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

Theorem smflimsuplem4 43441
 Description: If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem4.1 𝑛𝜑
smflimsuplem4.m (𝜑𝑀 ∈ ℤ)
smflimsuplem4.z 𝑍 = (ℤ𝑀)
smflimsuplem4.s (𝜑𝑆 ∈ SAlg)
smflimsuplem4.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem4.e 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem4.h 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem4.n (𝜑𝑁𝑍)
smflimsuplem4.i (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
smflimsuplem4.c (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
Assertion
Ref Expression
smflimsuplem4 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)
Distinct variable groups:   𝑛,𝐸,𝑥   𝑚,𝐹,𝑛,𝑥   𝑛,𝐻   𝑚,𝑀   𝑚,𝑁,𝑛   𝑚,𝑍,𝑛   𝜑,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐸(𝑚)   𝐻(𝑥,𝑚)   𝑀(𝑥,𝑛)   𝑁(𝑥)   𝑍(𝑥)

Proof of Theorem smflimsuplem4
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 nfv 1915 . . . 4 𝑚𝜑
2 smflimsuplem4.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 smflimsuplem4.z . . . . 5 𝑍 = (ℤ𝑀)
4 smflimsuplem4.n . . . . 5 (𝜑𝑁𝑍)
53, 4eluzelz2d 42037 . . . 4 (𝜑𝑁 ∈ ℤ)
6 eqid 2801 . . . 4 (ℤ𝑁) = (ℤ𝑁)
7 fvexd 6664 . . . 4 ((𝜑𝑚𝑍) → ((𝐹𝑚)‘𝑥) ∈ V)
8 fvexd 6664 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 42358 . . 3 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))))
10 smflimsuplem4.s . . . . . . . 8 (𝜑𝑆 ∈ SAlg)
1110adantr 484 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑆 ∈ SAlg)
123, 4uzssd2 42041 . . . . . . . . 9 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
1312sselda 3918 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
14 smflimsuplem4.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1514ffvelrnda 6832 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
1613, 15syldan 594 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2801 . . . . . . 7 dom (𝐹𝑚) = dom (𝐹𝑚)
1811, 16, 17smff 43353 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
19 smflimsuplem4.e . . . . . . . 8 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
20 smflimsuplem4.h . . . . . . . 8 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
213, 19, 20, 13smflimsuplem1 43438 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → dom (𝐻𝑚) ⊆ dom (𝐹𝑚))
22 smflimsuplem4.i . . . . . . . . 9 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
2322adantr 484 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
24 simpr 488 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁))
25 fveq2 6649 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐻𝑛) = (𝐻𝑚))
2625dmeqd 5742 . . . . . . . . 9 (𝑛 = 𝑚 → dom (𝐻𝑛) = dom (𝐻𝑚))
2726eleq2d 2878 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻𝑛) ↔ 𝑥 ∈ dom (𝐻𝑚)))
2823, 24, 27eliind 41692 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐻𝑚))
2921, 28sseldd 3919 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐹𝑚))
3018, 29ffvelrnd 6833 . . . . 5 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
3130rexrd 10684 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 42346 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
339, 32eqtrd 2836 . 2 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
34 smflimsuplem4.1 . . 3 𝑛𝜑
3512adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → (ℤ𝑁) ⊆ 𝑍)
36 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛 ∈ (ℤ𝑁))
3735, 36sseldd 3919 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
3820a1i 11 . . . . . . . . . . . . 13 (𝜑𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))))
39 fvex 6662 . . . . . . . . . . . . . . 15 (𝐸𝑛) ∈ V
4039mptex 6967 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V
4140a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑛𝑍) → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V)
4238, 41fvmpt2d 6762 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4337, 42syldan 594 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4443dmeqd 5742 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
45 xrltso 12526 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 8915 . . . . . . . . . . . 12 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V
47 eqid 2801 . . . . . . . . . . . 12 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
4846, 47dmmpti 6468 . . . . . . . . . . 11 dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝐸𝑛)
4948a1i 11 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝐸𝑛))
5044, 49eqtrd 2836 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = (𝐸𝑛))
5134, 50iineq2d 4907 . . . . . . . 8 (𝜑 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛) = 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5222, 51eleqtrd 2895 . . . . . . 7 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5352adantr 484 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
54 eliinid 41734 . . . . . 6 ((𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛) ∧ 𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5553, 36, 54syl2anc 587 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5646a1i 11 . . . . . 6 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V)
5743, 56fvmpt2d 6762 . . . . 5 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
5855, 57mpdan 686 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
59 eqid 2801 . . . . . . . . . 10 {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
603eluzelz2 42027 . . . . . . . . . . . . 13 (𝑛𝑍𝑛 ∈ ℤ)
61 eqid 2801 . . . . . . . . . . . . 13 (ℤ𝑛) = (ℤ𝑛)
6260, 61uzn0d 42049 . . . . . . . . . . . 12 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
63 fvex 6662 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
6463dmex 7602 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
6564rgenw 3121 . . . . . . . . . . . . 13 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
