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Theorem smflimsuplem4 46270
Description: If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem4.1 β„²π‘›πœ‘
smflimsuplem4.m (πœ‘ β†’ 𝑀 ∈ β„€)
smflimsuplem4.z 𝑍 = (β„€β‰₯β€˜π‘€)
smflimsuplem4.s (πœ‘ β†’ 𝑆 ∈ SAlg)
smflimsuplem4.f (πœ‘ β†’ 𝐹:π‘βŸΆ(SMblFnβ€˜π‘†))
smflimsuplem4.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})
smflimsuplem4.h 𝐻 = (𝑛 ∈ 𝑍 ↦ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
smflimsuplem4.n (πœ‘ β†’ 𝑁 ∈ 𝑍)
smflimsuplem4.i (πœ‘ β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›))
smflimsuplem4.c (πœ‘ β†’ (𝑛 ∈ 𝑍 ↦ ((π»β€˜π‘›)β€˜π‘₯)) ∈ dom ⇝ )
Assertion
Ref Expression
smflimsuplem4 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘₯))) ∈ ℝ)
Distinct variable groups:   𝑛,𝐸,π‘₯   π‘š,𝐹,𝑛,π‘₯   𝑛,𝐻   π‘š,𝑀   π‘š,𝑁,𝑛   π‘š,𝑍,𝑛   πœ‘,π‘š
Allowed substitution hints:   πœ‘(π‘₯,𝑛)   𝑆(π‘₯,π‘š,𝑛)   𝐸(π‘š)   𝐻(π‘₯,π‘š)   𝑀(π‘₯,𝑛)   𝑁(π‘₯)   𝑍(π‘₯)

Proof of Theorem smflimsuplem4
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 nfv 1909 . . . 4 β„²π‘šπœ‘
2 smflimsuplem4.m . . . 4 (πœ‘ β†’ 𝑀 ∈ β„€)
3 smflimsuplem4.z . . . . 5 𝑍 = (β„€β‰₯β€˜π‘€)
4 smflimsuplem4.n . . . . 5 (πœ‘ β†’ 𝑁 ∈ 𝑍)
53, 4eluzelz2d 44854 . . . 4 (πœ‘ β†’ 𝑁 ∈ β„€)
6 eqid 2725 . . . 4 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
7 fvexd 6905 . . . 4 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
8 fvexd 6905 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 45176 . . 3 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘₯))) = (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘₯))))
10 smflimsuplem4.s . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ SAlg)
1110adantr 479 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑆 ∈ SAlg)
123, 4uzssd2 44858 . . . . . . . . 9 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
1312sselda 3973 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ 𝑍)
14 smflimsuplem4.f . . . . . . . . 9 (πœ‘ β†’ 𝐹:π‘βŸΆ(SMblFnβ€˜π‘†))
1514ffvelcdmda 7087 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
1613, 15syldan 589 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
17 eqid 2725 . . . . . . 7 dom (πΉβ€˜π‘š) = dom (πΉβ€˜π‘š)
1811, 16, 17smff 46179 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š):dom (πΉβ€˜π‘š)βŸΆβ„)
19 smflimsuplem4.e . . . . . . . 8 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})
20 smflimsuplem4.h . . . . . . . 8 𝐻 = (𝑛 ∈ 𝑍 ↦ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
213, 19, 20, 13smflimsuplem1 46267 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π»β€˜π‘š) βŠ† dom (πΉβ€˜π‘š))
22 smflimsuplem4.i . . . . . . . . 9 (πœ‘ β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›))
2322adantr 479 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›))
24 simpr 483 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
25 fveq2 6890 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π»β€˜π‘›) = (π»β€˜π‘š))
2625dmeqd 5903 . . . . . . . . 9 (𝑛 = π‘š β†’ dom (π»β€˜π‘›) = dom (π»β€˜π‘š))
2726eleq2d 2811 . . . . . . . 8 (𝑛 = π‘š β†’ (π‘₯ ∈ dom (π»β€˜π‘›) ↔ π‘₯ ∈ dom (π»β€˜π‘š)))
2823, 24, 27eliind 44496 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (π»β€˜π‘š))
2921, 28sseldd 3974 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (πΉβ€˜π‘š))
3018, 29ffvelcdmd 7088 . . . . 5 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ)
3130rexrd 11289 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 45164 . . 3 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ (β„€β‰₯β€˜π‘) ↦ ((πΉβ€˜π‘š)β€˜π‘₯))) = inf(ran (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )), ℝ*, < ))
339, 32eqtrd 2765 . 2 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘₯))) = inf(ran (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )), ℝ*, < ))
34 smflimsuplem4.1 . . 3 β„²π‘›πœ‘
3512adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
36 simpr 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
3735, 36sseldd 3974 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ 𝑍)
3820a1i 11 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐻 = (𝑛 ∈ 𝑍 ↦ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))))
39 fvex 6903 . . . . . . . . . . . . . . 15 (πΈβ€˜π‘›) ∈ V
4039mptex 7229 . . . . . . . . . . . . . 14 (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ V
4140a1i 11 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ V)
4238, 41fvmpt2d 7011 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ (π»β€˜π‘›) = (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
4337, 42syldan 589 . . . . . . . . . . 11 ((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*, < )))
4443dmeqd 5903 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π»β€˜π‘›) = dom (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )))
