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Theorem smflimsuplem4 46381
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 1909 . . . 4 𝑚𝜑
2 smflimsuplem4.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 smflimsuplem4.z . . . . 5 𝑍 = (ℤ𝑀)
4 smflimsuplem4.n . . . . 5 (𝜑𝑁𝑍)
53, 4eluzelz2d 44965 . . . 4 (𝜑𝑁 ∈ ℤ)
6 eqid 2725 . . . 4 (ℤ𝑁) = (ℤ𝑁)
7 fvexd 6915 . . . 4 ((𝜑𝑚𝑍) → ((𝐹𝑚)‘𝑥) ∈ V)
8 fvexd 6915 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 45287 . . 3 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))))
10 smflimsuplem4.s . . . . . . . 8 (𝜑𝑆 ∈ SAlg)
1110adantr 479 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑆 ∈ SAlg)
123, 4uzssd2 44969 . . . . . . . . 9 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
1312sselda 3978 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
14 smflimsuplem4.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1514ffvelcdmda 7097 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
1613, 15syldan 589 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2725 . . . . . . 7 dom (𝐹𝑚) = dom (𝐹𝑚)
1811, 16, 17smff 46290 . . . . . 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 46378 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → dom (𝐻𝑚) ⊆ dom (𝐹𝑚))
22 smflimsuplem4.i . . . . . . . . 9 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
2322adantr 479 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
24 simpr 483 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁))
25 fveq2 6900 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐻𝑛) = (𝐻𝑚))
2625dmeqd 5911 . . . . . . . . 9 (𝑛 = 𝑚 → dom (𝐻𝑛) = dom (𝐻𝑚))
2726eleq2d 2811 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻𝑛) ↔ 𝑥 ∈ dom (𝐻𝑚)))
2823, 24, 27eliind 44609 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐻𝑚))
2921, 28sseldd 3979 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐹𝑚))
3018, 29ffvelcdmd 7098 . . . . 5 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
3130rexrd 11310 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 45275 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
339, 32eqtrd 2765 . 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 479 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → (ℤ𝑁) ⊆ 𝑍)
36 simpr 483 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛 ∈ (ℤ𝑁))
3735, 36sseldd 3979 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
3820a1i 11 . . . . . . . . . . . . 13 (𝜑𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))))
39 fvex 6913 . . . . . . . . . . . . . . 15 (𝐸𝑛) ∈ V
4039mptex 7239 . . . . . . . . . . . . . 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 7021 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4337, 42syldan 589 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4443dmeqd 5911 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
45 xrltso 13169 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9502 . . . . . . . . . . . 12 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V
47 eqid 2725 . . . . . . . . . . . 12 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
4846, 47dmmpti 6704 . . . . . . . . . . 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 2765 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = (𝐸𝑛))
5134, 50iineq2d 5023 . . . . . . . 8 (𝜑 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛) = 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5222, 51eleqtrd 2827 . . . . . . 7 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5352adantr 479 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
54 eliinid 44649 . . . . . 6 ((𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛) ∧ 𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5553, 36, 54syl2anc 582 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5646a1i 11 . . . . . 6 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V)
5743, 56fvmpt2d 7021 . . . . 5 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
5855, 57mpdan 685 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
59 eqid 2725 . . . . . . . . . 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 44955 . . . . . . . . . . . . 13 (𝑛𝑍𝑛 ∈ ℤ)
61 eqid 2725 . . . . . . . . . . . . 13 (ℤ𝑛) = (ℤ𝑛)
6260, 61uzn0d 44977 . . . . . . . . . . . 12 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
63 fvex 6913 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
6463dmex 7921 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
6564rgenw 3054 . . . . . . . . . . . . 13 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
6665a1i 11 . . . . . . . . . . . 12 (𝑛𝑍 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6762, 66iinexd 44671 . . . . . . . . . . 11 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6867adantl 480 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6959, 68rabexd 5339 . . . . . . . . 9 ((𝜑𝑛𝑍) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7037, 69syldan 589 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7119fvmpt2 7019 . . . . . . . 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 582 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
7355, 72eleqtrd 2827 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
74 rabid 3439 . . . . . 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 217 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
7675simprd 494 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ)
7758, 76eqeltrd 2825 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ ℝ)
