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

Theorem smflimsuplem2 41348
 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 2651 . . . . . 6 (ℤ𝑛) = (ℤ𝑛)
4 smflimsuplem2.n . . . . . . . . . . . . 13 (𝜑𝑛𝑍)
5 smflimsuplem2.z . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
64, 5syl6eleq 2740 . . . . . . . . . . . 12 (𝜑𝑛 ∈ (ℤ𝑀))
7 uzss 11746 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (ℤ𝑛) ⊆ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . 11 (𝜑 → (ℤ𝑛) ⊆ (ℤ𝑀))
98, 5syl6sseqr 3685 . . . . . . . . . 10 (𝜑 → (ℤ𝑛) ⊆ 𝑍)
109adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → (ℤ𝑛) ⊆ 𝑍)
11 simpr 476 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛))
1210, 11sseldd 3637 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 smflimsuplem2.s . . . . . . . . . 10 (𝜑𝑆 ∈ SAlg)
1413adantr 480 . . . . . . . . 9 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
15 smflimsuplem2.f . . . . . . . . . 10 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1615ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2651 . . . . . . . . 9 dom (𝐹𝑚) = dom (𝐹𝑚)
1814, 16, 17smff 41262 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1912, 18syldan 486 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
20 iinss2 4604 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
2120adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
221adantr 480 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
2321, 22sseldd 3637 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 ∈ dom (𝐹𝑚))
2419, 23ffvelrnd 6400 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
25 nfmpt1 4780 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
26 nfmpt1 4780 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
27 eluzelz 11735 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
286, 27syl 17 . . . . . . . . 9 (𝜑𝑛 ∈ ℤ)
29 eqid 2651 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
302, 24, 29fmptdf 6427 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)):(ℤ𝑛)⟶ℝ)
3130ffnd 6084 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑛))
32 smflimsuplem2.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
33 nfcv 2793 . . . . . . . . . 10 𝑚(ℤ𝑀)
34 fvexd 6241 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑀)) → ((𝐹𝑚)‘𝑋) ∈ V)
3533, 2, 34mptfnd 39765 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑀))
3629a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
37 fvexd 6241 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ V)
3836, 37fvmpt2d 6332 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
3912, 5syl6eleq 2740 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑀))
40 eqid 2651 . . . . . . . . . . . 12 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
4140fvmpt2 6330 . . . . . . . . . . 11 ((𝑚 ∈ (ℤ𝑀) ∧ ((𝐹𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4239, 37, 41syl2anc 694 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4338, 42eqtr4d 2688 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚))
442, 25, 26, 28, 31, 32, 35, 28, 43limsupequz 40273 . . . . . . . 8 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))))
455eqcomi 2660 . . . . . . . . . . 11 (ℤ𝑀) = 𝑍
4645mpteq1i 4772 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))
4746fveq2i 6232 . . . . . . . . 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 2685 . . . . . . 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 10128 . . . . . . 7 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
5249, 51eqnetrd 2890 . . . . . 6 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
532, 3, 24, 52limsupubuzmpt 40269 . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦)
54 uzid 11740 . . . . . . 7 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
55 ne0i 3954 . . . . . . 7 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
5628, 54, 553syl 18 . . . . . 6 (𝜑 → (ℤ𝑛) ≠ ∅)
572, 56, 24supxrre3rnmpt 39969 . . . . 5 (𝜑 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦))
5853, 57mpbird 247 . . . 4 (𝜑 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
591, 58jca 553 . . 3 (𝜑 → (𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60 fveq2 6229 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6160mpteq2dv 4778 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6261rneqd 5385 . . . . . . . 8 (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6362supeq1d 8393 . . . . . . 7 (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
6463eleq1d 2715 . . . . . 6 (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ))
6564cbvrabv 3230 . . . . 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 2722 . . . 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 6229 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
6867mpteq2dv 4778 . . . . . . . 8 (𝑦 = 𝑋 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
6968rneqd 5385 . . . . . . 7 (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
7069supeq1d 8393 . . . . . 6 (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
7170eleq1d 2715 . . . . 5 (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
7271elrab 3396 . . . 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 264 . . 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 224 . 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 2793 . . . . . . . . . . 11 𝑥𝑍
80 nfrab1 3152 . . . . . . . . . . 11 𝑥{𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
8179, 80nfmpt 4779 . . . . . . . . . 10 𝑥(𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
8278, 81nfcxfr 2791 . . . . . . . . 9 𝑥𝐸
83 nfcv 2793 . . . . . . . . 9 𝑥𝑛
8482, 83nffv 6236 . . . . . . . 8 𝑥(𝐸𝑛)
85 fvex 6239 . . . . . . . 8 (𝐸𝑛) ∈ V
8684, 85mptexf 39758 . . . . . . 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 6332 . . . . 5 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
8975, 4, 88syl2anc 694 . . . 4 (𝜑 → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
9089dmeqd 5358 . . 3 (𝜑 → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
91 nfcv 2793 . . . . 5 𝑦(𝐸𝑛)
92 nfcv 2793 . . . . 5 𝑦sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )
93 nfcv 2793 . . . . 5 𝑥sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < )
9484, 91, 92, 93, 63cbvmptf 4781 . . . 4 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
95 xrltso 12012 . . . . . 6 < Or ℝ*
9695supex 8410 . . . . 5 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V
9796a1i 11 . . . 4 ((𝜑𝑦 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V)
9894, 97dmmptd 6062 . . 3 (𝜑 → dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝐸𝑛))
99 eqid 2651 . . . . 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 6239 . . . . . . . . 9 (𝐹𝑚) ∈ V
101100dmex 7141 . . . . . . . 8 dom (𝐹𝑚) ∈ V
102101rgenw 2953 . . . . . . 7 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
103102a1i 11 . . . . . 6 (𝜑 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10456, 103iinexd 39632 . . . . 5 (𝜑 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10599, 104rabexd 4846 . . . 4 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
10678fvmpt2 6330 . . . 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 694 . . 3 (𝜑 → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
10890, 98, 1073eqtrrd 2690 . 2 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = dom (𝐻𝑛))
10974, 108eleqtrd 2732 1 (𝜑𝑋 ∈ dom (𝐻𝑛))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  ∩ ciin 4553   class class class wbr 4685   ↦ cmpt 4762  dom cdm 5143  ran crn 5144  ⟶wf 5922  ‘cfv 5926  supcsup 8387  ℝcr 9973  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113  ℤcz 11415  ℤ≥cuz 11725  lim supclsp 14245  SAlgcsalg 40846  SMblFncsmblfn 41230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-ioo 12217  df-ico 12219  df-fz 12365  df-fl 12633  df-ceil 12634  df-limsup 14246  df-smblfn 41231 This theorem is referenced by:  smflimsuplem7  41353
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