45 xrltso 13147 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9481 . . . . . . . . . . . 12 sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ V
47 eqid 2725 . . . . . . . . . . . 12 (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) = (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))
4846, 47dmmpti 6694 . . . . . . . . . . 11 dom (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) = (πΈβ€˜π‘›)
4948a1i 11 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π‘₯ ∈ (πΈβ€˜π‘›) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) = (πΈβ€˜π‘›))
5044, 49eqtrd 2765 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π»β€˜π‘›) = (πΈβ€˜π‘›))
5134, 50iineq2d 5015 . . . . . . . 8 (πœ‘ β†’ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›) = ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5222, 51eleqtrd 2827 . . . . . . 7 (πœ‘ β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5352adantr 479 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
54 eliinid 44538 . . . . . 6 ((π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ (πΈβ€˜π‘›))
5553, 36, 54syl2anc 582 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ (πΈβ€˜π‘›))
5646a1i 11 . . . . . 6 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘₯ ∈ (πΈβ€˜π‘›)) β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ V)
5743, 56fvmpt2d 7011 . . . . 5 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘₯ ∈ (πΈβ€˜π‘›)) β†’ ((π»β€˜π‘›)β€˜π‘₯) = sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ))
5855, 57mpdan 685 . . . 4 ((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*, < ))
59 eqid 2725 . . . . . . . . . 10 {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}
603eluzelz2 44844 . . . . . . . . . . . . 13 (𝑛 ∈ 𝑍 β†’ 𝑛 ∈ β„€)
61 eqid 2725 . . . . . . . . . . . . 13 (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘›)
6260, 61uzn0d 44866 . . . . . . . . . . . 12 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
63 fvex 6903 . . . . . . . . . . . . . . 15 (πΉβ€˜π‘š) ∈ V
6463dmex 7911 . . . . . . . . . . . . . 14 dom (πΉβ€˜π‘š) ∈ V
6564rgenw 3055 . . . . . . . . . . . . 13 βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V
6665a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ 𝑍 β†’ βˆ€π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6762, 66iinexd 44560 . . . . . . . . . . 11 (𝑛 ∈ 𝑍 β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6867adantl 480 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6959, 68rabexd 5331 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ 𝑍) 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)
7037, 69syldan 589 . . . . . . . 8 ((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)
7119fvmpt2 7009 . . . . . . . 8 ((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})
7237, 70, 71syl2anc 582 . . . . . . 7 ((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})
7355, 72eleqtrd 2827 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 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})
74 rabid 3440 . . . . . 6 (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} ↔ (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))
7573, 74sylib 217 . . . . 5 ((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))
7675simprd 494 . . . 4 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ ℝ)
7758, 76eqeltrd 2825 . . 3 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ ℝ)
7834, 58mpteq2da 5242 . . . 4 (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*, < )))
79 nfv 1909 . . . . 5 β„²π‘˜πœ‘
80 fveq2 6890 . . . . . . . 8 (𝑛 = π‘˜ β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘˜))
8180mpteq1d 5239 . . . . . . 7 (𝑛 = 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‘₯)))
8281rneqd 5935 . . . . . 6 (𝑛 = 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‘₯)))
8382supeq1d 9464 . . . . 5 (𝑛 = 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*, < ))
84 nfv 1909 . . . . . . . 8 β„²π‘š(𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1))
85 eluzelz 12857 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑛 ∈ β„€)
8685adantr 479 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ β„€)
87 simpr 483 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ = (𝑛 + 1))
8886peano2zd 12694 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) ∈ β„€)
8987, 88eqeltrd 2825 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ β„€)
9086zred 12691 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ ℝ)
9189zred 12691 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ ℝ)
9290ltp1d 12169 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < (𝑛 + 1))
9387eqcomd 2731 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) = π‘˜)
