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Theorem evth 25005
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 23419 . . . . . . . . . 10 (𝐽 ∈ Comp → 𝐽 ∈ Top)
64, 5syl 17 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ Top)
71toptopon 22939 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
86, 7sylib 218 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 𝐽 ∈ (TopOn‘𝑋))
9 eqid 2735 . . . . . . . . . . 11 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
109cnfldtopon 24819 . . . . . . . . . 10 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
1110a1i 11 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ))
12 1cnd 11254 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → 1 ∈ ℂ)
138, 11, 12cnmptc 23686 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
14 bndth.4 . . . . . . . . . . . . . . . . 17 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
15 uniretop 24799 . . . . . . . . . . . . . . . . . . 19 ℝ = (topGen‘ran (,))
162unieqi 4924 . . . . . . . . . . . . . . . . . . 19 𝐾 = (topGen‘ran (,))
1715, 16eqtr4i 2766 . . . . . . . . . . . . . . . . . 18 ℝ = 𝐾
181, 17cnf 23270 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶ℝ)
1914, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝑋⟶ℝ)
2019frnd 6745 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ⊆ ℝ)
2119fdmd 6747 . . . . . . . . . . . . . . . . 17 (𝜑 → dom 𝐹 = 𝑋)
22 evth.5 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 ≠ ∅)
2321, 22eqnetrd 3006 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐹 ≠ ∅)
24 dm0rn0 5938 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = ∅ ↔ ran 𝐹 = ∅)
2524necon3bii 2991 . . . . . . . . . . . . . . . 16 (dom 𝐹 ≠ ∅ ↔ ran 𝐹 ≠ ∅)
2623, 25sylib 218 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ≠ ∅)
271, 2, 3, 14bndth 25004 . . . . . . . . . . . . . . . 16 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥)
2819ffnd 6738 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 Fn 𝑋)
29 breq1 5151 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝐹𝑦) → (𝑧𝑥 ↔ (𝐹𝑦) ≤ 𝑥))
3029ralrn 7108 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑋 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3128, 30syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3231rexbidv 3177 . . . . . . . . . . . . . . . 16 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦𝑋 (𝐹𝑦) ≤ 𝑥))
3327, 32mpbird 257 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥)
3420, 26, 333jca 1127 . . . . . . . . . . . . . 14 (𝜑 → (ran 𝐹 ⊆ ℝ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝐹 𝑧𝑥))
35 suprcl 12226 . . . . . . . . . . . . . 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 11287 . . . . . . . . . . . 12 (𝜑 → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
3837adantr 480 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
398, 11, 38cnmptc 23686 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4019feqmptd 6977 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑧𝑋 ↦ (𝐹𝑧)))
419cnfldtop 24820 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) ∈ Top
42 cnrest2r 23311 . . . . . . . . . . . . . 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 24839 . . . . . . . . . . . . . . . 16 (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ)
452, 44eqtri 2763 . . . . . . . . . . . . . . 15 𝐾 = ((TopOpen‘ℂfld) ↾t ℝ)
4645oveq2i 7442 . . . . . . . . . . . . . 14 (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ))
4714, 46eleqtrdi 2849 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
4843, 47sselid 3993 . . . . . . . . . . . 12 (𝜑𝐹 ∈ (𝐽 Cn (TopOpen‘ℂfld)))
4940, 48eqeltrrd 2840 . . . . . . . . . . 11 (𝜑 → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
5049adantr 480 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (𝐹𝑧)) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
519subcn 24902 . . . . . . . . . . 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 23690 . . . . . . . . 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 ffvelcdm 7101 . . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . . . 17 ((𝐹𝑧) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5856, 57sylib 218 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → ((𝐹𝑧) ∈ ℝ ∧ (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < )))
5958simpld 494 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℝ)
6054, 59resubcld 11689 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℝ)
6160recnd 11287 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ∈ ℂ)
6254recnd 11287 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ∈ ℂ)
6359recnd 11287 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ∈ ℂ)
6458simprd 495 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (𝐹𝑧) ≠ sup(ran 𝐹, ℝ, < ))
6564necomd 2994 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑧))
6662, 63, 65subne0d 11627 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) ≠ 0)
