Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimsuplem4 Structured version   Visualization version   GIF version

Theorem smflimsuplem4 45154
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 1918 . . . 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 43738 . . . 4 (πœ‘ β†’ 𝑁 ∈ β„€)
6 eqid 2733 . . . 4 (β„€β‰₯β€˜π‘) = (β„€β‰₯β€˜π‘)
7 fvexd 6861 . . . 4 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
8 fvexd 6861 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 44060 . . 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 482 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑆 ∈ SAlg)
123, 4uzssd2 43742 . . . . . . . . 9 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
1312sselda 3948 . . . . . . . 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 7039 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ 𝑍) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
1613, 15syldan 592 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ (πΉβ€˜π‘š) ∈ (SMblFnβ€˜π‘†))
17 eqid 2733 . . . . . . 7 dom (πΉβ€˜π‘š) = dom (πΉβ€˜π‘š)
1811, 16, 17smff 45063 . . . . . 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 45151 . . . . . . 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 482 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›))
24 simpr 486 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘š ∈ (β„€β‰₯β€˜π‘))
25 fveq2 6846 . . . . . . . . . 10 (𝑛 = π‘š β†’ (π»β€˜π‘›) = (π»β€˜π‘š))
2625dmeqd 5865 . . . . . . . . 9 (𝑛 = π‘š β†’ dom (π»β€˜π‘›) = dom (π»β€˜π‘š))
2726eleq2d 2820 . . . . . . . 8 (𝑛 = π‘š β†’ (π‘₯ ∈ dom (π»β€˜π‘›) ↔ π‘₯ ∈ dom (π»β€˜π‘š)))
2823, 24, 27eliind 43371 . . . . . . 7 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (π»β€˜π‘š))
2921, 28sseldd 3949 . . . . . 6 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ dom (πΉβ€˜π‘š))
3018, 29ffvelcdmd 7040 . . . . 5 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ)
3130rexrd 11213 . . . 4 ((πœ‘ ∧ π‘š ∈ (β„€β‰₯β€˜π‘)) β†’ ((πΉβ€˜π‘š)β€˜π‘₯) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 44048 . . 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 2773 . 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 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ (β„€β‰₯β€˜π‘) βŠ† 𝑍)
36 simpr 486 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ 𝑛 ∈ (β„€β‰₯β€˜π‘))
3735, 36sseldd 3949 . . . . . . . . . . . 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 6859 . . . . . . . . . . . . . . 15 (πΈβ€˜π‘›) ∈ V
4039mptex 7177 . . . . . . . . . . . . . 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 6965 . . . . . . . . . . . 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 592 . . . . . . . . . . 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 5865 . . . . . . . . . 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 13069 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9407 . . . . . . . . . . . 12 sup(ran (π‘š ∈ (β„€β‰₯β€˜π‘›) ↦ ((πΉβ€˜π‘š)β€˜π‘₯)), ℝ*, < ) ∈ V
47 eqid 2733 . . . . . . . . . . . 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 6649 . . . . . . . . . . 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 2773 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ dom (π»β€˜π‘›) = (πΈβ€˜π‘›))
5134, 50iineq2d 4981 . . . . . . . 8 (πœ‘ β†’ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)dom (π»β€˜π‘›) = ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5222, 51eleqtrd 2836 . . . . . . 7 (πœ‘ β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
5352adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›))
54 eliinid 43413 . . . . . 6 ((π‘₯ ∈ ∩ 𝑛 ∈ (β„€β‰₯β€˜π‘)(πΈβ€˜π‘›) ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ π‘₯ ∈ (πΈβ€˜π‘›))
5553, 36, 54syl2anc 585 . . . . 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 6965 . . . . 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 2733 . . . . . . . . . 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 43728 . . . . . . . . . . . . 13 (𝑛 ∈ 𝑍 β†’ 𝑛 ∈ β„€)
61 eqid 2733 . . . . . . . . . . . . 13 (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘›)
6260, 61uzn0d 43750 . . . . . . . . . . . 12 (𝑛 ∈ 𝑍 β†’ (β„€β‰₯β€˜π‘›) β‰  βˆ…)
63 fvex 6859 . . . . . . . . . . . . . . 15 (πΉβ€˜π‘š) ∈ V
6463dmex 7852 . . . . . . . . . . . . . 14 dom (πΉβ€˜π‘š) ∈ V
6564rgenw 3065 . . . . . . . . . . . . 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 43435 . . . . . . . . . . 11 (𝑛 ∈ 𝑍 β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6867adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ∩ π‘š ∈ (β„€β‰₯β€˜π‘›)dom (πΉβ€˜π‘š) ∈ V)
