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Theorem evth 24475
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 = βˆͺ 𝐽
bndth.2 𝐾 = (topGenβ€˜ran (,))
bndth.3 (πœ‘ β†’ 𝐽 ∈ Comp)
bndth.4 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
evth.5 (πœ‘ β†’ 𝑋 β‰  βˆ…)
Assertion
Ref Expression
evth (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯))
Distinct variable groups:   π‘₯,𝑦,𝐹   𝑦,𝐾   πœ‘,π‘₯,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝐽,𝑦
Allowed substitution hint:   𝐾(π‘₯)
Proof of Theorem evth
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bndth.1 . . . . 5 𝑋 = βˆͺ 𝐽
2 bndth.2 . . . . 5 𝐾 = (topGenβ€˜ran (,))
3 bndth.3 . . . . . 6 (πœ‘ β†’ 𝐽 ∈ Comp)
43adantr 482 . . . . 5 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ 𝐽 ∈ Comp)
5 cmptop 22899 . . . . . . . . . 10 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
64, 5syl 17 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ 𝐽 ∈ Top)
71toptopon 22419 . . . . . . . . 9 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
86, 7sylib 217 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 eqid 2733 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
109cnfldtopon 24299 . . . . . . . . . 10 (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚)
1110a1i 11 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚))
12 1cnd 11209 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ 1 ∈ β„‚)
138, 11, 12cnmptc 23166 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ 1) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
14 bndth.4 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
15 uniretop 24279 . . . . . . . . . . . . . . . . . . 19 ℝ = βˆͺ (topGenβ€˜ran (,))
162unieqi 4922 . . . . . . . . . . . . . . . . . . 19 βˆͺ 𝐾 = βˆͺ (topGenβ€˜ran (,))
1715, 16eqtr4i 2764 . . . . . . . . . . . . . . . . . 18 ℝ = βˆͺ 𝐾
181, 17cnf 22750 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:π‘‹βŸΆβ„)
1914, 18syl 17 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„)
2019frnd 6726 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
2119fdmd 6729 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ dom 𝐹 = 𝑋)
22 evth.5 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝑋 β‰  βˆ…)
2321, 22eqnetrd 3009 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ dom 𝐹 β‰  βˆ…)
24 dm0rn0 5925 . . . . . . . . . . . . . . . . 17 (dom 𝐹 = βˆ… ↔ ran 𝐹 = βˆ…)
2524necon3bii 2994 . . . . . . . . . . . . . . . 16 (dom 𝐹 β‰  βˆ… ↔ ran 𝐹 β‰  βˆ…)
2623, 25sylib 217 . . . . . . . . . . . . . . 15 (πœ‘ β†’ ran 𝐹 β‰  βˆ…)
271, 2, 3, 14bndth 24474 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ π‘₯)
2819ffnd 6719 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐹 Fn 𝑋)
29 breq1 5152 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (πΉβ€˜π‘¦) β†’ (𝑧 ≀ π‘₯ ↔ (πΉβ€˜π‘¦) ≀ π‘₯))
3029ralrn 7090 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn 𝑋 β†’ (βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯ ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ π‘₯))
3128, 30syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯ ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ π‘₯))
3231rexbidv 3179 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯ ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ π‘₯))
3327, 32mpbird 257 . . . . . . . . . . . . . . 15 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯)
3420, 26, 333jca 1129 . . . . . . . . . . . . . 14 (πœ‘ β†’ (ran 𝐹 βŠ† ℝ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯))
35 suprcl 12174 . . . . . . . . . . . . . 14 ((ran 𝐹 βŠ† ℝ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯) β†’ sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3634, 35syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ sup(ran 𝐹, ℝ, < ) ∈ ℝ)
3736recnd 11242 . . . . . . . . . . . 12 (πœ‘ β†’ sup(ran 𝐹, ℝ, < ) ∈ β„‚)
3837adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ sup(ran 𝐹, ℝ, < ) ∈ β„‚)
398, 11, 38cnmptc 23166 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ sup(ran 𝐹, ℝ, < )) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
