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Theorem limsupvaluz2 44752
Description: The superior limit, when the domain of a real-valued function is a set of upper integers, and the superior limit is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupvaluz2.m (πœ‘ β†’ 𝑀 ∈ β„€)
limsupvaluz2.z 𝑍 = (β„€β‰₯β€˜π‘€)
limsupvaluz2.f (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
limsupvaluz2.r (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ)
Assertion
Ref Expression
limsupvaluz2 (πœ‘ β†’ (lim supβ€˜πΉ) = inf(ran (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < )), ℝ, < ))
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝑍
Allowed substitution hints:   πœ‘(π‘˜)   𝑀(π‘˜)

Proof of Theorem limsupvaluz2
Dummy variables 𝑖 𝑗 π‘₯ 𝑛 π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupvaluz2.m . . 3 (πœ‘ β†’ 𝑀 ∈ β„€)
2 limsupvaluz2.z . . 3 𝑍 = (β„€β‰₯β€˜π‘€)
3 limsupvaluz2.f . . . 4 (πœ‘ β†’ 𝐹:π‘βŸΆβ„)
43frexr 44393 . . 3 (πœ‘ β†’ 𝐹:π‘βŸΆβ„*)
51, 2, 4limsupvaluz 44722 . 2 (πœ‘ β†’ (lim supβ€˜πΉ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ*, < ))
63adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ 𝐹:π‘βŸΆβ„)
7 id 22 . . . . . . . . . . 11 (𝑛 ∈ 𝑍 β†’ 𝑛 ∈ 𝑍)
82, 7uzssd2 44425 . . . . . . . . . 10 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) βŠ† 𝑍)
98adantl 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘›) βŠ† 𝑍)
106, 9feqresmpt 6960 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ (πΉβ€˜π‘š)))
1110rneqd 5936 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ (πΉβ€˜π‘š)))
1211supeq1d 9443 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ (πΉβ€˜π‘š)), ℝ*, < ))
13 nfcv 2901 . . . . . . . . . 10 β„²π‘šπΉ
14 limsupvaluz2.r . . . . . . . . . . 11 (πœ‘ β†’ (lim supβ€˜πΉ) ∈ ℝ)
1514renepnfd 11269 . . . . . . . . . 10 (πœ‘ β†’ (lim supβ€˜πΉ) β‰  +∞)
1613, 2, 3, 15limsupubuz 44727 . . . . . . . . 9 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯)
1716adantr 479 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯)
18 ssralv 4049 . . . . . . . . . . 11 ((β„€β‰₯β€˜π‘›) βŠ† 𝑍 β†’ (βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯ β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯))
198, 18syl 17 . . . . . . . . . 10 (𝑛 ∈ 𝑍 β†’ (βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯ β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯))
2019adantl 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯ β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯))
2120reximdv 3168 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ 𝑍 (πΉβ€˜π‘š) ≀ π‘₯ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯))
2217, 21mpd 15 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯)
23 nfv 1915 . . . . . . . 8 β„²π‘š(πœ‘ ∧ 𝑛 ∈ 𝑍)
242eluzelz2 44411 . . . . . . . . . 10 (𝑛 ∈ 𝑍 β†’ 𝑛 ∈ β„€)
25 uzid 12841 . . . . . . . . . 10 (𝑛 ∈ β„€ β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘›))
26 ne0i 4333 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘›) β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
2827adantl 480 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
296adantr 479 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝐹:π‘βŸΆβ„)
309sselda 3981 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ 𝑍)
3129, 30ffvelcdmd 7086 . . . . . . . 8 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ (πΉβ€˜π‘š) ∈ ℝ)
3223, 28, 31supxrre3rnmpt 44437 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ (πΉβ€˜π‘š)), ℝ*, < ) ∈ ℝ ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)(πΉβ€˜π‘š) ≀ π‘₯))
3322, 32mpbird 256 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ (πΉβ€˜π‘š)), ℝ*, < ) ∈ ℝ)
3412, 33eqeltrd 2831 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) ∈ ℝ)
3534fmpttd 7115 . . . 4 (πœ‘ β†’ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )):π‘βŸΆβ„)
3635frnd 6724 . . 3 (πœ‘ β†’ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) βŠ† ℝ)
37 nfv 1915 . . . 4 β„²π‘›πœ‘
38 eqid 2730 . . . 4 (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
391, 2uzn0d 44433 . . . 4 (πœ‘ β†’ 𝑍 β‰  βˆ…)
4037, 34, 38, 39rnmptn0 6242 . . 3 (πœ‘ β†’ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) β‰  βˆ…)
41 nfcv 2901 . . . . . . . . . 10 Ⅎ𝑗𝐹
4241, 1, 2, 4limsupre3uz 44750 . . . . . . . . 9 (πœ‘ β†’ ((lim supβ€˜πΉ) ∈ ℝ ↔ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—) ∧ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘– ∈ 𝑍 βˆ€π‘— ∈ (β„€β‰₯β€˜π‘–)(πΉβ€˜π‘—) ≀ π‘₯)))
4314, 42mpbid 231 . . . . . . . 8 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—) ∧ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘– ∈ 𝑍 βˆ€π‘— ∈ (β„€β‰₯β€˜π‘–)(πΉβ€˜π‘—) ≀ π‘₯))
