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Theorem smflimsuplem4 46124
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 1910 . . . 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 44708 . . . 4 (πœ‘ β†’ 𝑁 ∈ β„€)
6 eqid 2727 . . . 4 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
7 fvexd 6906 . . . 4 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
8 fvexd 6906 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 45030 . . 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 480 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑆 ∈ SAlg)
123, 4uzssd2 44712 . . . . . . . . 9 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
1312sselda 3978 . . . . . . . 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 7088 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
1613, 15syldan 590 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
17 eqid 2727 . . . . . . 7 dom (πΉβ€˜π‘š) = dom (πΉβ€˜π‘š)
1811, 16, 17smff 46033 . . . . . 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 46121 . . . . . . 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 480 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›))
24 simpr 484 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
25 fveq2 6891 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π»β€˜π‘›) = (π»β€˜π‘š))
2625dmeqd 5902 . . . . . . . . 9 (𝑛 = π‘š β†’ dom (π»β€˜π‘›) = dom (π»β€˜π‘š))
2726eleq2d 2814 . . . . . . . 8 (𝑛 = π‘š β†’ (π‘₯ ∈ dom (π»β€˜π‘›) ↔ π‘₯ ∈ dom (π»β€˜π‘š)))
2823, 24, 27eliind 44348 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (π»β€˜π‘š))
2921, 28sseldd 3979 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (πΉβ€˜π‘š))
3018, 29ffvelcdmd 7089 . . . . 5 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ)
3130rexrd 11280 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 45018 . . 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 2767 . 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 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
36 simpr 484 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
3735, 36sseldd 3979 . . . . . . . . . . . 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 6904 . . . . . . . . . . . . . . 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 7012 . . . . . . . . . . . 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 590 . . . . . . . . . . 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 5902 . . . . . . . . . 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 13138 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9472 . . . . . . . . . . . 12 sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ V
47 eqid 2727 . . . . . . . . . . . 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 6693 . . . . . . . . . . 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 2767 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π»β€˜π‘›) = (πΈβ€˜π‘›))
5134, 50iineq2d 5014 . . . . . . . 8 (πœ‘ β†’ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›) = ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5222, 51eleqtrd 2830 . . . . . . 7 (πœ‘ β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5352adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
54 eliinid 44390 . . . . . 6 ((π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ (πΈβ€˜π‘›))
5553, 36, 54syl2anc 583 . . . . 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 7012 . . . . 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 686 . . . 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 2727 . . . . . . . . . 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 44698 . . . . . . . . . . . . 13 (𝑛 ∈ 𝑍 β†’ 𝑛 ∈ β„€)
61 eqid 2727 . . . . . . . . . . . . 13 (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘›)
6260, 61uzn0d 44720 . . . . . . . . . . . 12 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
63 fvex 6904 . . . . . . . . . . . . . . 15 (πΉβ€˜π‘š) ∈ V
6463dmex 7909 . . . . . . . . . . . . . 14 dom (πΉβ€˜π‘š) ∈ V
6564rgenw 3060 . . . . . . . . . . . . 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 44412 . . . . . . . . . . 11 (𝑛 ∈ 𝑍 β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6867adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6959, 68rabexd 5329 . . . . . . . . 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 590 . . . . . . . 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 7010 . . . . . . . 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 583 . . . . . . 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 2830 . . . . . 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 3447 . . . . . 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 495 . . . 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 2828 . . 3 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ ℝ)
7834, 58mpteq2da 5240 . . . 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 1910 . . . . 5 β„²π‘˜πœ‘
80 fveq2 6891 . . . . . . . 8 (𝑛 = π‘˜ β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘˜))
8180mpteq1d 5237 . . . . . . 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 5934 . . . . . 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 9455 . . . . 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 1910 . . . . . . . 8 β„²π‘š(𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1))
85 eluzelz 12848 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑛 ∈ β„€)
8685adantr 480 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ β„€)
87 simpr 484 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ = (𝑛 + 1))
8886peano2zd 12685 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) ∈ β„€)
8987, 88eqeltrd 2828 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ β„€)
9086zred 12682 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ ℝ)
9189zred 12682 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ ℝ)
9290ltp1d 12160 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < (𝑛 + 1))
9387eqcomd 2733 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) = π‘˜)
9492, 93breqtrd 5168 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < π‘˜)
9590, 91, 94ltled 11378 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ≀ π‘˜)
9661, 86, 89, 95eluzd 44704 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘›))
97 uzss 12861 . . . . . . . . 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 6906 . . . . . . . 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 44546 . . . . . . 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 1128 . . . . . 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 1910 . . . . . . . . 9 β„²π‘š(πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘))
103 eqid 2727 . . . . . . . . 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 766 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑛 ∈ 𝑍) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
10537, 104syldanl 601 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
1066uztrn2 12857 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
107106adantll 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
108105, 107, 30syl2anc 583 . . . . . . . . 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 44482 . . . . . . . 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 11274 . . . . . . . . 9 ℝ βŠ† ℝ*
111110a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ℝ βŠ† ℝ*)
112109, 111sstrd 3988 . . . . . . 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 1130 . . . . . 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 13329 . . . . . 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 583 . . . . 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 6905 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝑍 ∈ V)
119 fvexd 6906 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
120 fvexd 6906 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) ∈ V)
12134, 36ssdf 44354 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘))
122 fvexd 6906 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
123 eqidd 2728 . . . . . . . 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 44969 . . . . . . 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 2829 . . . . 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 45015 . . . 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 5164 . . 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 44985 . 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 2828 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 395   ∧ w3a 1085   = wceq 1534  β„²wnf 1778   ∈ wcel 2099  βˆ€wral 3056  {crab 3427  Vcvv 3469   βŠ† wss 3944  βˆ© ciin 4992   class class class wbr 5142   ↦ cmpt 5225  dom cdm 5672  ran crn 5673  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414  supcsup 9449  infcinf 9450  β„cr 11123  1c1 11125   + caddc 11127  β„*cxr 11263   < clt 11264   ≀ cle 11265  β„€cz 12574  β„€β‰₯cuz 12838  lim supclsp 15432   ⇝ cli 15446  SAlgcsalg 45609  SMblFncsmblfn 45996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-q 12949  df-rp 12993  df-ioo 13346  df-ico 13348  df-fz 13503  df-fl 13775  df-seq 13985  df-exp 14045  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-limsup 15433  df-clim 15450  df-rlim 15451  df-smblfn 45997
This theorem is referenced by:  smflimsuplem7  46127
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