4019feqmptd 6961 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 = (𝑧 ∈ 𝑋 ↦ (πΉβ€˜π‘§)))
419cnfldtop 24300 . . . . . . . . . . . . . 14 (TopOpenβ€˜β„‚fld) ∈ Top
42 cnrest2r 22791 . . . . . . . . . . . . . 14 ((TopOpenβ€˜β„‚fld) ∈ Top β†’ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)) βŠ† (𝐽 Cn (TopOpenβ€˜β„‚fld)))
4341, 42ax-mp 5 . . . . . . . . . . . . 13 (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)) βŠ† (𝐽 Cn (TopOpenβ€˜β„‚fld))
449tgioo2 24319 . . . . . . . . . . . . . . . 16 (topGenβ€˜ran (,)) = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
452, 44eqtri 2761 . . . . . . . . . . . . . . 15 𝐾 = ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)
4645oveq2i 7420 . . . . . . . . . . . . . 14 (𝐽 Cn 𝐾) = (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ))
4714, 46eleqtrdi 2844 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)))
4843, 47sselid 3981 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
4940, 48eqeltrrd 2835 . . . . . . . . . . 11 (πœ‘ β†’ (𝑧 ∈ 𝑋 ↦ (πΉβ€˜π‘§)) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
5049adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (πΉβ€˜π‘§)) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
519subcn 24382 . . . . . . . . . . 11 βˆ’ ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld))
5251a1i 11 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ βˆ’ ∈ (((TopOpenβ€˜β„‚fld) Γ—t (TopOpenβ€˜β„‚fld)) Cn (TopOpenβ€˜β„‚fld)))
538, 39, 50, 52cnmpt12f 23170 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
5436ad2antrr 725 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ sup(ran 𝐹, ℝ, < ) ∈ ℝ)
55 ffvelcdm 7084 . . . . . . . . . . . . . . . . . 18 ((𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}))
5655adantll 713 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}))
57 eldifsn 4791 . . . . . . . . . . . . . . . . 17 ((πΉβ€˜π‘§) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ ((πΉβ€˜π‘§) ∈ ℝ ∧ (πΉβ€˜π‘§) β‰  sup(ran 𝐹, ℝ, < )))
5856, 57sylib 217 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ ((πΉβ€˜π‘§) ∈ ℝ ∧ (πΉβ€˜π‘§) β‰  sup(ran 𝐹, ℝ, < )))
5958simpld 496 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) ∈ ℝ)
6054, 59resubcld 11642 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) ∈ ℝ)
6160recnd 11242 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) ∈ β„‚)
6254recnd 11242 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ sup(ran 𝐹, ℝ, < ) ∈ β„‚)
6359recnd 11242 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) ∈ β„‚)
6458simprd 497 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (πΉβ€˜π‘§) β‰  sup(ran 𝐹, ℝ, < ))
6564necomd 2997 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ sup(ran 𝐹, ℝ, < ) β‰  (πΉβ€˜π‘§))
6662, 63, 65subne0d 11580 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) β‰  0)
67 eldifsn 4791 . . . . . . . . . . . . 13 ((sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) ∈ (β„‚ βˆ– {0}) ↔ ((sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) ∈ β„‚ ∧ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) β‰  0))
6861, 66, 67sylanbrc 584 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) ∈ (β„‚ βˆ– {0}))
6968fmpttd 7115 . . . . . . . . . . 11 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))):π‘‹βŸΆ(β„‚ βˆ– {0}))
7069frnd 6726 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) βŠ† (β„‚ βˆ– {0}))
71 difssd 4133 . . . . . . . . . 10 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (β„‚ βˆ– {0}) βŠ† β„‚)
72 cnrest2 22790 . . . . . . . . . 10 (((TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚) ∧ ran (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) βŠ† (β„‚ βˆ– {0}) ∧ (β„‚ βˆ– {0}) βŠ† β„‚) β†’ ((𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)) ↔ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0})))))
7311, 70, 71, 72syl3anc 1372 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ ((𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)) ↔ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0})))))
7453, 73mpbid 231 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0}))))
