
12831, 127sylan 487 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
129 suprub 10863 . . . . . . . . . . . . . . . . . 18 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) ∧ (𝐹𝑦) ∈ ran 𝐹) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
130126, 128, 129syl2anc 691 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
131130ad2ant2rl 781 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
132123simprd 478 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < ))
133132necomd 2837 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑦))
134124, 119ltlend 10061 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) < sup(ran 𝐹, ℝ, < ) ↔ ((𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ) ∧ sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑦))))
135131, 133, 134mpbir2and 959 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) < sup(ran 𝐹, ℝ, < ))
136124, 119posdifd 10493 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
137135, 136mpbid 221 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
138137gt0ne0d 10471 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ≠ 0)
139125, 138rereccld 10731 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ ℝ)
140116, 98, 99sylancl 693 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
141 letr 10010 . . . . . . . . . . . 12 (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ) → (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
142139, 116, 140, 141syl3anc 1318 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
143118, 142mpan2d 706 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
144 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
145144oveq2d 6565 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
146145oveq2d 6565 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
147 ovex 6577 . . . . . . . . . . . . 13 (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ V
148146, 85, 147fvmpt 6191 . . . . . . . . . . . 12 (𝑦𝑋 → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
149148breq1d 4593 . . . . . . . . . . 11 (𝑦𝑋 → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥))
150149ad2antll 761 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥))
151109adantrr 749 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
152107adantrr 749 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
153140, 152recgt0d 10837 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
154 lerec 10785 . . . . . . . . . . . 12 ((((1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ ∧ 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∧ ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ∈ ℝ ∧ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
155151, 153, 125, 137, 154syl22anc 1319 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
156 lesub 10386 . . . . . . . . . . . 12 (((1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ (𝐹𝑦) ∈ ℝ) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ↔ (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
157151, 119, 124, 156syl3anc 1318 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((1 / if(1 ≤ 𝑥, 𝑥, 1)) ≤ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ↔ (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
158140recnd 9947 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
159108adantrr 749 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
160158, 159recrecd 10677 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) = if(1 ≤ 𝑥, 𝑥, 1))
161160breq2d 4595 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
162155, 157, 1613bitr3d 297 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ if(1 ≤ 𝑥, 𝑥, 1)))
163143, 150, 1623imtr4d 282 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 → (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
164163anassrs 678 . . . . . . . 8 ((((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) ∧ 𝑦𝑋) → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 → (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
165164ralimdva 2945 . . . . . . 7 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 → ∀𝑦𝑋 (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
16637ad2antrr 758 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
167 suprleub 10866 . . . . . . . . 9 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) ∧ (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)))))
168166, 113, 167syl2anc 691 . . . . . . . 8 (((𝜑𝐹: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)))))
16931ad2antrr 758 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
170 breq1 4586 . . . . . . . . . 10 (𝑧 = (𝐹𝑦) → (𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
171170ralrn 6270 . . . . . . . . 9 (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
172169, 171syl 17 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑧 ∈ ran 𝐹 𝑧 ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
173168, 172bitrd 267 . . . . . . 7 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
174165, 173sylibrd 248 . . . . . 6 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 → sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1)))))
175115, 174mtod 188 . . . . 5 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
176175nrexdv 2984 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ¬ ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
17795, 176pm2.65da 598 . . 3 (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
178130ralrimiva 2949 . . . . . . . . 9 (𝜑 → ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
179 breq2 4587 . . . . . . . . . 10 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑦) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
180179ralbidv 2969 . . . . . . . . 9 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
181178, 180syl5ibrcom 236 . . . . . . . 8 (𝜑 → ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥)))
182181necon3bd 2796 . . . . . . 7 (𝜑 → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
183182adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
18419ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ)
185 eldifsn 4260 . . . . . . . 8 ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
186185baib 942 . . . . . . 7 ((𝐹𝑥) ∈ ℝ → ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
187184, 186syl 17 . . . . . 6 ((𝜑𝑥𝑋) → ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
188183, 187sylibrd 248 . . . . 5 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
189188ralimdva 2945 . . . 4 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
190 ffnfv 6295 . . . . . 6 (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
191190baib 942 . . . . 5 (𝐹 Fn 𝑋 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
19231, 191syl 17 . . . 4 (𝜑 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
193189, 192sylibrd 248 . . 3 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
194177, 193mtod 188 . 2 (𝜑 → ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
195 dfrex2 2979 . 2 (∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
196194, 195sylibr 223 1 (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  (,)cioo 12046   ↾t crest 15904  TopOpenctopn 15905  topGenctg 15921  ℂfldccnfld 19567  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  Compccmp 20999   ×t ctx 21173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-cmp 21000  df-tx 21175  df-hmeo 21368  df-xms 21935  df-ms 21936  df-tms 21937 This theorem is referenced by:  evth2  22567  evthicc  23035  evthf  38209  cncmpmax  38214
 Copyright terms: Public domain W3C validator