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Theorem smflimsuplem4 46778
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 1911 . . . 4 𝑚𝜑
2 smflimsuplem4.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 smflimsuplem4.z . . . . 5 𝑍 = (ℤ𝑀)
4 smflimsuplem4.n . . . . 5 (𝜑𝑁𝑍)
53, 4eluzelz2d 45362 . . . 4 (𝜑𝑁 ∈ ℤ)
6 eqid 2734 . . . 4 (ℤ𝑁) = (ℤ𝑁)
7 fvexd 6921 . . . 4 ((𝜑𝑚𝑍) → ((𝐹𝑚)‘𝑥) ∈ V)
8 fvexd 6921 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ V)
91, 2, 5, 3, 6, 7, 8limsupequzmpt 45684 . . 3 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))))
10 smflimsuplem4.s . . . . . . . 8 (𝜑𝑆 ∈ SAlg)
1110adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑆 ∈ SAlg)
123, 4uzssd2 45366 . . . . . . . . 9 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
1312sselda 3994 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
14 smflimsuplem4.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1514ffvelcdmda 7103 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
1613, 15syldan 591 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2734 . . . . . . 7 dom (𝐹𝑚) = dom (𝐹𝑚)
1811, 16, 17smff 46687 . . . . . 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 46775 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → dom (𝐻𝑚) ⊆ dom (𝐹𝑚))
22 smflimsuplem4.i . . . . . . . . 9 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
2322adantr 480 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛))
24 simpr 484 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁))
25 fveq2 6906 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐻𝑛) = (𝐻𝑚))
2625dmeqd 5918 . . . . . . . . 9 (𝑛 = 𝑚 → dom (𝐻𝑛) = dom (𝐻𝑚))
2726eleq2d 2824 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻𝑛) ↔ 𝑥 ∈ dom (𝐻𝑚)))
2823, 24, 27eliind 45010 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐻𝑚))
2921, 28sseldd 3995 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐹𝑚))
3018, 29ffvelcdmd 7104 . . . . 5 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
3130rexrd 11308 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 45672 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
339, 32eqtrd 2774 . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → (ℤ𝑁) ⊆ 𝑍)
36 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛 ∈ (ℤ𝑁))
3735, 36sseldd 3995 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
3820a1i 11 . . . . . . . . . . . . 13 (𝜑𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))))
39 fvex 6919 . . . . . . . . . . . . . . 15 (𝐸𝑛) ∈ V
4039mptex 7242 . . . . . . . . . . . . . 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 7028 . . . . . . . . . . . 12 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4337, 42syldan 591 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
4443dmeqd 5918 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
45 xrltso 13179 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9500 . . . . . . . . . . . 12 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V
47 eqid 2734 . . . . . . . . . . . 12 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
4846, 47dmmpti 6712 . . . . . . . . . . 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 2774 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = (𝐸𝑛))
5134, 50iineq2d 5019 . . . . . . . 8 (𝜑 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛) = 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5222, 51eleqtrd 2840 . . . . . . 7 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5352adantr 480 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
54 eliinid 45050 . . . . . 6 ((𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛) ∧ 𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5553, 36, 54syl2anc 584 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ (𝐸𝑛))
5646a1i 11 . . . . . 6 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V)
5743, 56fvmpt2d 7028 . . . . 5 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑥 ∈ (𝐸𝑛)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
5855, 57mpdan 687 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
59 eqid 2734 . . . . . . . . . 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 45352 . . . . . . . . . . . . 13 (𝑛𝑍𝑛 ∈ ℤ)
61 eqid 2734 . . . . . . . . . . . . 13 (ℤ𝑛) = (ℤ𝑛)
6260, 61uzn0d 45374 . . . . . . . . . . . 12 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
63 fvex 6919 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
6463dmex 7931 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
6564rgenw 3062 . . . . . . . . . . . . 13 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
6665a1i 11 . . . . . . . . . . . 12 (𝑛𝑍 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6762, 66iinexd 45072 . . . . . . . . . . 11 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6867adantl 481 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6959, 68rabexd 5345 . . . . . . . . 9 ((𝜑𝑛𝑍) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7037, 69syldan 591 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
7119fvmpt2 7026 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