75 eqid 2733 . . . . . . . . . 10 ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0})) = ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0}))
769, 75divcn 24384 . . . . . . . . 9 / ∈ (((TopOpenβ€˜β„‚fld) Γ—t ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0}))) Cn (TopOpenβ€˜β„‚fld))
7776a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ / ∈ (((TopOpenβ€˜β„‚fld) Γ—t ((TopOpenβ€˜β„‚fld) β†Ύt (β„‚ βˆ– {0}))) Cn (TopOpenβ€˜β„‚fld)))
788, 13, 74, 77cnmpt12f 23170 . . . . . . 7 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
7960, 66rereccld 12041 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ 𝑧 ∈ 𝑋) β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) ∈ ℝ)
8079fmpttd 7115 . . . . . . . . 9 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))):π‘‹βŸΆβ„)
8180frnd 6726 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) βŠ† ℝ)
82 ax-resscn 11167 . . . . . . . . 9 ℝ βŠ† β„‚
8382a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ ℝ βŠ† β„‚)
84 cnrest2 22790 . . . . . . . 8 (((TopOpenβ€˜β„‚fld) ∈ (TopOnβ€˜β„‚) ∧ ran (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) βŠ† ℝ ∧ ℝ βŠ† β„‚) β†’ ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)) ↔ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ))))
8511, 81, 83, 84syl3anc 1372 . . . . . . 7 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)) ↔ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ))))
8678, 85mpbid 231 . . . . . 6 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn ((TopOpenβ€˜β„‚fld) β†Ύt ℝ)))
8786, 46eleqtrrdi 2845 . . . . 5 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) ∈ (𝐽 Cn 𝐾))
881, 2, 4, 87bndth 24474 . . . 4 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯)
8936ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ sup(ran 𝐹, ℝ, < ) ∈ ℝ)
90 simpr 486 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ∈ ℝ)
91 1re 11214 . . . . . . . . . . 11 1 ∈ ℝ
92 ifcl 4574 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 1 ∈ ℝ) β†’ if(1 ≀ π‘₯, π‘₯, 1) ∈ ℝ)
9390, 91, 92sylancl 587 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ if(1 ≀ π‘₯, π‘₯, 1) ∈ ℝ)
94 0red 11217 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 0 ∈ ℝ)
9591a1i 11 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 1 ∈ ℝ)
96 0lt1 11736 . . . . . . . . . . . . 13 0 < 1
9796a1i 11 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 0 < 1)
98 max1 13164 . . . . . . . . . . . . 13 ((1 ∈ ℝ ∧ π‘₯ ∈ ℝ) β†’ 1 ≀ if(1 ≀ π‘₯, π‘₯, 1))
9991, 90, 98sylancr 588 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 1 ≀ if(1 ≀ π‘₯, π‘₯, 1))
10094, 95, 93, 97, 99ltletrd 11374 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 0 < if(1 ≀ π‘₯, π‘₯, 1))
101100gt0ne0d 11778 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ if(1 ≀ π‘₯, π‘₯, 1) β‰  0)
10293, 101rereccld 12041 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (1 / if(1 ≀ π‘₯, π‘₯, 1)) ∈ ℝ)
10393, 100recgt0d 12148 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 0 < (1 / if(1 ≀ π‘₯, π‘₯, 1)))
104102, 103elrpd 13013 . . . . . . . 8 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (1 / if(1 ≀ π‘₯, π‘₯, 1)) ∈ ℝ+)
10589, 104ltsubrpd 13048 . . . . . . 7 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) < sup(ran 𝐹, ℝ, < ))
10689, 102resubcld 11642 . . . . . . . 8 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ∈ ℝ)
107106, 89ltnled 11361 . . . . . . 7 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ ((sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) < sup(ran 𝐹, ℝ, < ) ↔ Β¬ sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
108105, 107mpbid 231 . . . . . 6 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ Β¬ sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))))
109 simprl 770 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ π‘₯ ∈ ℝ)
110 max2 13166 . . . . . . . . . . . 12 ((1 ∈ ℝ ∧ π‘₯ ∈ ℝ) β†’ π‘₯ ≀ if(1 ≀ π‘₯, π‘₯, 1))
11191, 109, 110sylancr 588 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ π‘₯ ≀ if(1 ≀ π‘₯, π‘₯, 1))
11236ad2antrr 725 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ sup(ran 𝐹, ℝ, < ) ∈ ℝ)