67 eldifsn 4791 . . . . . . . . . . . . 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 7135 . . . . . . . . . . 11 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))):𝑋⟶(ℂ ∖ {0}))
7069frnd 6745 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ⊆ (ℂ ∖ {0}))
71 difssd 4147 . . . . . . . . . 10 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (ℂ ∖ {0}) ⊆ ℂ)
72 cnrest2 23310 . . . . . . . . . 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 1370 . . . . . . . . 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 232 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))))
75 eqid 2735 . . . . . . . . . 10 ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0})) = ((TopOpen‘ℂfld) ↾t (ℂ ∖ {0}))
769, 75divcn 24906 . . . . . . . . 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 23690 . . . . . . 7 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn (TopOpen‘ℂfld)))
7960, 66rereccld 12092 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧𝑋) → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) ∈ ℝ)
8079fmpttd 7135 . . . . . . . . 9 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))):𝑋⟶ℝ)
8180frnd 6745 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ran (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ⊆ ℝ)
82 ax-resscn 11210 . . . . . . . . 9 ℝ ⊆ ℂ
8382a1i 11 . . . . . . . 8 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → ℝ ⊆ ℂ)
84 cnrest2 23310 . . . . . . . 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 1370 . . . . . . 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 232 . . . . . 6 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn ((TopOpen‘ℂfld) ↾t ℝ)))
8786, 46eleqtrrdi 2850 . . . . 5 ((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) → (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) ∈ (𝐽 Cn 𝐾))
881, 2, 4, 87bndth 25004 . . . 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 484 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
91 1re 11259 . . . . . . . . . . 11 1 ∈ ℝ
92 ifcl 4576 . . . . . . . . . . 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 11262 . . . . . . . . . . . 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 11783 . . . . . . . . . . . . 13 0 < 1
9796a1i 11 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < 1)
98 max1 13224 . . . . . . . . . . . . 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 11419 . . . . . . . . . . 11 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < if(1 ≤ 𝑥, 𝑥, 1))
101100gt0ne0d 11825 . . . . . . . . . 10 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → if(1 ≤ 𝑥, 𝑥, 1) ≠ 0)
10293, 101rereccld 12092 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ)
10393, 100recgt0d 12200 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
104102, 103elrpd 13072 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (1 / if(1 ≤ 𝑥, 𝑥, 1)) ∈ ℝ+)
10589, 104ltsubrpd 13107 . . . . . . 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 11689 . . . . . . . 8 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → (sup(ran 𝐹, ℝ, < ) − (1 / if(1 ≤ 𝑥, 𝑥, 1))) ∈ ℝ)
107106, 89ltnled 11406 . . . . . . 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 232 . . . . . 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 13226 . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113 ffvelcdm 7101 . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
116114, 115sylib 218 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) ∈ ℝ ∧ (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < )))
117116simpld 494 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ∈ ℝ)
118112, 117resubcld 11689 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ∈ ℝ)
119 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝑋𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
12028, 119sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ ran 𝐹)
121 suprub 12227 . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . 17 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) ≠ sup(ran 𝐹, ℝ, < ))
125124necomd 2994 . . . . . . . . . . . . . . . 16 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → sup(ran 𝐹, ℝ, < ) ≠ (𝐹𝑦))
126117, 112, 123, 125leneltd 11413 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (𝐹𝑦) < sup(ran 𝐹, ℝ, < ))
127117, 112posdifd 11848 . . . . . . . . . . . . . . 15 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → ((𝐹𝑦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
128126, 127mpbid 232 . . . . . . . . . . . . . 14 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
129128gt0ne0d 11825 . . . . . . . . . . . . 13 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)) ≠ 0)
130118, 129rereccld 12092 . . . . . . . . . . . 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 11353 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝐹𝑧) = (𝐹𝑦))