9492, 93breqtrd 5170 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < π‘˜)
9590, 91, 94ltled 11387 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ≀ π‘˜)
9661, 86, 89, 95eluzd 44850 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘›))
97 uzss 12870 . . . . . . . . 9 (π‘˜ ∈ (β„€β‰₯β€˜π‘›) β†’ (β„€β‰₯β€˜π‘˜) βŠ† (β„€β‰₯β€˜π‘›))
9896, 97syl 17 . . . . . . . 8 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (β„€β‰₯β€˜π‘˜) βŠ† (β„€β‰₯β€˜π‘›))
99 fvexd 6905 . . . . . . . 8 (((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘˜)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
10084, 98, 99rnmptss2 44692 . . . . . . 7 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘˜) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)))
1011003adant1 1127 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘˜) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)))
102 nfv 1909 . . . . . . . . 9 β„²π‘š(πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘))
103 eqid 2725 . . . . . . . . 9 (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) = (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯))
104 simpll 765 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
10537, 104syldanl 600 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
1066uztrn2 12866 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
107106adantll 712 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
108105, 107, 30syl2anc 582 . . . . . . . . 9 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ)
109102, 103, 108rnmptssd 44629 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† ℝ)
110 ressxr 11283 . . . . . . . . 9 ℝ βŠ† ℝ*
111110a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ℝ βŠ† ℝ*)
112109, 111sstrd 3984 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† ℝ*)
1131123adant3 1129 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† ℝ*)
114 supxrss 13338 . . . . . 6 ((ran (π‘š ∈ (β„€β‰₯β€˜π‘˜) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)) βŠ† 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‘₯)) βŠ† ℝ*) 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*, < ))
115101, 113, 114syl2anc 582 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) 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*, < ))
116 smflimsuplem4.c . . . . . . 7 (πœ‘ β†’ (𝑛 ∈ 𝑍 ↦ ((π»β€˜π‘›)β€˜π‘₯)) ∈ dom ⇝ )
1173fvexi 6904 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝑍 ∈ V)
119 fvexd 6905 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
120 fvexd 6905 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) ∈ V)
12134, 36ssdf 44502 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘))
122 fvexd 6905 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
123 eqidd 2726 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) = ((π»β€˜π‘›)β€˜π‘₯))
12434, 5, 6, 118, 12, 119, 120, 121, 122, 123climeldmeqmpt 45115 . . . . . . 7 (πœ‘ β†’ ((𝑛 ∈ 𝑍 ↦ ((π»β€˜π‘›)β€˜π‘₯)) ∈ dom ⇝ ↔ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘₯)) ∈ dom ⇝ ))
125116, 124mpbid 231 . . . . . 6 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘₯)) ∈ dom ⇝ )
12678, 125eqeltrrd 2826 . . . . 5 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ∈ dom ⇝ )
12734, 79, 5, 6, 76, 83, 115, 126climinf2mpt 45161 . . . 4 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )) ⇝ inf(ran (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )), ℝ*, < ))
12878, 127eqbrtrd 5166 . . 3 (πœ‘ β†’ (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ ((π»β€˜π‘›)β€˜π‘₯)) ⇝ inf(ran (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )), ℝ*, < ))
12934, 5, 6, 77, 128climreclmpt 45131 . 2 (πœ‘ β†’ inf(ran (𝑛 ∈ (β„€β‰₯β€˜π‘) ↦ sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < )), ℝ*, < ) ∈ ℝ)
13033, 129eqeltrd 2825 1 (πœ‘ β†’ (lim supβ€˜(π‘š ∈ 𝑍 ↦ ((πΉβ€˜π‘š)β€˜π‘₯))) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  β„²wnf 1777   ∈ wcel 2098  βˆ€wral 3051  {crab 3419  Vcvv 3463   βŠ† wss 3941  βˆ© ciin 4993   class class class wbr 5144   ↦ cmpt 5227  dom cdm 5673  ran crn 5674  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413  supcsup 9458  infcinf 9459  β„cr 11132  1c1 11134   + caddc 11136  β„*cxr 11272   < clt 11273   ≀ cle 11274  β„€cz 12583  β„€β‰₯cuz 12847  lim supclsp 15441   ⇝ cli 15455  SAlgcsalg 45755  SMblFncsmblfn 46142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-pm 8841  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-q 12958  df-rp 13002  df-ioo 13355  df-ico 13357  df-fz 13512  df-fl 13784  df-seq 13994  df-exp 14054  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-limsup 15442  df-clim 15459  df-rlim 15460  df-smblfn 46143
This theorem is referenced by:  smflimsuplem7  46273
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