113 ffvelcdm 7084 . . . . . . . . . . . . . . . . 17 ((𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ∧ 𝑦 ∈ 𝑋) β†’ (πΉβ€˜π‘¦) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}))
114113ad2ant2l 745 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜π‘¦) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}))
115 eldifsn 4791 . . . . . . . . . . . . . . . 16 ((πΉβ€˜π‘¦) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ ((πΉβ€˜π‘¦) ∈ ℝ ∧ (πΉβ€˜π‘¦) β‰  sup(ran 𝐹, ℝ, < )))
116114, 115sylib 217 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘¦) ∈ ℝ ∧ (πΉβ€˜π‘¦) β‰  sup(ran 𝐹, ℝ, < )))
117116simpld 496 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜π‘¦) ∈ ℝ)
118112, 117resubcld 11642 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ∈ ℝ)
119 fnfvelrn 7083 . . . . . . . . . . . . . . . . . . 19 ((𝐹 Fn 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (πΉβ€˜π‘¦) ∈ ran 𝐹)
12028, 119sylan 581 . . . . . . . . . . . . . . . . . 18 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (πΉβ€˜π‘¦) ∈ ran 𝐹)
121 suprub 12175 . . . . . . . . . . . . . . . . . 18 (((ran 𝐹 βŠ† ℝ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯) ∧ (πΉβ€˜π‘¦) ∈ ran 𝐹) β†’ (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < ))
12234, 120, 121syl2an2r 684 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑦 ∈ 𝑋) β†’ (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < ))
123122ad2ant2rl 748 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < ))
124116simprd 497 . . . . . . . . . . . . . . . . 17 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜π‘¦) β‰  sup(ran 𝐹, ℝ, < ))
125124necomd 2997 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ sup(ran 𝐹, ℝ, < ) β‰  (πΉβ€˜π‘¦))
126117, 112, 123, 125leneltd 11368 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜π‘¦) < sup(ran 𝐹, ℝ, < ))
127117, 112posdifd 11801 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘¦) < sup(ran 𝐹, ℝ, < ) ↔ 0 < (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))))
128126, 127mpbid 231 . . . . . . . . . . . . . 14 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ 0 < (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)))
129128gt0ne0d 11778 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) β‰  0)
130118, 129rereccld 12041 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ∈ ℝ)
131109, 91, 92sylancl 587 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ if(1 ≀ π‘₯, π‘₯, 1) ∈ ℝ)
132 letr 11308 . . . . . . . . . . . 12 (((1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ∈ ℝ ∧ π‘₯ ∈ ℝ ∧ if(1 ≀ π‘₯, π‘₯, 1) ∈ ℝ) β†’ (((1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ π‘₯ ∧ π‘₯ ≀ if(1 ≀ π‘₯, π‘₯, 1)) β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ if(1 ≀ π‘₯, π‘₯, 1)))
133130, 109, 131, 132syl3anc 1372 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (((1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ π‘₯ ∧ π‘₯ ≀ if(1 ≀ π‘₯, π‘₯, 1)) β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ if(1 ≀ π‘₯, π‘₯, 1)))
134111, 133mpan2d 693 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ π‘₯ β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ if(1 ≀ π‘₯, π‘₯, 1)))
135 fveq2 6892 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 β†’ (πΉβ€˜π‘§) = (πΉβ€˜π‘¦))
136135oveq2d 7425 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 β†’ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)) = (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)))
137136oveq2d 7425 . . . . . . . . . . . . 13 (𝑧 = 𝑦 β†’ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))) = (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))))
138 eqid 2733 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§)))) = (𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))
139 ovex 7442 . . . . . . . . . . . . 13 (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ∈ V
140137, 138, 139fvmpt 6999 . . . . . . . . . . . 12 (𝑦 ∈ 𝑋 β†’ ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) = (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))))
141140breq1d 5159 . . . . . . . . . . 11 (𝑦 ∈ 𝑋 β†’ (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ π‘₯))
