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

Theorem evth 23856
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 484 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Comp)
5 cmptop 22292 . . . . . . . . . 10 (𝐽 ∈ Comp → 𝐽 ∈ Top)
64, 5syl 17 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
71toptopon 21814 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
86, 7sylib 221 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9 eqid 2737 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
109cnfldtopon 23680 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
1110a1i 11 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
12 1cnd 10828 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
138, 11, 12cnmptc 22559 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
14 bndth.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
15 uniretop 23660 . . . . . . . . . . . . . . . . . . 19 ℝ = (topGen‘ran (,))
162unieqi 4832 . . . . . . . . . . . . . . . . . . 19 𝐾 = (topGen‘ran (,))
1715, 16eqtr4i 2768 . . . . . . . . . . . . . . . . . 18 ℝ = 𝐾
181, 17cnf 22143 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
1914, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℝ)
2019frnd 6553 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ ℝ)
2119fdmd 6556 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑋)
22 evth.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
2321, 22eqnetrd 3008 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 ≠ ∅)
24 dm0rn0 5794 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2524necon3bii 2993 . . . . . . . . . . . . . . . 16 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2623, 25sylib 221 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ≠ ∅)
271, 2, 3, 14bndth 23855 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
2819ffnd 6546 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝑋)
29 breq1 5056 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐹𝑦) → (𝑧𝑥 ↔ (𝐹𝑦) ≤ 𝑥))
3029ralrn 6907 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3128, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3231rexbidv 3216 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3327, 32mpbird 260 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥)
3420, 26, 333jca 1130 . . . . . . . . . . . . . 14 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
35 suprcl 11792 . . . . . . . . . . . . . 14 ((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3634, 35syl 17 . . . . . . . . . . . . 13 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3736recnd 10861 . . . . . . . . . . . 12 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
3837adantr 484 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
398, 11, 38cnmptc 22559 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4019feqmptd 6780 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑧𝑋 ↦ (𝐹𝑧)))
419cnfldtop 23681 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ Top
42 cnrest2r 22184 . . . . . . . . . . . . . 14 ((TopOpen‘ℂfld) ∈ Top → (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂfld)))
4341, 42ax-mp 5 . . . . . . . . . . . . 13 (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)) ⊆ (𝐽 Cn (TopOpen‘ℂfld))
449tgioo2 23700 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
452, 44eqtri 2765 . . . . . . . . . . . . . . 15 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ)
4645oveq2i 7224 . . . . . . . . . . . . . 14 (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))
4714, 46eleqtrdi 2848 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
4843, 47sseldi 3899 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4940, 48eqeltrrd 2839 . . . . . . . . . . 11 (𝜑 → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5049adantr 484 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
519subcn 23763 . . . . . . . . . . 11 − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))
5251a1i 11 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → − ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)))
538, 39, 50, 52cnmpt12f 22563 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5436ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
55 ffvelrn 6902 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
5655adantll 714 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
57 eldifsn 4700 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5856, 57sylib 221 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5958simpld 498 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
6054, 59resubcld 11260 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℝ)
6160recnd 10861 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ)
6254recnd 10861 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
6359recnd 10861 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
6458simprd 499 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < ))
6564necomd 2996 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑧))
6662, 63, 65subne0d 11198 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0)
67 eldifsn 4700 . . . . . . . . . . . . 13 ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}) ↔ ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0))
6861, 66, 67sylanbrc 586 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}))
6968fmpttd 6932 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))):𝑋⟶(ℂ ∖ {0}))
7069frnd 6553 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}))
71 difssd 4047 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
72 cnrest2 22183 . . . . . . . . . 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})))))
7311, 70, 71, 72syl3anc 1373 . . . . . . . . 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})))))
7453, 73mpbid 235 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))))
75 eqid 2737 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
769, 75divcn 23765 . . . . . . . . 9 / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld))
7776a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → / ∈ (((TopOpen‘ℂfld) ×t ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))) Cn (TopOpen‘ℂfld)))
788, 13, 74, 77cnmpt12f 22563 . . . . . . 7 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
7960, 66rereccld 11659 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ ℝ)
8079fmpttd 6932 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))):𝑋⟶ℝ)
8180frnd 6553 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ)
82 ax-resscn 10786 . . . . . . . . 9 ℝ ⊆ ℂ
8382a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
84 cnrest2 22183 . . . . . . . 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))))
8511, 81, 83, 84syl3anc 1373 . . . . . . 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))))
8678, 85mpbid 235 . . . . . 6 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
8786, 46eleqtrrdi 2849 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn 𝐾))
881, 2, 4, 87bndth 23855 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
8936ad2antrr 726 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
90 simpr 488 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
91 1re 10833 . . . . . . . . . . 11 1 ∈ ℝ
92 ifcl 4484 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
9390, 91, 92sylancl 589 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
94 0red 10836 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 ∈ ℝ)
9591a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
96 0lt1 11354 . . . . . . . . . . . . 13 0 < 1
9796a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
98 max1 12775 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
9991, 90, 98sylancr 590 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
10094, 95, 93, 97, 99ltletrd 10992 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
101100gt0ne0d 11396 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
10293, 101rereccld 11659 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
10393, 100recgt0d 11766 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
104102, 103elrpd 12625 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ+)
10589, 104ltsubrpd 12660 . . . . . . 7 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) < sup(ran 𝐹, ℝ, < ))
