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Theorem smflimsuplem5 43381
Description: 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem5.a 𝑛𝜑
smflimsuplem5.b 𝑚𝜑
smflimsuplem5.m (𝜑𝑀 ∈ ℤ)
smflimsuplem5.z 𝑍 = (ℤ𝑀)
smflimsuplem5.s (𝜑𝑆 ∈ SAlg)
smflimsuplem5.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem5.e 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem5.h 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem5.r (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
smflimsuplem5.n (𝜑𝑁𝑍)
smflimsuplem5.x (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))
Assertion
Ref Expression
smflimsuplem5 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))
Distinct variable groups:   𝑛,𝐹,𝑥   𝑚,𝑀   𝑚,𝑁,𝑛   𝑚,𝑋,𝑛   𝑚,𝑍,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐸(𝑥,𝑚,𝑛)   𝐹(𝑚)   𝐻(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑛)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem smflimsuplem5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 smflimsuplem5.a . . 3 𝑛𝜑
2 smflimsuplem5.n . . . . . . . 8 (𝜑𝑁𝑍)
3 smflimsuplem5.z . . . . . . . . . . . 12 𝑍 = (ℤ𝑀)
43eleq2i 2907 . . . . . . . . . . 11 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
54biimpi 219 . . . . . . . . . 10 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
6 uzss 12262 . . . . . . . . . 10 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
75, 6syl 17 . . . . . . . . 9 (𝑁𝑍 → (ℤ𝑁) ⊆ (ℤ𝑀))
87, 3sseqtrrdi 4004 . . . . . . . 8 (𝑁𝑍 → (ℤ𝑁) ⊆ 𝑍)
92, 8syl 17 . . . . . . 7 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
109sselda 3953 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛𝑍)
11 smflimsuplem5.e . . . . . . . . . 10 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
12 nfcv 2982 . . . . . . . . . . 11 𝑥𝑍
13 nfrab1 3375 . . . . . . . . . . 11 𝑥{𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
1412, 13nfmpt 5149 . . . . . . . . . 10 𝑥(𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
1511, 14nfcxfr 2980 . . . . . . . . 9 𝑥𝐸
16 nfcv 2982 . . . . . . . . 9 𝑥𝑛
1715, 16nffv 6671 . . . . . . . 8 𝑥(𝐸𝑛)
18 fvex 6674 . . . . . . . 8 (𝐸𝑛) ∈ V
1917, 18mptexf 41798 . . . . . . 7 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V
2019a1i 11 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V)
21 smflimsuplem5.h . . . . . . 7 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
2221fvmpt2 6770 . . . . . 6 ((𝑛𝑍 ∧ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
2310, 20, 22syl2anc 587 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
2423fveq1d 6663 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑋) = ((𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))‘𝑋))
25 nfcv 2982 . . . . . 6 𝑦(𝐸𝑛)
26 nfcv 2982 . . . . . 6 𝑦sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )
27 nfcv 2982 . . . . . 6 𝑥sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < )
28 fveq2 6661 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
2928mpteq2dv 5148 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
3029rneqd 5795 . . . . . . 7 (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
3130supeq1d 8907 . . . . . 6 (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
3217, 25, 26, 27, 31cbvmptf 5151 . . . . 5 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
33 simpl 486 . . . . . . . . 9 ((𝑦 = 𝑋𝑚 ∈ (ℤ𝑛)) → 𝑦 = 𝑋)
3433fveq2d 6665 . . . . . . . 8 ((𝑦 = 𝑋𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
3534mpteq2dva 5147 . . . . . . 7 (𝑦 = 𝑋 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
3635rneqd 5795 . . . . . 6 (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
3736supeq1d 8907 . . . . 5 (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
3837eleq1d 2900 . . . . . . . 8 (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
39 uzss 12262 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (ℤ𝑛) ⊆ (ℤ𝑁))
40 iinss1 4920 . . . . . . . . . . 11 ((ℤ𝑛) ⊆ (ℤ𝑁) → 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ⊆ 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4139, 40syl 17 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑁) → 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ⊆ 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
4241adantl 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ⊆ 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
43 smflimsuplem5.x . . . . . . . . . 10 (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))
4443adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))
4542, 44sseldd 3954 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
46 smflimsuplem5.b . . . . . . . . . . 11 𝑚𝜑
47 nfv 1916 . . . . . . . . . . 11 𝑚 𝑛 ∈ (ℤ𝑁)
4846, 47nfan 1901 . . . . . . . . . 10 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
49 eqid 2824 . . . . . . . . . 10 (ℤ𝑛) = (ℤ𝑛)
50 simpll 766 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
5139sselda 3953 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
5251adantll 713 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
53 smflimsuplem5.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
5453adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑆 ∈ SAlg)
55 simpl 486 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝜑)
569sselda 3953 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
57 smflimsuplem5.f . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
5857ffvelrnda 6842 . . . . . . . . . . . . . 14 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
5955, 56, 58syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
60 eqid 2824 . . . . . . . . . . . . 13 dom (𝐹𝑚) = dom (𝐹𝑚)
6154, 59, 60smff 43292 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
