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Theorem smflimsuplem4 46838
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 1914 . . . 4 𝑚𝜑
2 smflimsuplem4.m . . . 4 (𝜑𝑀 ∈ ℤ)
3 smflimsuplem4.z . . . . 5 𝑍 = (ℤ𝑀)
4 smflimsuplem4.n . . . . 5 (𝜑𝑁𝑍)
53, 4eluzelz2d 45424 . . . 4 (𝜑𝑁 ∈ ℤ)
6 eqid 2737 . . . 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 45744 . . 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 45428 . . . . . . . . 9 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
1312sselda 3983 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
14 smflimsuplem4.f . . . . . . . . 9 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1514ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
1613, 15syldan 591 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2737 . . . . . . 7 dom (𝐹𝑚) = dom (𝐹𝑚)
1811, 16, 17smff 46747 . . . . . 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 46835 . . . . . . 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 5916 . . . . . . . . 9 (𝑛 = 𝑚 → dom (𝐻𝑛) = dom (𝐻𝑚))
2726eleq2d 2827 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻𝑛) ↔ 𝑥 ∈ dom (𝐻𝑚)))
2823, 24, 27eliind 45076 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐻𝑚))
2921, 28sseldd 3984 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑥 ∈ dom (𝐹𝑚))
3018, 29ffvelcdmd 7105 . . . . 5 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ)
3130rexrd 11311 . . . 4 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑥) ∈ ℝ*)
321, 5, 6, 31limsupvaluzmpt 45732 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
339, 32eqtrd 2777 . 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 3984 . . . . . . . . . . . 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 7243 . . . . . . . . . . . . . 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 7029 . . . . . . . . . . . 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 5916 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
45 xrltso 13183 . . . . . . . . . . . . 13 < Or ℝ*
4645supex 9503 . . . . . . . . . . . 12 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ V
47 eqid 2737 . . . . . . . . . . . 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 2777 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → dom (𝐻𝑛) = (𝐸𝑛))
5134, 50iineq2d 5015 . . . . . . . 8 (𝜑 𝑛 ∈ (ℤ𝑁)dom (𝐻𝑛) = 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5222, 51eleqtrd 2843 . . . . . . 7 (𝜑𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
5352adantr 480 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 𝑛 ∈ (ℤ𝑁)(𝐸𝑛))
54 eliinid 45116 . . . . . 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 7029 . . . . 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 2737 . . . . . . . . . 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 45414 . . . . . . . . . . . . 13 (𝑛𝑍𝑛 ∈ ℤ)
61 eqid 2737 . . . . . . . . . . . . 13 (ℤ𝑛) = (ℤ𝑛)
6260, 61uzn0d 45436 . . . . . . . . . . . 12 (𝑛𝑍 → (ℤ𝑛) ≠ ∅)
63 fvex 6919 . . . . . . . . . . . . . . 15 (𝐹𝑚) ∈ V
6463dmex 7931 . . . . . . . . . . . . . 14 dom (𝐹𝑚) ∈ V
6564rgenw 3065 . . . . . . . . . . . . 13 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
6665a1i 11 . . . . . . . . . . . 12 (𝑛𝑍 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6762, 66iinexd 45138 . . . . . . . . . . 11 (𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6867adantl 481 . . . . . . . . . 10 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
6959, 68rabexd 5340 . . . . . . . . 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 7027 . . . . . . . 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 2843 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑥 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
74 rabid 3458 . . . . . 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 2841 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ ℝ)
7834, 58mpteq2da 5240 . . . 4 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
79 nfv 1914 . . . . 5 𝑘𝜑
80 fveq2 6906 . . . . . . . 8 (𝑛 = 𝑘 → (ℤ𝑛) = (ℤ𝑘))
8180mpteq1d 5237 . . . . . . 7 (𝑛 = 𝑘 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8281rneqd 5949 . . . . . 6 (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)))
8382supeq1d 9486 . . . . 5 (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
84 nfv 1914 . . . . . . . 8 𝑚(𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1))
85 eluzelz 12888 . . . . . . . . . . 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 12725 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
8987, 88eqeltrd 2841 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
9086zred 12722 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
9189zred 12722 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
9290ltp1d 12198 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
9387eqcomd 2743 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
9492, 93breqtrd 5169 . . . . . . . . . . 11 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
9590, 91, 94ltled 11409 . . . . . . . . . 10 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛𝑘)
9661, 86, 89, 95eluzd 45420 . . . . . . . . 9 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ𝑛))
97 uzss 12901 . . . . . . . . 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 45264 . . . . . . 7 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
1011003adant1 1131 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑘) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
102 nfv 1914 . . . . . . . . 9 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
103 eqid 2737 . . . . . . . . 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 12897 . . . . . . . . . . 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 45201 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ)
110 ressxr 11305 . . . . . . . . 9 ℝ ⊆ ℝ*
111110a1i 11 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ℝ ⊆ ℝ*)
112109, 111sstrd 3994 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
1131123adant3 1133 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) ⊆ ℝ*)
114 supxrss 13374 . . . . . 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 45080 . . . . . . . 8 (𝜑 → (ℤ𝑁) ⊆ (ℤ𝑁))
122 fvexd 6921 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) ∈ V)
123 eqidd 2738 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑥) = ((𝐻𝑛)‘𝑥))
12434, 5, 6, 118, 12, 119, 120, 121, 122, 123climeldmeqmpt 45683 . . . . . . 7 (𝜑 → ((𝑛𝑍 ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ ))
125116, 124mpbid 232 . . . . . 6 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ∈ dom ⇝ )
12678, 125eqeltrrd 2842 . . . . 5 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ dom ⇝ )
12734, 79, 5, 6, 76, 83, 115, 126climinf2mpt 45729 . . . 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 5165 . . 3 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑥)) ⇝ inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ))
12934, 5, 6, 77, 128climreclmpt 45699 . 2 (𝜑 → inf(ran (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )), ℝ*, < ) ∈ ℝ)
13033, 129eqeltrd 2841 1 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wnf 1783  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  wss 3951   ciin 4992   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  supcsup 9480  infcinf 9481  cr 11154  1c1 11156   + caddc 11158  *cxr 11294   < clt 11295  cle 11296  cz 12613  cuz 12878  lim supclsp 15506  cli 15520  SAlgcsalg 46323  SMblFncsmblfn 46710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-smblfn 46711
This theorem is referenced by:  smflimsuplem7  46841
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