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Theorem smflimsup 42965
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
smflimsup.n 𝑚𝐹
smflimsup.x 𝑥𝐹
smflimsup.m (𝜑𝑀 ∈ ℤ)
smflimsup.z 𝑍 = (ℤ𝑀)
smflimsup.s (𝜑𝑆 ∈ SAlg)
smflimsup.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsup.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
smflimsup.g 𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
Assertion
Ref Expression
smflimsup (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝑛,𝐹   𝑥,𝑍,𝑚   𝑛,𝑍,𝑚   𝑥,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐹(𝑥,𝑚)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑚,𝑛)

Proof of Theorem smflimsup
Dummy variables 𝑗 𝑘 𝑞 𝑤 𝑖 𝑙 𝑝 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smflimsup.m . 2 (𝜑𝑀 ∈ ℤ)
2 smflimsup.z . 2 𝑍 = (ℤ𝑀)
3 smflimsup.s . 2 (𝜑𝑆 ∈ SAlg)
4 smflimsup.f . 2 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
5 smflimsup.d . . 3 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
6 fveq2 6666 . . . . . . . . 9 (𝑛 = 𝑗 → (ℤ𝑛) = (ℤ𝑗))
76iineq1d 41218 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚))
8 nfcv 2981 . . . . . . . . . 10 𝑞dom (𝐹𝑚)
9 smflimsup.n . . . . . . . . . . . 12 𝑚𝐹
10 nfcv 2981 . . . . . . . . . . . 12 𝑚𝑞
119, 10nffv 6676 . . . . . . . . . . 11 𝑚(𝐹𝑞)
1211nfdm 5821 . . . . . . . . . 10 𝑚dom (𝐹𝑞)
13 fveq2 6666 . . . . . . . . . . 11 (𝑚 = 𝑞 → (𝐹𝑚) = (𝐹𝑞))
1413dmeqd 5772 . . . . . . . . . 10 (𝑚 = 𝑞 → dom (𝐹𝑚) = dom (𝐹𝑞))
158, 12, 14cbviin 4958 . . . . . . . . 9 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1615a1i 11 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
177, 16eqtrd 2860 . . . . . . 7 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
1817cbviunv 4961 . . . . . 6 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1918eleq2i 2908 . . . . 5 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
20 nfcv 2981 . . . . . . . 8 𝑞((𝐹𝑚)‘𝑥)
21 nfcv 2981 . . . . . . . . 9 𝑚𝑥
2211, 21nffv 6676 . . . . . . . 8 𝑚((𝐹𝑞)‘𝑥)
2313fveq1d 6668 . . . . . . . 8 (𝑚 = 𝑞 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑞)‘𝑥))
2420, 22, 23cbvmpt 5163 . . . . . . 7 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))
2524fveq2i 6669 . . . . . 6 (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
2625eleq1i 2907 . . . . 5 ((lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ)
2719, 26anbi12i 626 . . . 4 ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ) ↔ (𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞) ∧ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ))
2827rabbia2 3482 . . 3 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ} = {𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞) ∣ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ}
29 nfcv 2981 . . . . 5 𝑥𝑍
30 nfcv 2981 . . . . . 6 𝑥(ℤ𝑗)
31 smflimsup.x . . . . . . . 8 𝑥𝐹
32 nfcv 2981 . . . . . . . 8 𝑥𝑞
3331, 32nffv 6676 . . . . . . 7 𝑥(𝐹𝑞)
3433nfdm 5821 . . . . . 6 𝑥dom (𝐹𝑞)
3530, 34nfiin 4946 . . . . 5 𝑥 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
3629, 35nfiun 4945 . . . 4 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
37 nfcv 2981 . . . 4 𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
38 nfv 1908 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ
39 nfcv 2981 . . . . . 6 𝑥lim sup
40 nfcv 2981 . . . . . . . 8 𝑥𝑤
4133, 40nffv 6676 . . . . . . 7 𝑥((𝐹𝑞)‘𝑤)
4229, 41nfmpt 5159 . . . . . 6 𝑥(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))
4339, 42nffv 6676 . . . . 5 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
44 nfcv 2981 . . . . 5 𝑥
4543, 44nfel 2996 . . . 4 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ
46 fveq2 6666 . . . . . . 7 (𝑥 = 𝑤 → ((𝐹𝑞)‘𝑥) = ((𝐹𝑞)‘𝑤))
4746mpteq2dv 5158 . . . . . 6 (𝑥 = 𝑤 → (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
4847fveq2d 6670 . . . . 5 (𝑥 = 𝑤 → (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
4948eleq1d 2901 . . . 4 (𝑥 = 𝑤 → ((lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ))
5036, 37, 38, 45, 49cbvrab 3495 . . 3 {𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞) ∣ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ} = {𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞) ∣ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ}
