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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  wnf 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  cr 9973  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113  cz 11415  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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