4443simpld 493 . . . . . . 7 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—))
45 simp-4r 780 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ π‘₯ ∈ ℝ)
4645rexrd 11268 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ π‘₯ ∈ ℝ*)
4743ad2ant1 1131 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ 𝐹:π‘βŸΆβ„*)
482uztrn2 12845 . . . . . . . . . . . . . 14 ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ 𝑗 ∈ 𝑍)
49483adant1 1128 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ 𝑗 ∈ 𝑍)
5047, 49ffvelcdmd 7086 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ (πΉβ€˜π‘—) ∈ ℝ*)
5150ad5ant134 1365 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ (πΉβ€˜π‘—) ∈ ℝ*)
52 rnresss 6016 . . . . . . . . . . . . . . . 16 ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) βŠ† ran 𝐹
5352a1i 11 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) βŠ† ran 𝐹)
543frnd 6724 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ ran 𝐹 βŠ† ℝ)
5554adantr 479 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ ran 𝐹 βŠ† ℝ)
5653, 55sstrd 3991 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) βŠ† ℝ)
57 ressxr 11262 . . . . . . . . . . . . . . 15 ℝ βŠ† ℝ*
5857a1i 11 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ ℝ βŠ† ℝ*)
5956, 58sstrd 3991 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) βŠ† ℝ*)
6059supxrcld 44097 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ∈ ℝ*)
6160ad5ant13 753 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ∈ ℝ*)
62 simpr 483 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ π‘₯ ≀ (πΉβ€˜π‘—))
63593adant3 1130 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) βŠ† ℝ*)
64 fvres 6909 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (β„€β‰₯β€˜π‘–) β†’ ((𝐹 β†Ύ (β„€β‰₯β€˜π‘–))β€˜π‘—) = (πΉβ€˜π‘—))
6564eqcomd 2736 . . . . . . . . . . . . . . 15 (𝑗 ∈ (β„€β‰₯β€˜π‘–) β†’ (πΉβ€˜π‘—) = ((𝐹 β†Ύ (β„€β‰₯β€˜π‘–))β€˜π‘—))
66653ad2ant3 1133 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ (πΉβ€˜π‘—) = ((𝐹 β†Ύ (β„€β‰₯β€˜π‘–))β€˜π‘—))
673ffnd 6717 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐹 Fn 𝑍)
6867adantr 479 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ 𝐹 Fn 𝑍)
69 id 22 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ 𝑍 β†’ 𝑖 ∈ 𝑍)
702, 69uzssd2 44425 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘–) βŠ† 𝑍)
7170adantl 480 . . . . . . . . . . . . . . . . 17 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ (β„€β‰₯β€˜π‘–) βŠ† 𝑍)
72 fnssres 6672 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑍 ∧ (β„€β‰₯β€˜π‘–) βŠ† 𝑍) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) Fn (β„€β‰₯β€˜π‘–))
7368, 71, 72syl2anc 582 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑖 ∈ 𝑍) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) Fn (β„€β‰₯β€˜π‘–))
74733adant3 1130 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) Fn (β„€β‰₯β€˜π‘–))
75 simp3 1136 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ 𝑗 ∈ (β„€β‰₯β€˜π‘–))
76 fnfvelrn 7081 . . . . . . . . . . . . . . 15 (((𝐹 β†Ύ (β„€β‰₯β€˜π‘–)) Fn (β„€β‰₯β€˜π‘–) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ ((𝐹 β†Ύ (β„€β‰₯β€˜π‘–))β€˜π‘—) ∈ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)))
7774, 75, 76syl2anc 582 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ ((𝐹 β†Ύ (β„€β‰₯β€˜π‘–))β€˜π‘—) ∈ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)))
7866, 77eqeltrd 2831 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ (πΉβ€˜π‘—) ∈ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)))
79 eqid 2730 . . . . . . . . . . . . 13 sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )
8063, 78, 79supxrubd 44103 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) β†’ (πΉβ€˜π‘—) ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
8180ad5ant134 1365 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ (πΉβ€˜π‘—) ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
8246, 51, 61, 62, 81xrletrd 13145 . . . . . . . . . 10 (((((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (β„€β‰₯β€˜π‘–)) ∧ π‘₯ ≀ (πΉβ€˜π‘—)) β†’ π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
8382rexlimdva2 3155 . . . . . . . . 9 (((πœ‘ ∧ π‘₯ ∈ ℝ) ∧ 𝑖 ∈ 𝑍) β†’ (βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—) β†’ π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )))
8483ralimdva 3165 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ ℝ) β†’ (βˆ€π‘– ∈ 𝑍 βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—) β†’ βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )))
