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Theorem smflimsuplem5 45218
Description: 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem5.a β„²π‘›πœ‘
smflimsuplem5.b β„²π‘šπœ‘
smflimsuplem5.m (πœ‘ β†’ 𝑀 ∈ β„€)
smflimsuplem5.z 𝑍 = (β„€β‰₯β€˜π‘€)
smflimsuplem5.s (πœ‘ β†’ 𝑆 ∈ SAlg)
smflimsuplem5.f (πœ‘ β†’ 𝐹:π‘βŸΆ(SMblFnβ€˜π‘†))
smflimsuplem5.e 𝐸 = (𝑛 ∈ 𝑍 ↦ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
smflimsuplem5.h 𝐻 = (𝑛 ∈ 𝑍 ↦ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
smflimsuplem5.r (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) ∈ ℝ)
smflimsuplem5.n (πœ‘ β†’ 𝑁 ∈ 𝑍)
smflimsuplem5.x (πœ‘ β†’ 𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š))
Assertion
Ref Expression
smflimsuplem5 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘‹)) ⇝ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))))
Distinct variable groups:   𝑛,𝐹,π‘₯   π‘š,𝑀   π‘š,𝑁,𝑛   π‘š,𝑋,𝑛   π‘š,𝑍,𝑛,π‘₯
Allowed substitution hints:   πœ‘(π‘₯,π‘š,𝑛)   𝑆(π‘₯,π‘š,𝑛)   𝐸(π‘₯,π‘š,𝑛)   𝐹(π‘š)   𝐻(π‘₯,π‘š,𝑛)   𝑀(π‘₯,𝑛)   𝑁(π‘₯)   𝑋(π‘₯)

Proof of Theorem smflimsuplem5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 smflimsuplem5.a . . 3 β„²π‘›πœ‘
2 smflimsuplem5.n . . . . . . . 8 (πœ‘ β†’ 𝑁 ∈ 𝑍)
3 smflimsuplem5.z . . . . . . . . . . . 12 𝑍 = (β„€β‰₯β€˜π‘€)
43eleq2i 2824 . . . . . . . . . . 11 (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
54biimpi 215 . . . . . . . . . 10 (𝑁 ∈ 𝑍 β†’ 𝑁 ∈ (β„€β‰₯β€˜π‘€))
6 uzss 12810 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜π‘€) β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘€))
75, 6syl 17 . . . . . . . . 9 (𝑁 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘€))
87, 3sseqtrrdi 4013 . . . . . . . 8 (𝑁 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
92, 8syl 17 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
109sselda 3962 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ 𝑍)
11 smflimsuplem5.e . . . . . . . . . 10 𝐸 = (𝑛 ∈ 𝑍 ↦ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
12 nfcv 2902 . . . . . . . . . . 11 β„²π‘₯𝑍
13 nfrab1 3437 . . . . . . . . . . 11 β„²π‘₯{π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ}
1412, 13nfmpt 5232 . . . . . . . . . 10 β„²π‘₯(𝑛 ∈ 𝑍 ↦ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
1511, 14nfcxfr 2900 . . . . . . . . 9 β„²π‘₯𝐸
16 nfcv 2902 . . . . . . . . 9 β„²π‘₯𝑛
1715, 16nffv 6872 . . . . . . . 8 β„²π‘₯(πΈβ€˜π‘›)
18 fvex 6875 . . . . . . . 8 (πΈβ€˜π‘›) ∈ V
1917, 18mptexf 43617 . . . . . . 7 (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ V
2019a1i 11 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ V)
21 smflimsuplem5.h . . . . . . 7 𝐻 = (𝑛 ∈ 𝑍 ↦ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
2221fvmpt2 6979 . . . . . 6 ((𝑛 ∈ 𝑍 ∧ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ V) β†’ (π»β€˜π‘›) = (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
2310, 20, 22syl2anc 584 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (π»β€˜π‘›) = (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
2423fveq1d 6864 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘‹) = ((π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))β€˜π‘‹))
25 nfcv 2902 . . . . . 6 Ⅎ𝑦(πΈβ€˜π‘›)
26 nfcv 2902 . . . . . 6 Ⅎ𝑦sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )
27 nfcv 2902 . . . . . 6 β„²π‘₯sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < )
28 fveq2 6862 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ((πΉβ€˜π‘š)β€˜π‘₯) = ((πΉβ€˜π‘š)β€˜π‘¦))
2928mpteq2dv 5227 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) = (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)))
3029rneqd 5913 . . . . . . 7 (π‘₯ = 𝑦 β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) = ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)))
3130supeq1d 9406 . . . . . 6 (π‘₯ = 𝑦 β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ))
3217, 25, 26, 27, 31cbvmptf 5234 . . . . 5 (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) = (𝑦 ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ))
33 simpl 483 . . . . . . . . 9 ((𝑦 = 𝑋 ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝑦 = 𝑋)
3433fveq2d 6866 . . . . . . . 8 ((𝑦 = 𝑋 ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ ((πΉβ€˜π‘š)β€˜π‘¦) = ((πΉβ€˜π‘š)β€˜π‘‹))
3534mpteq2dva 5225 . . . . . . 7 (𝑦 = 𝑋 β†’ (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)) = (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)))
3635rneqd 5913 . . . . . 6 (𝑦 = 𝑋 β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)) = ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)))
3736supeq1d 9406 . . . . 5 (𝑦 = 𝑋 β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ))
3837eleq1d 2817 . . . . . . . 8 (𝑦 = 𝑋 β†’ (sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ) ∈ ℝ))
39 uzss 12810 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (β„€β‰₯β€˜π‘›) βŠ† (β„€β‰₯β€˜π‘))