6665a1i 11 . . . . . . . . . . . 12 (𝑛𝑍 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6762, 66iinexd 41756 . . . . . . . . . . 11 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6867adantl 485 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6959, 68rabexd 5203 . . . . . . . . 9 ((𝜑𝑛𝑍) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7037, 69syldan 594 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7119fvmpt2 6760 . . . . . . . 8 ((𝑛𝑍 ∧ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
7237, 70, 71syl2anc 587 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
7355, 72eleqtrd 2895 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
74 rabid 3334 . . . . . 6 (𝑥 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↔ (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
7573, 74sylib 221 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
7675simprd 499 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ)
7758, 76eqeltrd 2893 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ ℝ)
7834, 58mpteq2da 5127 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
79 nfv 1915 . . . . 5 𝑘𝜑
80 fveq2 6649 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
8180mpteq1d 5122 . . . . . . 7 (𝑛 = 𝑘 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8281rneqd 5776 . . . . . 6 (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8382supeq1d 8898 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
84 nfv 1915 . . . . . . . 8 𝑚(𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1))
85 eluzelz 12245 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ ℤ)
8685adantr 484 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87 simpr 488 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
8886peano2zd 12082 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
8987, 88eqeltrd 2893 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
9086zred 12079 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
9189zred 12079 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
9290ltp1d 11563 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
9387eqcomd 2807 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
9492, 93breqtrd 5059 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
9590, 91, 94ltled 10781 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛𝑘)
9661, 86, 89, 95eluzd 42033 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ𝑛))
97 uzss 12257 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑛) → (ℤ𝑘) ⊆ (ℤ𝑛))
9896, 97syl 17 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (ℤ𝑘) ⊆ (ℤ𝑛))
99 fvexd 6664 . . . . . . . 8 (((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ𝑘)) → ((𝐹𝑚)‘𝑥) ∈ V)
10084, 98, 99rnmptss2 41882 . . . . . . 7 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
1011003adant1 1127 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
102 nfv 1915 . . . . . . . . 9 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
103 eqid 2801 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥))
104 simpll 766 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
10537, 104syldanl 604 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
1066uztrn2 12254 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
107106adantll 713 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
108105, 107, 30syl2anc 587 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
109102, 103, 108rnmptssd 41811 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ)
110 ressxr 10678 . . . . . . . . 9 ℝ ⊆ ℝ*
111110a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ℝ ⊆ ℝ*)
112109, 111sstrd 3928 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
1131123adant3 1129 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
114 supxrss 12717 . . . . . 6 ((ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ∧ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*) → sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ≤ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
115101, 113, 114syl2anc 587 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ≤ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
116 smflimsuplem4.c . . . . . . 7 (𝜑 → (𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
1173fvexi 6663 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (𝜑𝑍 ∈ V)
119 fvexd 6664 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐻𝑛)‘𝑥) ∈ V)
120 fvexd 6664 . . . . . . . 8 (𝜑 → (ℤ𝑁) ∈ V)
12134, 36ssdf 41698 . . . . . . . 8 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑁))
122 fvexd 6664 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ V)
123 eqidd 2802 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = ((𝐻𝑛)‘𝑥))
12434, 5, 6, 118, 12, 119, 120, 121, 122, 123climeldmeqmpt 42297 . . . . . . 7 (𝜑 → ((𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ))
125116, 124mpbid 235 . . . . . 6 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
12678, 125eqeltrrd 2894 . . . . 5 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ dom ⇝ )
12734, 79, 5, 6, 76, 83, 115, 126climinf2mpt 42343 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ⇝ inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
12878, 127eqbrtrd 5055 . . 3 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ⇝ inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
12934, 5, 6, 77, 128climreclmpt 42313 . 2 (𝜑 → inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ) ∈ ℝ)
13033, 129eqeltrd 2893 1 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2112  ∀wral 3109  {crab 3113  Vcvv 3444   ⊆ wss 3884  ∩ ciin 4885   class class class wbr 5033   ↦ cmpt 5113  dom cdm 5523  ran crn 5524  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  supcsup 8892  infcinf 8893  ℝcr 10529  1c1 10531   + caddc 10533  ℝ*cxr 10667   < clt 10668   ≤ cle 10669  ℤcz 11973  ℤ≥cuz 12235  lim supclsp 14822   ⇝ cli 14836  SAlgcsalg 42937  SMblFncsmblfn 43321 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-ioo 12734  df-ico 12736  df-fz 12890  df-fl 13161  df-seq 13369  df-exp 13430  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-limsup 14823  df-clim 14840  df-rlim 14841  df-smblfn 43322 This theorem is referenced by:  smflimsuplem7  43444
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