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Theorem smflimsup 41355
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 6229 . . . . . . . . 9 (𝑛 = 𝑗 → (ℤ𝑛) = (ℤ𝑗))
76iineq1d 39581 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚))
8 nfcv 2793 . . . . . . . . . 10 𝑞dom (𝐹𝑚)
9 smflimsup.n . . . . . . . . . . . 12 𝑚𝐹
10 nfcv 2793 . . . . . . . . . . . 12 𝑚𝑞
119, 10nffv 6236 . . . . . . . . . . 11 𝑚(𝐹𝑞)
1211nfdm 5399 . . . . . . . . . 10 𝑚dom (𝐹𝑞)
13 fveq2 6229 . . . . . . . . . . 11 (𝑚 = 𝑞 → (𝐹𝑚) = (𝐹𝑞))
1413dmeqd 5358 . . . . . . . . . 10 (𝑚 = 𝑞 → dom (𝐹𝑚) = dom (𝐹𝑞))
158, 12, 14cbviin 4590 . . . . . . . . 9 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1615a1i 11 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
177, 16eqtrd 2685 . . . . . . 7 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
1817cbviunv 4591 . . . . . 6 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1918eleq2i 2722 . . . . 5 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
20 nfcv 2793 . . . . . . . 8 𝑞((𝐹𝑚)‘𝑥)
21 nfcv 2793 . . . . . . . . 9 𝑚𝑥
2211, 21nffv 6236 . . . . . . . 8 𝑚((𝐹𝑞)‘𝑥)
2313fveq1d 6231 . . . . . . . 8 (𝑚 = 𝑞 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑞)‘𝑥))
2420, 22, 23cbvmpt 4782 . . . . . . 7 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))
2524fveq2i 6232 . . . . . 6 (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
2625eleq1i 2721 . . . . 5 ((lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ)
2719, 26anbi12i 733 . . . 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 3218 . . 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 2793 . . . . 5 𝑥𝑍
30 nfcv 2793 . . . . . 6 𝑥(ℤ𝑗)
31 smflimsup.x . . . . . . . 8 𝑥𝐹
32 nfcv 2793 . . . . . . . 8 𝑥𝑞
3331, 32nffv 6236 . . . . . . 7 𝑥(𝐹𝑞)
3433nfdm 5399 . . . . . 6 𝑥dom (𝐹𝑞)
3530, 34nfiin 4581 . . . . 5 𝑥 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
3629, 35nfiun 4580 . . . 4 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
37 nfcv 2793 . . . 4 𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
38 nfv 1883 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ
39 nfcv 2793 . . . . . 6 𝑥lim sup
40 nfcv 2793 . . . . . . . 8 𝑥𝑤
4133, 40nffv 6236 . . . . . . 7 𝑥((𝐹𝑞)‘𝑤)
4229, 41nfmpt 4779 . . . . . 6 𝑥(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))
4339, 42nffv 6236 . . . . 5 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
44 nfcv 2793 . . . . 5 𝑥
4543, 44nfel 2806 . . . 4 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ
46 fveq2 6229 . . . . . . 7 (𝑥 = 𝑤 → ((𝐹𝑞)‘𝑥) = ((𝐹𝑞)‘𝑤))
4746mpteq2dv 4778 . . . . . 6 (𝑥 = 𝑤 → (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
4847fveq2d 6233 . . . . 5 (𝑥 = 𝑤 → (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
4948eleq1d 2715 . . . 4 (𝑥 = 𝑤 → ((lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ))
5036, 37, 38, 45, 49cbvrab 3229 . . 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 2677 . 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 4774 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))))
54 nfrab1 3152 . . . . 5 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
555, 54nfcxfr 2791 . . . 4 𝑥𝐷
56 nfcv 2793 . . . 4 𝑤𝐷
57 nfcv 2793 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
5855, 56, 57, 43, 48cbvmptf 4781 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))) = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
5952, 53, 583eqtri 2677 . 2 𝐺 = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
60 nfcv 2793 . . . . . . 7 𝑥(ℤ𝑖)
6160, 34nfiin 4581 . . . . . 6 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
62 nfcv 2793 . . . . . 6 𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
63 nfv 1883 . . . . . 6 𝑤sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
6460, 41nfmpt 4779 . . . . . . . . 9 𝑥(𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
6564nfrn 5400 . . . . . . . 8 𝑥ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
66 nfcv 2793 . . . . . . . 8 𝑥*
67 nfcv 2793 . . . . . . . 8 𝑥 <
6865, 66, 67nfsup 8398 . . . . . . 7 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
6968, 44nfel 2806 . . . . . 6 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
7046mpteq2dv 4778 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7170rneqd 5385 . . . . . . . 8 (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7271supeq1d 8393 . . . . . . 7 (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
7372eleq1d 2715 . . . . . 6 (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
7461, 62, 63, 69, 73cbvrab 3229 . . . . 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 6229 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7776iineq1d 39581 . . . . . . 7 (𝑖 = 𝑘 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) = 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞))
