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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  wnf 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  cr 10529  1c1 10531   + caddc 10533  *cxr 10667   < clt 10668  cle 10669  cz 11973  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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