6959, 68rabexd 5294 . . . . . . . . 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 592 . . . . . . . 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 6963 . . . . . . . 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 585 . . . . . . 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 2836 . . . . . 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 3426 . . . . . 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 497 . . . 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 2834 . . 3 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ ℝ)
7834, 58mpteq2da 5207 . . . 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 1918 . . . . 5 β„²π‘˜πœ‘
80 fveq2 6846 . . . . . . . 8 (𝑛 = π‘˜ β†’ (β„€β‰₯β€˜π‘›) = (β„€β‰₯β€˜π‘˜))
8180mpteq1d 5204 . . . . . . 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 5897 . . . . . 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 9390 . . . . 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 1918 . . . . . . . 8 β„²π‘š(𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1))
85 eluzelz 12781 . . . . . . . . . . 11 (𝑛 ∈ (β„€β‰₯β€˜π‘) β†’ 𝑛 ∈ β„€)
8685adantr 482 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ β„€)
87 simpr 486 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ = (𝑛 + 1))
8886peano2zd 12618 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) ∈ β„€)
8987, 88eqeltrd 2834 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ β„€)
9086zred 12615 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ∈ ℝ)
9189zred 12615 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ ℝ)
9290ltp1d 12093 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < (𝑛 + 1))
9387eqcomd 2739 . . . . . . . . . . . 12 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ (𝑛 + 1) = π‘˜)
9492, 93breqtrd 5135 . . . . . . . . . . 11 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 < π‘˜)
9590, 91, 94ltled 11311 . . . . . . . . . 10 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ 𝑛 ≀ π‘˜)
9661, 86, 89, 95eluzd 43734 . . . . . . . . 9 ((𝑛 ∈ (β„€β‰₯β€˜π‘) ∧ π‘˜ = (𝑛 + 1)) β†’ π‘˜ ∈ (β„€β‰₯β€˜π‘›))
97 uzss 12794 . . . . . . . . 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 6861 . . . . . . . 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 43576 . . . . . . 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 1131 . . . . . 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 1918 . . . . . . . . 9 β„²π‘š(πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘))
103 eqid 2733 . . . . . . . . 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 603 . . . . . . . . . 10 (((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) ∧ π‘š ∈ (β„€β‰₯β€˜π‘›)) β†’ πœ‘)
1066uztrn2 12790 . . . . . . . . . . 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 585 . . . . . . . . 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 43508 . . . . . . . 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 11207 . . . . . . . . 9 ℝ βŠ† ℝ*
111110a1i 11 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ℝ βŠ† ℝ*)
112109, 111sstrd 3958 . . . . . . 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 1133 . . . . . 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 13260 . . . . . 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 585 . . . . 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 6860 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝑍 ∈ V)
119 fvexd 6861 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ 𝑍) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
120 fvexd 6861 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) ∈ V)
12134, 36ssdf 43377 . . . . . . . 8 (πœ‘ β†’ (β„€β‰₯β€˜π‘) βŠ† (β„€β‰₯β€˜π‘))
122 fvexd 6861 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ (β„€β‰₯β€˜π‘)) β†’ ((π»β€˜π‘›)β€˜π‘₯) ∈ V)
123 eqidd 2734 . . . . . . . 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 43999 . . . . . . 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 2835 . . . . 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 44045 . . . 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 5131 . . 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 44015 . 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 2834 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 397   ∧ w3a 1088   = wceq 1542  β„²wnf 1786   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆ© ciin 4959   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5637  ran crn 5638  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  supcsup 9384  infcinf 9385  β„cr 11058  1c1 11060   + caddc 11062  β„*cxr 11196   < clt 11197   ≀ cle 11198  β„€cz 12507  β„€β‰₯cuz 12771  lim supclsp 15361   ⇝ cli 15375  SAlgcsalg 44639  SMblFncsmblfn 45026
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-ioo 13277  df-ico 13279  df-fz 13434  df-fl 13706  df-seq 13916  df-exp 13977  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-limsup 15362  df-clim 15379  df-rlim 15380  df-smblfn 45027
This theorem is referenced by:  smflimsuplem7  45157
  Copyright terms: Public domain W3C validator