7355, 72eleqtrd 2840 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
74 rabid 3454 . . . . . 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 218 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
7675simprd 495 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ)
7758, 76eqeltrd 2838 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ ℝ)
7834, 58mpteq2da 5245 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
79 nfv 1911 . . . . 5 𝑘𝜑
80 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
8180mpteq1d 5242 . . . . . . 7 (𝑛 = 𝑘 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8281rneqd 5951 . . . . . 6 (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8382supeq1d 9483 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
84 nfv 1911 . . . . . . . 8 𝑚(𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1))
85 eluzelz 12885 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ ℤ)
8685adantr 480 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87 simpr 484 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
8886peano2zd 12722 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
8987, 88eqeltrd 2838 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
9086zred 12719 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
9189zred 12719 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
9290ltp1d 12195 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
9387eqcomd 2740 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
9492, 93breqtrd 5173 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
9590, 91, 94ltled 11406 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛𝑘)
9661, 86, 89, 95eluzd 45358 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ𝑛))
97 uzss 12898 . . . . . . . . 9 (𝑘 ∈ (ℤ𝑛) → (ℤ𝑘) ⊆ (ℤ𝑛))
9896, 97syl 17 . . . . . . . 8 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (ℤ𝑘) ⊆ (ℤ𝑛))
99 fvexd 6921 . . . . . . . 8 (((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ𝑘)) → ((𝐹𝑚)‘𝑥) ∈ V)
10084, 98, 99rnmptss2 45201 . . . . . . 7 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
1011003adant1 1129 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
102 nfv 1911 . . . . . . . . 9 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
103 eqid 2734 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥))
104 simpll 767 . . . . . . . . . . 11 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
10537, 104syldanl 602 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
1066uztrn2 12894 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
107106adantll 714 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
108105, 107, 30syl2anc 584 . . . . . . . . 9 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
109102, 103, 108rnmptssd 45138 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ)
110 ressxr 11302 . . . . . . . . 9 ℝ ⊆ ℝ*
111110a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ℝ ⊆ ℝ*)
112109, 111sstrd 4005 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
1131123adant3 1131 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
114 supxrss 13370 . . . . . 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 584 . . . . 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 6920 . . . . . . . . 9 𝑍 ∈ V
118117a1i 11 . . . . . . . 8 (𝜑𝑍 ∈ V)
119 fvexd 6921 . . . . . . . 8 ((𝜑𝑛𝑍) → ((𝐻𝑛)‘𝑥) ∈ V)
120 fvexd 6921 . . . . . . . 8 (𝜑 → (ℤ𝑁) ∈ V)
12134, 36ssdf 45014 . . . . . . . 8 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑁))
122 fvexd 6921 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ V)
123 eqidd 2735 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = ((𝐻𝑛)‘𝑥))
12434, 5, 6, 118, 12, 119, 120, 121, 122, 123climeldmeqmpt 45623 . . . . . . 7 (𝜑 → ((𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ))
125116, 124mpbid 232 . . . . . 6 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
12678, 125eqeltrrd 2839 . . . . 5 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ dom ⇝ )
12734, 79, 5, 6, 76, 83, 115, 126climinf2mpt 45669 . . . 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 5169 . . 3 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ⇝ inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
12934, 5, 6, 77, 128climreclmpt 45639 . 2 (𝜑 → inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ) ∈ ℝ)
13033, 129eqeltrd 2838 1 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wnf 1779  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  wss 3962   ciin 4996   class class class wbr 5147  cmpt 5230  dom cdm 5688  ran crn 5689  wf 6558  cfv 6562  (class class class)co 7430  supcsup 9477  infcinf 9478  cr 11151  1c1 11153   + caddc 11155  *cxr 11291   < clt 11292  cle 11293  cz 12610  cuz 12875  lim supclsp 15502  cli 15516  SAlgcsalg 46263  SMblFncsmblfn 46650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-ioo 13387  df-ico 13389  df-fz 13544  df-fl 13828  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-smblfn 46651
This theorem is referenced by:  smflimsuplem7  46781
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