142141ad2antll 728 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ π‘₯))
143102adantrr 716 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (1 / if(1 ≀ π‘₯, π‘₯, 1)) ∈ ℝ)
144100adantrr 716 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ 0 < if(1 ≀ π‘₯, π‘₯, 1))
145131, 144recgt0d 12148 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ 0 < (1 / if(1 ≀ π‘₯, π‘₯, 1)))
146 lerec 12097 . . . . . . . . . . . 12 ((((1 / if(1 ≀ π‘₯, π‘₯, 1)) ∈ ℝ ∧ 0 < (1 / if(1 ≀ π‘₯, π‘₯, 1))) ∧ ((sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ∈ ℝ ∧ 0 < (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)))) β†’ ((1 / if(1 ≀ π‘₯, π‘₯, 1)) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ (1 / (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
147143, 145, 118, 128, 146syl22anc 838 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((1 / if(1 ≀ π‘₯, π‘₯, 1)) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ (1 / (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
148 lesub 11693 . . . . . . . . . . . 12 (((1 / if(1 ≀ π‘₯, π‘₯, 1)) ∈ ℝ ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ (πΉβ€˜π‘¦) ∈ ℝ) β†’ ((1 / if(1 ≀ π‘₯, π‘₯, 1)) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ↔ (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
149143, 112, 117, 148syl3anc 1372 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((1 / if(1 ≀ π‘₯, π‘₯, 1)) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦)) ↔ (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
150131recnd 11242 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ if(1 ≀ π‘₯, π‘₯, 1) ∈ β„‚)
151101adantrr 716 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ if(1 ≀ π‘₯, π‘₯, 1) β‰  0)
152150, 151recrecd 11987 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (1 / (1 / if(1 ≀ π‘₯, π‘₯, 1))) = if(1 ≀ π‘₯, π‘₯, 1))
153152breq2d 5161 . . . . . . . . . . 11 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ (1 / (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ if(1 ≀ π‘₯, π‘₯, 1)))
154147, 149, 1533bitr3d 309 . . . . . . . . . 10 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ ((πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘¦))) ≀ if(1 ≀ π‘₯, π‘₯, 1)))
155134, 142, 1543imtr4d 294 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ (π‘₯ ∈ ℝ ∧ 𝑦 ∈ 𝑋)) β†’ (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ β†’ (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
156155anassrs 469 . . . . . . . 8 ((((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) ∧ 𝑦 ∈ 𝑋) β†’ (((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ β†’ (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
157156ralimdva 3168 . . . . . . 7 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ β†’ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
15834ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (ran 𝐹 βŠ† ℝ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯))
159 suprleub 12180 . . . . . . . . 9 (((ran 𝐹 βŠ† ℝ ∧ ran 𝐹 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ π‘₯) ∧ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ∈ ℝ) β†’ (sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
160158, 106, 159syl2anc 585 . . . . . . . 8 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
16128ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ 𝐹 Fn 𝑋)
162 breq1 5152 . . . . . . . . . 10 (𝑧 = (πΉβ€˜π‘¦) β†’ (𝑧 ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
163162ralrn 7090 . . . . . . . . 9 (𝐹 Fn 𝑋 β†’ (βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
164161, 163syl 17 . . . . . . . 8 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘§ ∈ ran 𝐹 𝑧 ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
165160, 164bitrd 279 . . . . . . 7 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1))) ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
166157, 165sylibrd 259 . . . . . 6 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘¦ ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯ β†’ sup(ran 𝐹, ℝ, < ) ≀ (sup(ran 𝐹, ℝ, < ) βˆ’ (1 / if(1 ≀ π‘₯, π‘₯, 1)))))
