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

Theorem evth 24322
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 481 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Comp)
5 cmptop 22746 . . . . . . . . . 10 (𝐽 ∈ Comp → 𝐽 ∈ Top)
64, 5syl 17 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
71toptopon 22266 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
86, 7sylib 217 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9 eqid 2736 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
109cnfldtopon 24146 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
1110a1i 11 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
12 1cnd 11150 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
138, 11, 12cnmptc 23013 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
14 bndth.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
15 uniretop 24126 . . . . . . . . . . . . . . . . . . 19 ℝ = (topGen‘ran (,))
162unieqi 4878 . . . . . . . . . . . . . . . . . . 19 𝐾 = (topGen‘ran (,))
1715, 16eqtr4i 2767 . . . . . . . . . . . . . . . . . 18 ℝ = 𝐾
181, 17cnf 22597 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
1914, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℝ)
2019frnd 6676 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ ℝ)
2119fdmd 6679 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑋)
22 evth.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
2321, 22eqnetrd 3011 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 ≠ ∅)
24 dm0rn0 5880 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2524necon3bii 2996 . . . . . . . . . . . . . . . 16 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2623, 25sylib 217 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ≠ ∅)
271, 2, 3, 14bndth 24321 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
2819ffnd 6669 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝑋)
29 breq1 5108 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐹𝑦) → (𝑧𝑥 ↔ (𝐹𝑦) ≤ 𝑥))
3029ralrn 7038 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3128, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3231rexbidv 3175 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3327, 32mpbird 256 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥)
3420, 26, 333jca 1128 . . . . . . . . . . . . . 14 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
35 suprcl 12115 . . . . . . . . . . . . . 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 11183 . . . . . . . . . . . 12 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
3837adantr 481 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
398, 11, 38cnmptc 23013 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4019feqmptd 6910 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑧𝑋 ↦ (𝐹𝑧)))
419cnfldtop 24147 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ Top
42 cnrest2r 22638 . . . . . . . . . . . . . 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 24166 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
452, 44eqtri 2764 . . . . . . . . . . . . . . 15 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ)
4645oveq2i 7368 . . . . . . . . . . . . . 14 (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))
4714, 46eleqtrdi 2848 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
4843, 47sselid 3942 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4940, 48eqeltrrd 2839 . . . . . . . . . . 11 (𝜑 → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5049adantr 481 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
519subcn 24229 . . . . . . . . . . 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 23017 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5436ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
55 ffvelcdm 7032 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
5655adantll 712 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
57 eldifsn 4747 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5856, 57sylib 217 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5958simpld 495 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
6054, 59resubcld 11583 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℝ)
6160recnd 11183 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ)
6254recnd 11183 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
6359recnd 11183 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
6458simprd 496 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < ))
6564necomd 2999 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑧))
6662, 63, 65subne0d 11521 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0)
67 eldifsn 4747 . . . . . . . . . . . . 13 ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}) ↔ ((sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ ∧ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0))
6861, 66, 67sylanbrc 583 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ (ℂ ∖ {0}))
6968fmpttd 7063 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))):𝑋⟶(ℂ ∖ {0}))
7069frnd 6676 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}))
71 difssd 4092 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
72 cnrest2 22637 . . . . . . . . . 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 1371 . . . . . . . . 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 231 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))))
75 eqid 2736 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
769, 75divcn 24231 . . . . . . . . 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 23017 . . . . . . 7 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
7960, 66rereccld 11982 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ ℝ)
8079fmpttd 7063 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))):𝑋⟶ℝ)
8180frnd 6676 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ)
82 ax-resscn 11108 . . . . . . . . 9 ℝ ⊆ ℂ
8382a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
84 cnrest2 22637 . . . . . . . 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 1371 . . . . . . 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 231 . . . . . 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 24321 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
8936ad2antrr 724 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
90 simpr 485 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
91 1re 11155 . . . . . . . . . . 11 1 ∈ ℝ
92 ifcl 4531 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
9390, 91, 92sylancl 586 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
94 0red 11158 . . . . . . . . . . . 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 11677 . . . . . . . . . . . . 13 0 < 1
9796a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
98 max1 13104 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
9991, 90, 98sylancr 587 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 1 ≤ if(1 ≤ 𝑥, 𝑥, 1))
10094, 95, 93, 97, 99ltletrd 11315 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
101100gt0ne0d 11719 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
10293, 101rereccld 11982 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
10393, 100recgt0d 12089 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