40 iinss1 4989 . . . . . . . . . . 11 ((β„€β‰₯β€˜π‘›) βŠ† (β„€β‰₯β€˜π‘) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) βŠ† ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š))
4139, 40syl 17 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) βŠ† ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š))
4241adantl 482 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) βŠ† ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š))
43 smflimsuplem5.x . . . . . . . . . 10 (πœ‘ β†’ 𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š))
4443adantr 481 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š))
4542, 44sseldd 3963 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š))
46 smflimsuplem5.b . . . . . . . . . . 11 β„²π‘šπœ‘
47 nfv 1917 . . . . . . . . . . 11 β„²π‘š 𝑛 ∈ (β„€β‰₯β€˜π‘)
4846, 47nfan 1902 . . . . . . . . . 10 β„²π‘š(πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘))
49 eqid 2731 . . . . . . . . . 10 (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘›)
50 simpll 765 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
5139sselda 3962 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
5251adantll 712 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
53 smflimsuplem5.s . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑆 ∈ SAlg)
5453adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑆 ∈ SAlg)
55 simpl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ πœ‘)
569sselda 3962 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ 𝑍)
57 smflimsuplem5.f . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐹:π‘βŸΆ(SMblFnβ€˜π‘†))
5857ffvelcdmda 7055 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
5955, 56, 58syl2anc 584 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
60 eqid 2731 . . . . . . . . . . . . 13 dom (πΉβ€˜π‘š) = dom (πΉβ€˜π‘š)
6154, 59, 60smff 45126 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š):dom (πΉβ€˜π‘š)βŸΆβ„)
62 eliin 4979 . . . . . . . . . . . . . . . 16 (𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) β†’ (𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) ↔ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ dom (πΉβ€˜π‘š)))
6343, 62syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ (𝑋 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘)dom (πΉβ€˜π‘š) ↔ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ dom (πΉβ€˜π‘š)))
6443, 63mpbid 231 . . . . . . . . . . . . . 14 (πœ‘ β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ dom (πΉβ€˜π‘š))
6564adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ dom (πΉβ€˜π‘š))
66 simpr 485 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
67 rspa 3242 . . . . . . . . . . . . 13 ((βˆ€π‘š ∈ (β„€β‰₯β€˜π‘)𝑋 ∈ dom (πΉβ€˜π‘š) ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ dom (πΉβ€˜π‘š))
6865, 66, 67syl2anc 584 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ dom (πΉβ€˜π‘š))
6961, 68ffvelcdmd 7056 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
7050, 52, 69syl2anc 584 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ ℝ)
71 eluzelz 12797 . . . . . . . . . . . . . 14 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑛 ∈ β„€)
7271adantl 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ β„€)
73 smflimsuplem5.m . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ β„€)
7473adantr 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑀 ∈ β„€)
75 fvex 6875 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘š)β€˜π‘‹) ∈ V
7675a1i 11 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ 𝑍) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ V)
7748, 72, 74, 49, 3, 70, 76limsupequzmpt 44123 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) = (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘‹))))
78 smflimsuplem5.r . . . . . . . . . . . . 13 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) ∈ ℝ)
7978adantr 481 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) ∈ ℝ)
8077, 79eqeltrd 2832 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) ∈ ℝ)
8180renepnfd 11230 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) β‰  +∞)
8248, 49, 70, 81limsupubuzmpt 44113 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ βˆƒπ‘¦ ∈ ℝ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ≀ 𝑦)
83 uzid2 43793 . . . . . . . . . . . 12 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘›))
8483ne0d 4315 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
8584adantl 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
8648, 85, 70supxrre3rnmpt 43817 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ) ∈ ℝ ↔ βˆƒπ‘¦ ∈ ℝ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)((πΉβ€˜π‘š)β€˜π‘‹) ≀ 𝑦))
8782, 86mpbird 256 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ) ∈ ℝ)
8838, 45, 87elrabd 3665 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ {𝑦 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) ∈ ℝ})
