Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimsuplem5 Structured version   Visualization version   GIF version

Theorem smflimsuplem5 42530
 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 2857 . . . . . . . . . . 11 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
54biimpi 208 . . . . . . . . . 10 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
6 uzss 12082 . . . . . . . . . 10 (𝑁 ∈ (ℤ𝑀) → (ℤ𝑁) ⊆ (ℤ𝑀))
75, 6syl 17 . . . . . . . . 9 (𝑁𝑍 → (ℤ𝑁) ⊆ (ℤ𝑀))
87, 3syl6sseqr 3910 . . . . . . . 8 (𝑁𝑍 → (ℤ𝑁) ⊆ 𝑍)
92, 8syl 17 . . . . . . 7 (𝜑 → (ℤ𝑁) ⊆ 𝑍)
109sselda 3860 . . . . . 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 2932 . . . . . . . . . . 11 𝑥𝑍
13 nfrab1 3324 . . . . . . . . . . 11 𝑥{𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
1412, 13nfmpt 5025 . . . . . . . . . 10 𝑥(𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
1511, 14nfcxfr 2930 . . . . . . . . 9 𝑥𝐸
16 nfcv 2932 . . . . . . . . 9 𝑥𝑛
1715, 16nffv 6511 . . . . . . . 8 𝑥(𝐸𝑛)
18 fvex 6514 . . . . . . . 8 (𝐸𝑛) ∈ V
1917, 18mptexf 40935 . . . . . . 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 6607 . . . . . 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 576 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
2423fveq1d 6503 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑋) = ((𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))‘𝑋))
25 nfcv 2932 . . . . . 6 𝑦(𝐸𝑛)
26 nfcv 2932 . . . . . 6 𝑦sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )
27 nfcv 2932 . . . . . 6 𝑥sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < )
28 fveq2 6501 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
2928mpteq2dv 5024 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
3029rneqd 5652 . . . . . . 7 (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
3130supeq1d 8707 . . . . . 6 (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
3217, 25, 26, 27, 31cbvmptf 5027 . . . . 5 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
33 simpl 475 . . . . . . . . 9 ((𝑦 = 𝑋𝑚 ∈ (ℤ𝑛)) → 𝑦 = 𝑋)
3433fveq2d 6505 . . . . . . . 8 ((𝑦 = 𝑋𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
3534mpteq2dva 5023 . . . . . . 7 (𝑦 = 𝑋 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
3635rneqd 5652 . . . . . 6 (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
3736supeq1d 8707 . . . . 5 (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
3837eleq1d 2850 . . . . . . . 8 (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
39 uzss 12082 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (ℤ𝑛) ⊆ (ℤ𝑁))
40 iinss1 4807 . . . . . . . . . . 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 474 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚) ⊆ 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
43 smflimsuplem5.x . . . . . . . . . 10 (𝜑𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))
4443adantr 473 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 𝑚 ∈ (ℤ𝑁)dom (𝐹𝑚))
4542, 44sseldd 3861 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
46 smflimsuplem5.b . . . . . . . . . . 11 𝑚𝜑
47 nfv 1873 . . . . . . . . . . 11 𝑚 𝑛 ∈ (ℤ𝑁)
4846, 47nfan 1862 . . . . . . . . . 10 𝑚(𝜑𝑛 ∈ (ℤ𝑁))
49 eqid 2778 . . . . . . . . . 10 (ℤ𝑛) = (ℤ𝑛)
50 simpll 754 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
5139sselda 3860 . . . . . . . . . . . 12 ((𝑛 ∈ (ℤ𝑁) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
5251adantll 701 . . . . . . . . . . 11 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑁))
53 smflimsuplem5.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
5453adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑆 ∈ SAlg)
55 simpl 475 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝜑)
569sselda 3860 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚𝑍)
57 smflimsuplem5.f . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
5857ffvelrnda 6678 . . . . . . . . . . . . . 14 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
5955, 56, 58syl2anc 576 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
60 eqid 2778 . . . . . . . . . . . . 13 dom (𝐹𝑚) = dom (𝐹𝑚)
6154, 59, 60smff 42441 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (ℤ𝑁)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
62 eliin 4798 . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚))
6564adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → ∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚))
66 simpr 477 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑁))
67 rspa 3156 . . . . . . . . . . . . 13 ((∀𝑚 ∈ (ℤ𝑁)𝑋 ∈ dom (𝐹𝑚) ∧ 𝑚 ∈ (ℤ𝑁)) → 𝑋 ∈ dom (𝐹𝑚))
6865, 66, 67syl2anc 576 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (ℤ𝑁)) → 𝑋 ∈ dom (𝐹𝑚))
6961, 68ffvelrnd 6679 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑁)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
7050, 52, 69syl2anc 576 . . . . . . . . . 10 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
71 eluzelz 12071 . . . . . . . . . . . . . 14 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ ℤ)
7271adantl 474 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑛 ∈ ℤ)
73 smflimsuplem5.m . . . . . . . . . . . . . 14 (𝜑𝑀 ∈ ℤ)
7473adantr 473 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑀 ∈ ℤ)
75 fvex 6514 . . . . . . . . . . . . . 14 ((𝐹𝑚)‘𝑋) ∈ V