7877eleq2d 2716 . . . . . 6 (𝑖 = 𝑘 → (𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ↔ 𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞)))
7976mpteq1d 4771 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8079rneqd 5385 . . . . . . . 8 (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8180supeq1d 8393 . . . . . . 7 (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
8281eleq1d 2715 . . . . . 6 (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
8378, 82anbi12d 747 . . . . 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 3217 . . . 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 2685 . . 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 4783 . 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 6229 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝐹𝑝)‘𝑦) = ((𝐹𝑝)‘𝑤))
8887mpteq2dv 4778 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
8988rneqd 5385 . . . . . . . 8 (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
9089supeq1d 8393 . . . . . . 7 (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
9190cbvmptv 4783 . . . . . 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 6229 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
9392dmeqd 5358 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → dom (𝐹𝑝) = dom (𝐹𝑞))
9493cbviinv 4592 . . . . . . . . . . . 12 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) = 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
9594eleq2i 2722 . . . . . . . . . . 11 (𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ↔ 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞))
96 nfcv 2793 . . . . . . . . . . . . . . 15 𝑞((𝐹𝑝)‘𝑥)
97 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑝(𝐹𝑞)
98 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑝𝑥
9997, 98nffv 6236 . . . . . . . . . . . . . . 15 𝑝((𝐹𝑞)‘𝑥)
10092fveq1d 6231 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑥) = ((𝐹𝑞)‘𝑥))
10196, 99, 100cbvmpt 4782 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
102101rneqi 5384 . . . . . . . . . . . . 13 ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
103102supeq1i 8394 . . . . . . . . . . . 12 sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < )
104103eleq1i 2721 . . . . . . . . . . 11 (sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
10595, 104anbi12i 733 . . . . . . . . . 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 3218 . . . . . . . . 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 4774 . . . . . . . 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 6230 . . . . . . 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 6231 . . . . . . . . . 10 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑤) = ((𝐹𝑞)‘𝑤))
110109cbvmptv 4783 . . . . . . . . 9 (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
111110rneqi 5384 . . . . . . . 8 ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
112111supeq1i 8394 . . . . . . 7 sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
113108, 112mpteq12i 4775 . . . . . 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 2673 . . . . 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 6229 . . . . 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 6229 . . . . . . . 8 (𝑙 = 𝑘 → (ℤ𝑙) = (ℤ𝑘))
118117mpteq1d 4771 . . . . . . 7 (𝑙 = 𝑘 → (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
119118rneqd 5385 . . . . . 6 (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
120119supeq1d 8393 . . . . 5 (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
121116, 120mpteq12dv 4766 . . . 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 2685 . . 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 4783 . 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 41354 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wnfc 2780  {crab 2945   ciun 4552   ciin 4553  cmpt 4762  dom cdm 5143  ran crn 5144  wf 5922  cfv 5926  supcsup 8387  cr 9973  *cxr 10111   < clt 10112  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-inf2 8576  ax-cc 9295  ax-ac2 9323  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-se 5103  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-isom 5935  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-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  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-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ioc 12218  df-ico 12219  df-fz 12365  df-fl 12633  df-ceil 12634  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-rest 16130  df-topgen 16151  df-top 20747  df-bases 20798  df-salg 40847  df-salgen 40851  df-smblfn 41231
This theorem is referenced by:  smflimsupmpt  41356
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