10689, 102resubcld 11260 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
107106, 89ltnled 10979 . . . . . . 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)))))
108105, 107mpbid 235 . . . . . 6 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ sup(ran 𝐹, ℝ, < ) ≤ (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))))
109 simprl 771 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ∈ ℝ)
110 max2 12777 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11191, 109, 110sylancr 590 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11236ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113 ffvelrn 6902 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
114113ad2ant2l 746 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
115 eldifsn 4700 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
116114, 115sylib 221 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
117116simpld 498 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ ℝ)
118112, 117resubcld 11260 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ∈ ℝ)
119 fnfvelrn 6901 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
12028, 119sylan 583 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
121 suprub 11793 . . . . . . . . . . . . . . . . . 18 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) ∧ (𝐹𝑦) ∈ ran 𝐹) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
12234, 120, 121syl2an2r 685 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
123122ad2ant2rl 749 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
124116simprd 499 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < ))
125124necomd 2996 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑦))
126117, 112, 123, 125leneltd 10986 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) < sup(ran 𝐹, ℝ, < ))
127117, 112posdifd 11419 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
128126, 127mpbid 235 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
129128gt0ne0d 11396 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ≠ 0)
130118, 129rereccld 11659 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ ℝ)
131109, 91, 92sylancl 589 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
132 letr 10926 . . . . . . . . . . . 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)))
133130, 109, 131, 132syl3anc 1373 . . . . . . . . . . 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)))
134111, 133mpan2d 694 . . . . . . . . . 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)))
135 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
136135oveq2d 7229 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
137136oveq2d 7229 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
138 eqid 2737 . . . . . . . . . . . . 13 (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) = (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))
139 ovex 7246 . . . . . . . . . . . . 13 (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ V
140137, 138, 139fvmpt 6818 . . . . . . . . . . . 12 (𝑦𝑋 → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
141140breq1d 5063 . . . . . . . . . . 11 (𝑦𝑋 → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥))
142141ad2antll 729 . . . . . . . . . 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))
143102adantrr 717 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
144100adantrr 717 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
145131, 144recgt0d 11766 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
146 lerec 11715 . . . . . . . . . . . 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)))))
147143, 145, 118, 128, 146syl22anc 839 . . . . . . . . . . 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)))))
148 lesub 11311 . . . . . . . . . . . 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)))))
149143, 112, 117, 148syl3anc 1373 . . . . . . . . . . 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)))))
150131recnd 10861 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
151101adantrr 717 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
152150, 151recrecd 11605 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) = if(1 ≤ 𝑥, 𝑥, 1))
153152breq2d 5065 . . . . . . . . . . 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)))
154147, 149, 1533bitr3d 312 . . . . . . . . . 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)))
155134, 142, 1543imtr4d 297 . . . . . . . . 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)))))
156155anassrs 471 . . . . . . . 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)))))
157156ralimdva 3100 . . . . . . 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)))))
15834ad2antrr 726 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
159 suprleub 11798 . . . . . . . . 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)))))
160158, 106, 159syl2anc 587 . . . . . . . 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)))))
16128ad2antrr 726 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
162 breq1 5056 . . . . . . . . . 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)))))
163162ralrn 6907 . . . . . . . . 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)))))
164161, 163syl 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)))))
165160, 164bitrd 282 . . . . . . 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)))))
166157, 165sylibrd 262 . . . . . 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)))))
167108, 166mtod 201 . . . . 5 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
168167nrexdv 3189 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ¬ ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
16988, 168pm2.65da 817 . . 3 (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
170122ralrimiva 3105 . . . . . . . . 9 (𝜑 → ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
171 breq2 5057 . . . . . . . . . 10 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑦) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
172171ralbidv 3118 . . . . . . . . 9 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
173170, 172syl5ibrcom 250 . . . . . . . 8 (𝜑 → ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥)))
174173necon3bd 2954 . . . . . . 7 (𝜑 → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
175174adantr 484 . . . . . 6 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
17619ffvelrnda 6904 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ)
177 eldifsn 4700 . . . . . . . 8 ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
178177baib 539 . . . . . . 7 ((𝐹𝑥) ∈ ℝ → ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
179176, 178syl 17 . . . . . 6 ((𝜑𝑥𝑋) → ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
180175, 179sylibrd 262 . . . . 5 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
181180ralimdva 3100 . . . 4 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
182 ffnfv 6935 . . . . . 6 (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
183182baib 539 . . . . 5 (𝐹 Fn 𝑋 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
18428, 183syl 17 . . . 4 (𝜑 → (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
185181, 184sylibrd 262 . . 3 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
186169, 185mtod 201 . 2 (𝜑 → ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
187 dfrex2 3161 . 2 (∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
188186, 187sylibr 237 1 (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  cdif 3863  wss 3866  c0 4237  ifcif 4439  {csn 4541   cuni 4819   class class class wbr 5053  cmpt 5135  dom cdm 5551  ran crn 5552   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  supcsup 9056  cc 10727  cr 10728  0cc0 10729  1c1 10730   < clt 10867  cle 10868  cmin 11062   / cdiv 11489  (,)cioo 12935  t crest 16925  TopOpenctopn 16926  topGenctg 16942  fldccnfld 20363  Topctop 21790  TopOnctopon 21807   Cn ccn 22121  Compccmp 22283   ×t ctx 22457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-ioo 12939  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cn 22124  df-cnp 22125  df-cmp 22284  df-tx 22459  df-hmeo 22652  df-xms 23218  df-ms 23219  df-tms 23220
This theorem is referenced by:  evth2  23857  evthicc  24356  evthf  42243  cncmpmax  42248
  Copyright terms: Public domain W3C validator