89 simpl 483 . . . . . . . . . . . . 13 ((𝑦 = π‘₯ ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ 𝑦 = π‘₯)
9089fveq2d 6866 . . . . . . . . . . . 12 ((𝑦 = π‘₯ ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ ((πΉβ€˜π‘š)β€˜π‘¦) = ((πΉβ€˜π‘š)β€˜π‘₯))
9190mpteq2dva 5225 . . . . . . . . . . 11 (𝑦 = π‘₯ β†’ (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)) = (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)))
9291rneqd 5913 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)) = ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)))
9392supeq1d 9406 . . . . . . . . 9 (𝑦 = π‘₯ β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))
9493eleq1d 2817 . . . . . . . 8 (𝑦 = π‘₯ β†’ (sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ))
9594cbvrabv 3428 . . . . . . 7 {𝑦 ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘¦)), ℝ*, < ) ∈ ℝ} = {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ}
9688, 95eleqtrdi 2842 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
97 eqid 2731 . . . . . . . 8 {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ} = {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ}
98 fvex 6875 . . . . . . . . . . . . 13 (πΉβ€˜π‘š) ∈ V
9998dmex 7868 . . . . . . . . . . . 12 dom (πΉβ€˜π‘š) ∈ V
10099rgenw 3064 . . . . . . . . . . 11 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V
101100a1i 11 . . . . . . . . . 10 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
10284, 101iinexd 43498 . . . . . . . . 9 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
103102adantl 482 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
10497, 103rabexd 5310 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ} ∈ V)
10511fvmpt2 6979 . . . . . . 7 ((𝑛 ∈ 𝑍 ∧ {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ} ∈ V) β†’ (πΈβ€˜π‘›) = {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
10610, 104, 105syl2anc 584 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (πΈβ€˜π‘›) = {π‘₯ ∈ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∣ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ})
10796, 106eleqtrrd 2835 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑋 ∈ (πΈβ€˜π‘›))
10832, 37, 107, 87fvmptd3 6991 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))β€˜π‘‹) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ))
10924, 108eqtrd 2771 . . 3 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘‹) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < ))
1101, 109mpteq2da 5223 . 2 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘‹)) = (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < )))
1113eluzelz2 43791 . . . 4 (𝑁 ∈ 𝑍 β†’ 𝑁 ∈ β„€)
1122, 111syl 17 . . 3 (πœ‘ β†’ 𝑁 ∈ β„€)
113 eqid 2731 . . 3 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
11475a1i 11 . . . . 5 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ V)
11575a1i 11 . . . . 5 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ ((πΉβ€˜π‘š)β€˜π‘‹) ∈ V)
11646, 112, 73, 113, 3, 114, 115limsupequzmpt 44123 . . . 4 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) = (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘‹))))
117116, 78eqeltrd 2832 . . 3 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))) ∈ ℝ)
11846, 112, 113, 69, 117supcnvlimsupmpt 44135 . 2 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘‹)), ℝ*, < )) ⇝ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))))
119110, 118eqbrtrd 5147 1 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘‹)) ⇝ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘‹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  β„²wnf 1785   ∈ wcel 2106   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  {crab 3418  Vcvv 3459   βŠ† wss 3928  βˆ…c0 4302  βˆ© ciin 4975   class class class wbr 5125   ↦ cmpt 5208  dom cdm 5653  ran crn 5654  βŸΆwf 6512  β€˜cfv 6516  supcsup 9400  β„cr 11074  β„*cxr 11212   < clt 11213   ≀ cle 11214  β„€cz 12523  β„€β‰₯cuz 12787  lim supclsp 15379   ⇝ cli 15393  SAlgcsalg 44702  SMblFncsmblfn 45089
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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3364  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-iin 4977  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-pm 8790  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-sup 9402  df-inf 9403  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-div 11837  df-nn 12178  df-2 12240  df-3 12241  df-n0 12438  df-z 12524  df-uz 12788  df-q 12898  df-rp 12940  df-ioo 13293  df-ico 13295  df-fz 13450  df-fl 13722  df-ceil 13723  df-seq 13932  df-exp 13993  df-cj 15011  df-re 15012  df-im 15013  df-sqrt 15147  df-abs 15148  df-limsup 15380  df-clim 15397  df-smblfn 45090
This theorem is referenced by:  smflimsuplem6  45219  smflimsuplem8  45221
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