515, 28, 503eqtri 2852 . 2 𝐷 = {𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞) ∣ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ}
52 smflimsup.g . . 3 𝐺 = (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
5325mpteq2i 5154 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))))
54 nfrab1 3389 . . . . 5 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
555, 54nfcxfr 2979 . . . 4 𝑥𝐷
56 nfcv 2981 . . . 4 𝑤𝐷
57 nfcv 2981 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
5855, 56, 57, 43, 48cbvmptf 5161 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))) = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
5952, 53, 583eqtri 2852 . 2 𝐺 = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
60 nfcv 2981 . . . . . . 7 𝑥(ℤ𝑖)
6160, 34nfiin 4946 . . . . . 6 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
62 nfcv 2981 . . . . . 6 𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
63 nfv 1908 . . . . . 6 𝑤sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
6460, 41nfmpt 5159 . . . . . . . . 9 𝑥(𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
6564nfrn 5822 . . . . . . . 8 𝑥ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
66 nfcv 2981 . . . . . . . 8 𝑥*
67 nfcv 2981 . . . . . . . 8 𝑥 <
6865, 66, 67nfsup 8907 . . . . . . 7 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
6968, 44nfel 2996 . . . . . 6 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
7046mpteq2dv 5158 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7170rneqd 5806 . . . . . . . 8 (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7271supeq1d 8902 . . . . . . 7 (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
7372eleq1d 2901 . . . . . 6 (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
7461, 62, 63, 69, 73cbvrab 3495 . . . . 5 {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ}
7574a1i 11 . . . 4 (𝑖 = 𝑘 → {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
76 fveq2 6666 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7776iineq1d 41218 . . . . . . 7 (𝑖 = 𝑘 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) = 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞))
7877eleq2d 2902 . . . . . 6 (𝑖 = 𝑘 → (𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ↔ 𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞)))
7976mpteq1d 5151 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8079rneqd 5806 . . . . . . . 8 (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8180supeq1d 8902 . . . . . . 7 (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
8281eleq1d 2901 . . . . . 6 (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
8378, 82anbi12d 630 . . . . 5 (𝑖 = 𝑘 → ((𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∧ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ) ↔ (𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞) ∧ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ)))
8483rabbidva2 3481 . . . 4 (𝑖 = 𝑘 → {𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ} = {𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
8575, 84eqtrd 2860 . . 3 (𝑖 = 𝑘 → {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
8685cbvmptv 5165 . 2 (𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ}) = (𝑘𝑍 ↦ {𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
87 fveq2 6666 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝐹𝑝)‘𝑦) = ((𝐹𝑝)‘𝑤))
8887mpteq2dv 5158 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
8988rneqd 5806 . . . . . . . 8 (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
9089supeq1d 8902 . . . . . . 7 (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
9190cbvmptv 5165 . . . . . 6 (𝑦 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
92 fveq2 6666 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
9392dmeqd 5772 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → dom (𝐹𝑝) = dom (𝐹𝑞))
9493cbviinv 4962 . . . . . . . . . . . 12 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) = 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