7834, 58mpteq2da 5250 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
79 nfv 1909 . . . . 5 𝑘𝜑
80 fveq2 6900 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
8180mpteq1d 5247 . . . . . . 7 (𝑛 = 𝑘 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8281rneqd 5943 . . . . . 6 (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8382supeq1d 9485 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
84 nfv 1909 . . . . . . . 8 𝑚(𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1))
85 eluzelz 12879 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ ℤ)
8685adantr 479 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87 simpr 483 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
8886peano2zd 12716 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
8987, 88eqeltrd 2825 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
9086zred 12713 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
9189zred 12713 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
9290ltp1d 12191 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
9387eqcomd 2731 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
9492, 93breqtrd 5178 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
9590, 91, 94ltled 11408 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛𝑘)
9661, 86, 89, 95eluzd 44961 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ𝑛))
97 uzss 12892 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑛) → (ℤ𝑘) ⊆ (ℤ𝑛))
9896, 97syl 17 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (ℤ𝑘) ⊆ (ℤ𝑛))
99 fvexd 6915 . . . . . . . 8 (((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ𝑘)) → ((𝐹𝑚)‘𝑥) ∈ V)
10084, 98, 99rnmptss2 44803 . . . . . . 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 1909 . . . . . . . . 9 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
103 eqid 2725 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥))
104 simpll 765 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
10537, 104syldanl 600 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
1066uztrn2 12888 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
107106adantll 712 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
108105, 107, 30syl2anc 582 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
109102, 103, 108rnmptssd 44740 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ)
110 ressxr 11304 . . . . . . . . 9 ℝ ⊆ ℝ*
111110a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ℝ ⊆ ℝ*)
112109, 111sstrd 3989 . . . . . . 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 13360 . . . . . 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 582 . . . . 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 6914 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (𝜑𝑍 ∈ V)
119 fvexd 6915 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐻𝑛)‘𝑥) ∈ V)
120 fvexd 6915 . . . . . . . 8 (𝜑 → (ℤ𝑁) ∈ V)
12134, 36ssdf 44613 . . . . . . . 8 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑁))
122 fvexd 6915 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ V)
123 eqidd 2726 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = ((𝐻𝑛)‘𝑥))
12434, 5, 6, 118, 12, 119, 120, 121, 122, 123climeldmeqmpt 45226 . . . . . . 7 (𝜑 → ((𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ))
125116, 124mpbid 231 . . . . . 6 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
12678, 125eqeltrrd 2826 . . . . 5 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ dom ⇝ )
12734, 79, 5, 6, 76, 83, 115, 126climinf2mpt 45272 . . . 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 5174 . . 3 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ⇝ inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
12934, 5, 6, 77, 128climreclmpt 45242 . 2 (𝜑 → inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ) ∈ ℝ)
13033, 129eqeltrd 2825 1 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wnf 1777  wcel 2098  wral 3050  {crab 3418  Vcvv 3461  wss 3946   ciin 5001   class class class wbr 5152  cmpt 5235  dom cdm 5681  ran crn 5682  wf 6549  cfv 6553  (class class class)co 7423  supcsup 9479  infcinf 9480  cr 11153  1c1 11155   + caddc 11157  *cxr 11293   < clt 11294  cle 11295  cz 12605  cuz 12869  lim supclsp 15467  cli 15481  SAlgcsalg 45866  SMblFncsmblfn 46253
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 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745  ax-cnex 11210  ax-resscn 11211  ax-1cn 11212  ax-icn 11213  ax-addcl 11214  ax-addrcl 11215  ax-mulcl 11216  ax-mulrcl 11217  ax-mulcom 11218  ax-addass 11219  ax-mulass 11220  ax-distr 11221  ax-i2m1 11222  ax-1ne0 11223  ax-1rid 11224  ax-rnegex 11225  ax-rrecex 11226  ax-cnre 11227  ax-pre-lttri 11228  ax-pre-lttrn 11229  ax-pre-ltadd 11230  ax-pre-mulgt0 11231  ax-pre-sup 11232
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 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-er 8733  df-pm 8857  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9481  df-inf 9482  df-pnf 11296  df-mnf 11297  df-xr 11298  df-ltxr 11299  df-le 11300  df-sub 11492  df-neg 11493  df-div 11918  df-nn 12260  df-2 12322  df-3 12323  df-n0 12520  df-z 12606  df-uz 12870  df-q 12980  df-rp 13024  df-ioo 13377  df-ico 13379  df-fz 13534  df-fl 13807  df-seq 14017  df-exp 14077  df-cj 15099  df-re 15100  df-im 15101  df-sqrt 15235  df-abs 15236  df-limsup 15468  df-clim 15485  df-rlim 15486  df-smblfn 46254
This theorem is referenced by:  smflimsuplem7  46384
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