8584reximdva 3166 . . . . . . 7 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 βˆƒπ‘— ∈ (β„€β‰₯β€˜π‘–)π‘₯ ≀ (πΉβ€˜π‘—) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )))
8644, 85mpd 15 . . . . . 6 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
8786idi 1 . . . . 5 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
88 fveq2 6890 . . . . . . . . . . . 12 (𝑛 = 𝑖 β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘–))
8988reseq2d 5980 . . . . . . . . . . 11 (𝑛 = 𝑖 β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)))
9089rneqd 5936 . . . . . . . . . 10 (𝑛 = 𝑖 β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)))
9190supeq1d 9443 . . . . . . . . 9 (𝑛 = 𝑖 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ))
92 eqcom 2737 . . . . . . . . . . 11 (𝑛 = 𝑖 ↔ 𝑖 = 𝑛)
9392imbi1i 348 . . . . . . . . . 10 ((𝑛 = 𝑖 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )) ↔ (𝑖 = 𝑛 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )))
94 eqcom 2737 . . . . . . . . . . 11 (sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ↔ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
9594imbi2i 335 . . . . . . . . . 10 ((𝑖 = 𝑛 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )) ↔ (𝑖 = 𝑛 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )))
9693, 95bitri 274 . . . . . . . . 9 ((𝑛 = 𝑖 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < )) ↔ (𝑖 = 𝑛 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )))
9791, 96mpbi 229 . . . . . . . 8 (𝑖 = 𝑛 β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
9897breq2d 5159 . . . . . . 7 (𝑖 = 𝑛 β†’ (π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ↔ π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )))
9998cbvralvw 3232 . . . . . 6 (βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ↔ βˆ€π‘› ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
10099rexbii 3092 . . . . 5 (βˆƒπ‘₯ ∈ ℝ βˆ€π‘– ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘–)), ℝ*, < ) ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘› ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
10187, 100sylib 217 . . . 4 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘› ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))
10234elexd 3493 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) ∈ V)
10337, 102rnmptbd2 44251 . . . 4 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘› ∈ 𝑍 π‘₯ ≀ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))π‘₯ ≀ 𝑦))
104101, 103mpbid 231 . . 3 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))π‘₯ ≀ 𝑦)
105 infxrre 13319 . . 3 ((ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) βŠ† ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ))π‘₯ ≀ 𝑦) β†’ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ, < ))
10636, 40, 104, 105syl3anc 1369 . 2 (πœ‘ β†’ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ, < ))
107 fveq2 6890 . . . . . . . . 9 (𝑛 = π‘˜ β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘˜))
108107reseq2d 5980 . . . . . . . 8 (𝑛 = π‘˜ β†’ (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)))
109108rneqd 5936 . . . . . . 7 (𝑛 = π‘˜ β†’ ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)) = ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)))
110109supeq1d 9443 . . . . . 6 (𝑛 = π‘˜ β†’ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < ) = sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < ))
111110cbvmptv 5260 . . . . 5 (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) = (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < ))
112111rneqi 5935 . . . 4 ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )) = ran (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < ))
113112infeq1i 9475 . . 3 inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ, < ) = inf(ran (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < )), ℝ, < )
114113a1i 11 . 2 (πœ‘ β†’ inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘›)), ℝ*, < )), ℝ, < ) = inf(ran (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < )), ℝ, < ))
1155, 106, 1143eqtrd 2774 1 (πœ‘ β†’ (lim supβ€˜πΉ) = inf(ran (π‘˜ ∈ 𝑍 ↦ sup(ran (𝐹 β†Ύ (β„€β‰₯β€˜π‘˜)), ℝ*, < )), ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  supcsup 9437  infcinf 9438  β„cr 11111  β„*cxr 11251   < clt 11252   ≀ cle 11253  β„€cz 12562  β„€β‰₯cuz 12826  lim supclsp 15418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-ico 13334  df-fz 13489  df-fl 13761  df-ceil 13762  df-limsup 15419
This theorem is referenced by:  supcnvlimsup  44754
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