7675a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ (ℤ𝑁)) ∧ 𝑚𝑍) → ((𝐹𝑚)‘𝑋) ∈ V)
7748, 72, 74, 49, 3, 70, 76limsupequzmpt 41442 . . . . . . . . . . . 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 473 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
8077, 79eqeltrd 2866 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
8180renepnfd 10493 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
8248, 49, 70, 81limsupubuzmpt 41432 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦)
83 uzid2 41109 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑁) → 𝑛 ∈ (ℤ𝑛))
8483ne0d 4189 . . . . . . . . . . 11 (𝑛 ∈ (ℤ𝑁) → (ℤ𝑛) ≠ ∅)
8584adantl 474 . . . . . . . . . 10 ((𝜑𝑛 ∈ (ℤ𝑁)) → (ℤ𝑛) ≠ ∅)
8648, 85, 70supxrre3rnmpt 41135 . . . . . . . . 9 ((𝜑𝑛 ∈ (ℤ𝑁)) → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦))
8782, 86mpbird 249 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
8838, 45, 87elrabd 3598 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ})
89 simpl 475 . . . . . . . . . . . . 13 ((𝑦 = 𝑥𝑚 ∈ (ℤ𝑛)) → 𝑦 = 𝑥)
9089fveq2d 6505 . . . . . . . . . . . 12 ((𝑦 = 𝑥𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑥))
9190mpteq2dva 5023 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
9291rneqd 5652 . . . . . . . . . 10 (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)))
9392supeq1d 8707 . . . . . . . . 9 (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))
9493eleq1d 2850 . . . . . . . 8 (𝑦 = 𝑥 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
9594cbvrabv 3412 . . . . . . 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, 95syl6eleq 2876 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
97 eqid 2778 . . . . . . . 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 6514 . . . . . . . . . . . . 13 (𝐹𝑚) ∈ V
9998dmex 7433 . . . . . . . . . . . 12 dom (𝐹𝑚) ∈ V
10099rgenw 3100 . . . . . . . . . . 11 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
101100a1i 11 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑁) → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10284, 101iinexd 40823 . . . . . . . . 9 (𝑛 ∈ (ℤ𝑁) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
103102adantl 474 . . . . . . . 8 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10497, 103rabexd 5093 . . . . . . 7 ((𝜑𝑛 ∈ (ℤ𝑁)) → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
10511fvmpt2 6607 . . . . . . 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 576 . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑁)) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
10796, 106eleqtrrd 2869 . . . . 5 ((𝜑𝑛 ∈ (ℤ𝑁)) → 𝑋 ∈ (𝐸𝑛))
10832, 37, 107, 87fvmptd3 6619 . . . 4 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))‘𝑋) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
10924, 108eqtrd 2814 . . 3 ((𝜑𝑛 ∈ (ℤ𝑁)) → ((𝐻𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
1101, 109mpteq2da 5022 . 2 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) = (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < )))
1113eluzelz2 41107 . . . 4 (𝑁𝑍𝑁 ∈ ℤ)
1122, 111syl 17 . . 3 (𝜑𝑁 ∈ ℤ)
113 eqid 2778 . . 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 41442 . . . 4 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
117116, 78eqeltrd 2866 . . 3 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
11846, 112, 113, 69, 117supcnvlimsupmpt 41454 . 2 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < )) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))
119110, 118eqbrtrd 4952 1 (𝜑 → (𝑛 ∈ (ℤ𝑁) ↦ ((𝐻𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ𝑁) ↦ ((𝐹𝑚)‘𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  Ⅎwnf 1746   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  {crab 3092  Vcvv 3415   ⊆ wss 3831  ∅c0 4180  ∩ ciin 4794   class class class wbr 4930   ↦ cmpt 5009  dom cdm 5408  ran crn 5409  ⟶wf 6186  ‘cfv 6190  supcsup 8701  ℝcr 10336  ℝ*cxr 10475   < clt 10476   ≤ cle 10477  ℤcz 11796  ℤ≥cuz 12061  lim supclsp 14691   ⇝ cli 14705  SAlgcsalg 42025  SMblFncsmblfn 42409 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-resscn 10394  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-addrcl 10398  ax-mulcl 10399  ax-mulrcl 10400  ax-mulcom 10401  ax-addass 10402  ax-mulass 10403  ax-distr 10404  ax-i2m1 10405  ax-1ne0 10406  ax-1rid 10407  ax-rnegex 10408  ax-rrecex 10409  ax-cnre 10410  ax-pre-lttri 10411  ax-pre-lttrn 10412  ax-pre-ltadd 10413  ax-pre-mulgt0 10414  ax-pre-sup 10415 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-iin 4796  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-riota 6939  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-er 8091  df-pm 8211  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312  df-sup 8703  df-inf 8704  df-pnf 10478  df-mnf 10479  df-xr 10480  df-ltxr 10481  df-le 10482  df-sub 10674  df-neg 10675  df-div 11101  df-nn 11442  df-2 11506  df-3 11507  df-n0 11711  df-z 11797  df-uz 12062  df-q 12166  df-rp 12208  df-ioo 12561  df-ico 12563  df-fz 12712  df-fl 12980  df-ceil 12981  df-seq 13188  df-exp 13248  df-cj 14322  df-re 14323  df-im 14324  df-sqrt 14458  df-abs 14459  df-limsup 14692  df-clim 14709  df-smblfn 42410 This theorem is referenced by:  smflimsuplem6  42531  smflimsuplem8  42533
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