Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimsuplem2 Structured version   Visualization version   GIF version

Theorem smflimsuplem2 42961
Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem2.p 𝑚𝜑
smflimsuplem2.m (𝜑𝑀 ∈ ℤ)
smflimsuplem2.z 𝑍 = (ℤ𝑀)
smflimsuplem2.s (𝜑𝑆 ∈ SAlg)
smflimsuplem2.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem2.e 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem2.h 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem2.n (𝜑𝑛𝑍)
smflimsuplem2.r (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
smflimsuplem2.x (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
Assertion
Ref Expression
smflimsuplem2 (𝜑𝑋 ∈ dom (𝐻𝑛))
Distinct variable groups:   𝑥,𝐹   𝑚,𝑀   𝑚,𝑋   𝑚,𝑍,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐸(𝑥,𝑚,𝑛)   𝐹(𝑚,𝑛)   𝐻(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑛)   𝑋(𝑥,𝑛)

Proof of Theorem smflimsuplem2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 smflimsuplem2.x . . . 4 (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
2 smflimsuplem2.p . . . . . 6 𝑚𝜑
3 eqid 2826 . . . . . 6 (ℤ𝑛) = (ℤ𝑛)
4 smflimsuplem2.n . . . . . . . . . . . . 13 (𝜑𝑛𝑍)
5 smflimsuplem2.z . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
64, 5syl6eleq 2928 . . . . . . . . . . . 12 (𝜑𝑛 ∈ (ℤ𝑀))
7 uzss 12254 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (ℤ𝑛) ⊆ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . 11 (𝜑 → (ℤ𝑛) ⊆ (ℤ𝑀))
98, 5sseqtrrdi 4022 . . . . . . . . . 10 (𝜑 → (ℤ𝑛) ⊆ 𝑍)
109adantr 481 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → (ℤ𝑛) ⊆ 𝑍)
11 simpr 485 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛))
1210, 11sseldd 3972 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 smflimsuplem2.s . . . . . . . . . 10 (𝜑𝑆 ∈ SAlg)
1413adantr 481 . . . . . . . . 9 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
15 smflimsuplem2.f . . . . . . . . . 10 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1615ffvelrnda 6847 . . . . . . . . 9 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2826 . . . . . . . . 9 dom (𝐹𝑚) = dom (𝐹𝑚)
1814, 16, 17smff 42875 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1912, 18syldan 591 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
20 iinss2 4978 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
2120adantl 482 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
221adantr 481 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
2321, 22sseldd 3972 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 ∈ dom (𝐹𝑚))
2419, 23ffvelrnd 6848 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
25 nfmpt1 5161 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
26 nfmpt1 5161 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
27 eluzelz 12242 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
286, 27syl 17 . . . . . . . . 9 (𝜑𝑛 ∈ ℤ)
29 eqid 2826 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
302, 24, 29fmptdf 6877 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)):(ℤ𝑛)⟶ℝ)
3130ffnd 6512 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑛))
32 smflimsuplem2.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
33 nfcv 2982 . . . . . . . . . 10 𝑚(ℤ𝑀)
34 fvexd 6682 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑀)) → ((𝐹𝑚)‘𝑋) ∈ V)
3533, 2, 34mptfnd 41377 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑀))
3629a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
37 fvexd 6682 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ V)
3836, 37fvmpt2d 6777 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
3912, 5syl6eleq 2928 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑀))
40 eqid 2826 . . . . . . . . . . . 12 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
4140fvmpt2 6775 . . . . . . . . . . 11 ((𝑚 ∈ (ℤ𝑀) ∧ ((𝐹𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4239, 37, 41syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4338, 42eqtr4d 2864 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚))
442, 25, 26, 28, 31, 32, 35, 28, 43limsupequz 41869 . . . . . . . 8 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))))
455eqcomi 2835 . . . . . . . . . . 11 (ℤ𝑀) = 𝑍
4645mpteq1i 5153 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))
4746fveq2i 6670 . . . . . . . . 9 (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
4847a1i 11 . . . . . . . 8 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
4944, 48eqtrd 2861 . . . . . . 7 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
50 smflimsuplem2.r . . . . . . . 8 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
5150renepnfd 10681 . . . . . . 7 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
5249, 51eqnetrd 3088 . . . . . 6 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
532, 3, 24, 52limsupubuzmpt 41865 . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦)
54 uzid 12247 . . . . . . 7 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