167108, 166mtod 197 . . . . 5 (((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) ∧ π‘₯ ∈ ℝ) β†’ Β¬ βˆ€π‘¦ ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯)
168167nrexdv 3150 . . . 4 ((πœ‘ ∧ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})) β†’ Β¬ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝑋 ((𝑧 ∈ 𝑋 ↦ (1 / (sup(ran 𝐹, ℝ, < ) βˆ’ (πΉβ€˜π‘§))))β€˜π‘¦) ≀ π‘₯)
16988, 168pm2.65da 816 . . 3 (πœ‘ β†’ Β¬ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}))
170122ralrimiva 3147 . . . . . . . . 9 (πœ‘ β†’ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < ))
171 breq2 5153 . . . . . . . . . 10 ((πΉβ€˜π‘₯) = sup(ran 𝐹, ℝ, < ) β†’ ((πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) ↔ (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < )))
172171ralbidv 3178 . . . . . . . . 9 ((πΉβ€˜π‘₯) = sup(ran 𝐹, ℝ, < ) β†’ (βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) ↔ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ sup(ran 𝐹, ℝ, < )))
173170, 172syl5ibrcom 246 . . . . . . . 8 (πœ‘ β†’ ((πΉβ€˜π‘₯) = sup(ran 𝐹, ℝ, < ) β†’ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯)))
174173necon3bd 2955 . . . . . . 7 (πœ‘ β†’ (Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) β†’ (πΉβ€˜π‘₯) β‰  sup(ran 𝐹, ℝ, < )))
175174adantr 482 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) β†’ (πΉβ€˜π‘₯) β‰  sup(ran 𝐹, ℝ, < )))
17619ffvelcdmda 7087 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (πΉβ€˜π‘₯) ∈ ℝ)
177 eldifsn 4791 . . . . . . . 8 ((πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ ((πΉβ€˜π‘₯) ∈ ℝ ∧ (πΉβ€˜π‘₯) β‰  sup(ran 𝐹, ℝ, < )))
178177baib 537 . . . . . . 7 ((πΉβ€˜π‘₯) ∈ ℝ β†’ ((πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ (πΉβ€˜π‘₯) β‰  sup(ran 𝐹, ℝ, < )))
179176, 178syl 17 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ (πΉβ€˜π‘₯) β‰  sup(ran 𝐹, ℝ, < )))
180175, 179sylibrd 259 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ (Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) β†’ (πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
181180ralimdva 3168 . . . 4 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) β†’ βˆ€π‘₯ ∈ 𝑋 (πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
182 ffnfv 7118 . . . . . 6 (𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ (𝐹 Fn 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 (πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
183182baib 537 . . . . 5 (𝐹 Fn 𝑋 β†’ (𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ βˆ€π‘₯ ∈ 𝑋 (πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
18428, 183syl 17 . . . 4 (πœ‘ β†’ (𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )}) ↔ βˆ€π‘₯ ∈ 𝑋 (πΉβ€˜π‘₯) ∈ (ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
185181, 184sylibrd 259 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) β†’ 𝐹:π‘‹βŸΆ(ℝ βˆ– {sup(ran 𝐹, ℝ, < )})))
186169, 185mtod 197 . 2 (πœ‘ β†’ Β¬ βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯))
187 dfrex2 3074 . 2 (βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯) ↔ Β¬ βˆ€π‘₯ ∈ 𝑋 Β¬ βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯))
188186, 187sylibr 233 1 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (πΉβ€˜π‘¦) ≀ (πΉβ€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {csn 4629  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  supcsup 9435  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444   / cdiv 11871  (,)cioo 13324   β†Ύt crest 17366  TopOpenctopn 17367  topGenctg 17383  β„‚fldccnfld 20944  Topctop 22395  TopOnctopon 22412   Cn ccn 22728  Compccmp 22890   Γ—t ctx 23064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-icc 13331  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-submnd 18672  df-mulg 18951  df-cntz 19181  df-cmn 19650  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cn 22731  df-cnp 22732  df-cmp 22891  df-tx 23066  df-hmeo 23259  df-xms 23826  df-ms 23827  df-tms 23828
This theorem is referenced by:  evth2  24476  evthicc  24976  evthf  43711  cncmpmax  43716
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