104102, 103elrpd 12954 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ+)
10589, 104ltsubrpd 12989 . . . . . . 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 11583 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
107106, 89ltnled 11302 . . . . . . 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 231 . . . . . 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 769 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ∈ ℝ)
110 max2 13106 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ 𝑥 ∈ ℝ) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11191, 109, 110sylancr 587 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 𝑥 ≤ if(1 ≤ 𝑥, 𝑥, 1))
11236ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113 ffvelcdm 7032 . . . . . . . . . . . . . . . . 17 ((𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
114113ad2ant2l 744 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
115 eldifsn 4747 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
116114, 115sylib 217 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
117116simpld 495 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ ℝ)
118112, 117resubcld 11583 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ∈ ℝ)
119 fnfvelrn 7031 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
12028, 119sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
121 suprub 12116 . . . . . . . . . . . . . . . . . 18 (((ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥) ∧ (𝐹𝑦) ∈ ran 𝐹) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
12234, 120, 121syl2an2r 683 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
123122ad2ant2rl 747 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
124116simprd 496 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < ))
125124necomd 2999 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑦))
126117, 112, 123, 125leneltd 11309 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) < sup(ran 𝐹, ℝ, < ))
127117, 112posdifd 11742 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
128126, 127mpbid 231 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
129128gt0ne0d 11719 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ≠ 0)
130118, 129rereccld 11982 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ ℝ)
131109, 91, 92sylancl 586 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℝ)
132 letr 11249 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 692 . . . . . . . . . 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 6842 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
136135oveq2d 7373 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
137136oveq2d 7373 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
138 eqid 2736 . . . . . . . . . . . . 13 (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) = (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))
139 ovex 7390 . . . . . . . . . . . . 13 (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ V
140137, 138, 139fvmpt 6948 . . . . . . . . . . . 12 (𝑦𝑋 → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
141140breq1d 5115 . . . . . . . . . . 11 (𝑦𝑋 → (((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥 ↔ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ≤ 𝑥))
142141ad2antll 727 . . . . . . . . . 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 715 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
144100adantrr 715 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
145131, 144recgt0d 12089 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
146 lerec 12038 . . . . . . . . . . . 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 837 . . . . . . . . . . 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 11634 . . . . . . . . . . . 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 1371 . . . . . . . . . . 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 11183 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ∈ ℂ)
151101adantrr 715 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
152150, 151recrecd 11928 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) = if(1 ≤ 𝑥, 𝑥, 1))
153152breq2d 5117 . . . . . . . . . . 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 308 . . . . . . . . . 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 293 . . . . . . . . 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 468 . . . . . . . 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 3164 . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
159 suprleub 12121 . . . . . . . . 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 584 . . . . . . . 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 724 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
162 breq1 5108 . . . . . . . . . 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 7038 . . . . . . . . 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 278 . . . . . . 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 258 . . . . . 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 197 . . . . 5 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
168167nrexdv 3146 . . . 4 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ¬ ∃𝑥 ∈ ℝ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
16988, 168pm2.65da 815 . . 3 (𝜑 → ¬ 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}))
170122ralrimiva 3143 . . . . . . . . 9 (𝜑 → ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
171 breq2 5109 . . . . . . . . . 10 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑦) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
172171ralbidv 3174 . . . . . . . . 9 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
173170, 172syl5ibrcom 246 . . . . . . . 8 (𝜑 → ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥)))
174173necon3bd 2957 . . . . . . 7 (𝜑 → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
175174adantr 481 . . . . . 6 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
17619ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ)
177 eldifsn 4747 . . . . . . . 8 ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
178177baib 536 . . . . . . 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 258 . . . . 5 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
181180ralimdva 3164 . . . 4 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
182 ffnfv 7066 . . . . . 6 (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
183182baib 536 . . . . 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 258 . . 3 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
186169, 185mtod 197 . 2 (𝜑 → ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
187 dfrex2 3076 . 2 (∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
188186, 187sylibr 233 1 (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  cdif 3907  wss 3910  c0 4282  ifcif 4486  {csn 4586   cuni 4865   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  (,)cioo 13264  t crest 17302  TopOpenctopn 17303  topGenctg 17319  fldccnfld 20796  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Compccmp 22737   ×t ctx 22911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675
This theorem is referenced by:  evth2  24323  evthicc  24823  evthf  43222  cncmpmax  43227
  Copyright terms: Public domain W3C validator