9594eleq2i 2908 . . . . . . . . . . 11 (𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ↔ 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞))
96 nfcv 2981 . . . . . . . . . . . . . . 15 𝑞((𝐹𝑝)‘𝑥)
97 nfcv 2981 . . . . . . . . . . . . . . . 16 𝑝(𝐹𝑞)
98 nfcv 2981 . . . . . . . . . . . . . . . 16 𝑝𝑥
9997, 98nffv 6676 . . . . . . . . . . . . . . 15 𝑝((𝐹𝑞)‘𝑥)
10092fveq1d 6668 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑥) = ((𝐹𝑞)‘𝑥))
10196, 99, 100cbvmpt 5163 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
102101rneqi 5805 . . . . . . . . . . . . 13 ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
103102supeq1i 8903 . . . . . . . . . . . 12 sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < )
104103eleq1i 2907 . . . . . . . . . . 11 (sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
10595, 104anbi12i 626 . . . . . . . . . 10 ((𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∧ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∧ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ))
106105rabbia2 3482 . . . . . . . . 9 {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ}
107106mpteq2i 5154 . . . . . . . 8 (𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ}) = (𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})
108107fveq1i 6667 . . . . . . 7 ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) = ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙)
10992fveq1d 6668 . . . . . . . . . 10 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑤) = ((𝐹𝑞)‘𝑤))
110109cbvmptv 5165 . . . . . . . . 9 (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
111110rneqi 5805 . . . . . . . 8 ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
112111supeq1i 8903 . . . . . . 7 sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
113108, 112mpteq12i 5155 . . . . . 6 (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
11491, 113eqtri 2848 . . . . 5 (𝑦 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
115114a1i 11 . . . 4 (𝑙 = 𝑘 → (𝑦 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )))
116 fveq2 6666 . . . . 5 (𝑙 = 𝑘 → ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) = ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘))
117 fveq2 6666 . . . . . . . 8 (𝑙 = 𝑘 → (ℤ𝑙) = (ℤ𝑘))
118117mpteq1d 5151 . . . . . . 7 (𝑙 = 𝑘 → (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
119118rneqd 5806 . . . . . 6 (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
120119supeq1d 8902 . . . . 5 (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
121116, 120mpteq12dv 5147 . . . 4 (𝑙 = 𝑘 → (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )))
122115, 121eqtrd 2860 . . 3 (𝑙 = 𝑘 → (𝑦 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )))
123122cbvmptv 5165 . 2 (𝑙𝑍 ↦ (𝑦 ∈ ((𝑖𝑍 ↦ {𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ∣ sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ))) = (𝑘𝑍 ↦ (𝑤 ∈ ((𝑖𝑍 ↦ {𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ∣ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )))
1241, 2, 3, 4, 51, 59, 86, 123smflimsuplem8 42964 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2106  wnfc 2965  {crab 3146   ciun 4916   ciin 4917  cmpt 5142  dom cdm 5553  ran crn 5554  wf 6347  cfv 6351  supcsup 8896  cr 10528  *cxr 10666   < clt 10667  cz 11973  cuz 12235  lim supclsp 14820  SAlgcsalg 42456  SMblFncsmblfn 42840
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-13 2385  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cc 9849  ax-ac2 9877  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-iin 4919  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8282  df-map 8401  df-pm 8402  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-acn 9363  df-ac 9534  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-ioo 12735  df-ioc 12736  df-ico 12737  df-fz 12886  df-fl 13155  df-ceil 13156  df-seq 13363  df-exp 13423  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-limsup 14821  df-clim 14838  df-rlim 14839  df-rest 16688  df-topgen 16709  df-top 21418  df-bases 21470  df-salg 42457  df-salgen 42461  df-smblfn 42841
This theorem is referenced by:  smflimsupmpt  42966
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