136135oveq2d 7447 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)) = (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦)))
137136oveq2d 7447 . . . . . . . . . . . . 13 (𝑧 = 𝑦 → (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
138 eqid 2735 . . . . . . . . . . . . 13 (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧)))) = (𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))
139 ovex 7464 . . . . . . . . . . . . 13 (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))) ∈ V
140137, 138, 139fvmpt 7016 . . . . . . . . . . . 12 (𝑦𝑋 → ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) = (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑦))))
141140breq1d 5158 . . . . . . . . . . 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 12200 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → 0 < (1 / if(1 ≤ 𝑥, 𝑥, 1)))
146 lerec 12149 . . . . . . . . . . . 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 11740 . . . . . . . . . . . 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 1370 . . . . . . . . . . 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 11287 . . . . . . . . . . . . 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 12038 . . . . . . . . . . . 12 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ (𝑥 ∈ ℝ ∧ 𝑦𝑋)) → (1 / (1 / if(1 ≤ 𝑥, 𝑥, 1))) = if(1 ≤ 𝑥, 𝑥, 1))
153152breq2d 5160 . . . . . . . . . . 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 309 . . . . . . . . . 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 294 . . . . . . . . 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 467 . . . . . . . 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 3165 . . . . . . 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 12232 . . . . . . . . 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 726 . . . . . . . . 9 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn 𝑋)
162 breq1 5151 . . . . . . . . . 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 7108 . . . . . . . . 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 279 . . . . . . 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 259 . . . . . 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 198 . . . . 5 (((𝜑𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})) ∧ 𝑥 ∈ ℝ) → ¬ ∀𝑦𝑋 ((𝑧𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) − (𝐹𝑧))))‘𝑦) ≤ 𝑥)
168167nrexdv 3147 . . . 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 3144 . . . . . . . . 9 (𝜑 → ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < ))
171 breq2 5152 . . . . . . . . . 10 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ((𝐹𝑦) ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
172171ralbidv 3176 . . . . . . . . 9 ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → (∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ∀𝑦𝑋 (𝐹𝑦) ≤ sup(ran 𝐹, ℝ, < )))
173170, 172syl5ibrcom 247 . . . . . . . 8 (𝜑 → ((𝐹𝑥) = sup(ran 𝐹, ℝ, < ) → ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥)))
174173necon3bd 2952 . . . . . . 7 (𝜑 → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
175174adantr 480 . . . . . 6 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
17619ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐹𝑥) ∈ ℝ)
177 eldifsn 4791 . . . . . . . 8 ((𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) ≠ sup(ran 𝐹, ℝ, < )))
178177baib 535 . . . . . . 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 259 . . . . 5 ((𝜑𝑥𝑋) → (¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
181180ralimdva 3165 . . . 4 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
182 ffnfv 7139 . . . . . 6 (𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ ∀𝑥𝑋 (𝐹𝑥) ∈ (ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
183182baib 535 . . . . 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 259 . . 3 (𝜑 → (∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) → 𝐹:𝑋⟶(ℝ ∖ {sup(ran 𝐹, ℝ, < )})))
186169, 185mtod 198 . 2 (𝜑 → ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
187 dfrex2 3071 . 2 (∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥) ↔ ¬ ∀𝑥𝑋 ¬ ∀𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
188186, 187sylibr 234 1 (𝜑 → ∃𝑥𝑋𝑦𝑋 (𝐹𝑦) ≤ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cdif 3960  wss 3963  c0 4339  ifcif 4531  {csn 4631   cuni 4912   class class class wbr 5148  cmpt 5231  dom cdm 5689  ran crn 5690   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  supcsup 9478  cc 11151  cr 11152  0cc0 11153  1c1 11154   < clt 11293  cle 11294  cmin 11490   / cdiv 11918  (,)cioo 13384  t crest 17467  TopOpenctopn 17468  topGenctg 17484  fldccnfld 21382  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  Compccmp 23410   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348
This theorem is referenced by:  evth2  25006  evthicc  25508  evthf  44965  cncmpmax  44970
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