62 eliin 4910 . . . . . . . . . . . . . . . 16 (𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) → (𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ↔ ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚)))
6343, 62syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ↔ ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚)))
6443, 63mpbid 235 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚))
6564adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚))
66 simpr 488 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁))
67 rspa 3201 . . . . . . . . . . . . 13 ((∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚) ∧ 𝑚 ∈ (ℤ𝑁)) → 𝑋 ∈ dom (𝐹𝑚))
6865, 66, 67syl2anc 587 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑋 ∈ dom (𝐹𝑚))
6961, 68ffvelrnd 6843 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
7050, 52, 69syl2anc 587 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
71 eluzelz 12250 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ ℤ)
7271adantl 485 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛 ∈ ℤ)
73 smflimsuplem5.m . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
7473adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑀 ∈ ℤ)
75 fvex 6674 . . . . . . . . . . . . . 14 ((𝐹𝑚)‘𝑋) ∈ V
7675a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚𝑍) → ((𝐹𝑚)‘𝑋) ∈ V)
7748, 72, 74, 49, 3, 70, 76limsupequzmpt 42297 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
78 smflimsuplem5.r . . . . . . . . . . . . 13 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
7978adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
8077, 79eqeltrd 2916 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
8180renepnfd 10690 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
8248, 49, 70, 81limsupubuzmpt 42287 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦)
83 uzid2 41968 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ (ℤ𝑛))
8483ne0d 4284 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (ℤ𝑛) ≠ ∅)
8584adantl 485 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → (ℤ𝑛) ≠ ∅)
8648, 85, 70supxrre3rnmpt 41992 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦))
8782, 86mpbird 260 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
8838, 45, 87elrabd 3668 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ})
89 simpl 486 . . . . . . . . . . . . 13 ((𝑦 = 𝑥𝑚 ∈ (ℤ𝑛)) → 𝑦 = 𝑥)
9089fveq2d 6665 . . . . . . . . . . . 12 ((𝑦 = 𝑥𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑥))
9190mpteq2dva 5147 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
9291rneqd 5795 . . . . . . . . . 10 (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
9392supeq1d 8907 . . . . . . . . 9 (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
9493eleq1d 2900 . . . . . . . 8 (𝑦 = 𝑥 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
9594cbvrabv 3477 . . . . . . 7 {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
9688, 95eleqtrdi 2926 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
97 eqid 2824 . . . . . . . 8 {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}
98 fvex 6674 . . . . . . . . . . . . 13 (𝐹𝑚) ∈ V
9998dmex 7611 . . . . . . . . . . . 12 dom (𝐹𝑚) ∈ V
10099rgenw 3145 . . . . . . . . . . 11 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
101100a1i 11 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑁) → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10284, 101iinexd 41691 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑁) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
103102adantl 485 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10497, 103rabexd 5222 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
10511fvmpt2 6770 . . . . . . 7 ((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})
10610, 104, 105syl2anc 587 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
10796, 106eleqtrrd 2919 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ (𝐸𝑛))
10832, 37, 107, 87fvmptd3 6782 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))‘𝑋) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
10924, 108eqtrd 2859 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
1101, 109mpteq2da 5146 . 2 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < )))
1113eluzelz2 41966 . . . 4 (𝑁𝑍𝑁 ∈ ℤ)
1122, 111syl 17 . . 3 (𝜑𝑁 ∈ ℤ)
113 eqid 2824 . . 3 (ℤ𝑁) = (ℤ𝑁)
11475a1i 11 . . . . 5 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑋) ∈ V)
11575a1i 11 . . . . 5 ((𝜑𝑚𝑍) → ((𝐹𝑚)‘𝑋) ∈ V)
11646, 112, 73, 113, 3, 114, 115limsupequzmpt 42297 . . . 4 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
117116, 78eqeltrd 2916 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
11846, 112, 113, 69, 117supcnvlimsupmpt 42309 . 2 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < )) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))
119110, 118eqbrtrd 5074 1 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wnf 1785  wcel 2115  wne 3014  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  wss 3919  c0 4276   ciin 4906   class class class wbr 5052  cmpt 5132  dom cdm 5542  ran crn 5543  wf 6339  cfv 6343  supcsup 8901  cr 10534  *cxr 10672   < clt 10673  cle 10674  cz 11978  cuz 12240  lim supclsp 14827  cli 14841  SAlgcsalg 42876  SMblFncsmblfn 43260
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-ioo 12739  df-ico 12741  df-fz 12895  df-fl 13166  df-ceil 13167  df-seq 13374  df-exp 13435  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-limsup 14828  df-clim 14845  df-smblfn 43261
This theorem is referenced by:  smflimsuplem6  43382  smflimsuplem8  43384
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