55 ne0i 4304 . . . . . . 7 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
5628, 54, 553syl 18 . . . . . 6 (𝜑 → (ℤ𝑛) ≠ ∅)
572, 56, 24supxrre3rnmpt 41568 . . . . 5 (𝜑 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦))
5853, 57mpbird 258 . . . 4 (𝜑 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
591, 58jca 512 . . 3 (𝜑 → (𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60 fveq2 6667 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6160mpteq2dv 5159 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6261rneqd 5807 . . . . . . . 8 (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6362supeq1d 8899 . . . . . . 7 (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
6463eleq1d 2902 . . . . . 6 (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ))
6564cbvrabv 3497 . . . . 5 {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ}
6665eleq2i 2909 . . . 4 (𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↔ 𝑋 ∈ {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ})
67 fveq2 6667 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
6867mpteq2dv 5159 . . . . . . . 8 (𝑦 = 𝑋 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
6968rneqd 5807 . . . . . . 7 (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
7069supeq1d 8899 . . . . . 6 (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
7170eleq1d 2902 . . . . 5 (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
7271elrab 3684 . . . 4 (𝑋 ∈ {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))
7366, 72bitri 276 . . 3 (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))
7459, 73sylibr 235 . 2 (𝜑𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
75 id 22 . . . . 5 (𝜑𝜑)
76 smflimsuplem2.h . . . . . . 7 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
7776a1i 11 . . . . . 6 (𝜑𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))))
78 smflimsuplem2.e . . . . . . . . . 10 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
79 nfcv 2982 . . . . . . . . . . 11 𝑥𝑍
80 nfrab1 3390 . . . . . . . . . . 11 𝑥{𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
8179, 80nfmpt 5160 . . . . . . . . . 10 𝑥(𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
8278, 81nfcxfr 2980 . . . . . . . . 9 𝑥𝐸
83 nfcv 2982 . . . . . . . . 9 𝑥𝑛
8482, 83nffv 6677 . . . . . . . 8 𝑥(𝐸𝑛)
85 fvex 6680 . . . . . . . 8 (𝐸𝑛) ∈ V
8684, 85mptexf 41372 . . . . . . 7 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V
8786a1i 11 . . . . . 6 ((𝜑𝑛𝑍) → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V)
8877, 87fvmpt2d 6777 . . . . 5 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
8975, 4, 88syl2anc 584 . . . 4 (𝜑 → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
9089dmeqd 5773 . . 3 (𝜑 → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
91 nfcv 2982 . . . . 5 𝑦(𝐸𝑛)
92 nfcv 2982 . . . . 5 𝑦sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )
93 nfcv 2982 . . . . 5 𝑥sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < )
9484, 91, 92, 93, 63cbvmptf 5162 . . . 4 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
95 xrltso 12524 . . . . . 6 < Or ℝ*
9695supex 8916 . . . . 5 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V
9796a1i 11 . . . 4 ((𝜑𝑦 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V)
9894, 97dmmptd 6490 . . 3 (𝜑 → dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝐸𝑛))
99 eqid 2826 . . . . 5 {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}
100 fvex 6680 . . . . . . . . 9 (𝐹𝑚) ∈ V
101100dmex 7604 . . . . . . . 8 dom (𝐹𝑚) ∈ V
102101rgenw 3155 . . . . . . 7 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
103102a1i 11 . . . . . 6 (𝜑 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10456, 103iinexd 41265 . . . . 5 (𝜑 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10599, 104rabexd 5233 . . . 4 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
10678fvmpt2 6775 . . . 4 ((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})
1074, 105, 106syl2anc 584 . . 3 (𝜑 → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
10890, 98, 1073eqtrrd 2866 . 2 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = dom (𝐻𝑛))
10974, 108eleqtrd 2920 1 (𝜑𝑋 ∈ dom (𝐻𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wnf 1777  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  wss 3940  c0 4295   ciin 4918   class class class wbr 5063  cmpt 5143  dom cdm 5554  ran crn 5555  wf 6348  cfv 6352  supcsup 8893  cr 10525  +∞cpnf 10661  *cxr 10663   < clt 10664  cle 10665  cz 11970  cuz 12232  lim supclsp 14817  SAlgcsalg 42459  SMblFncsmblfn 42843
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-n0 11887  df-z 11971  df-uz 12233  df-q 12338  df-ioo 12732  df-ico 12734  df-fz 12883  df-fl 13152  df-ceil 13153  df-limsup 14818  df-smblfn 42844
This theorem is referenced by:  smflimsuplem7  42966
  Copyright terms: Public domain W3C validator