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Theorem smflimsup 43818
 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 6659 . . . . . . . . 9 (𝑛 = 𝑗 → (ℤ𝑛) = (ℤ𝑗))
76iineq1d 42092 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚))
8 nfcv 2920 . . . . . . . . . 10 𝑞dom (𝐹𝑚)
9 smflimsup.n . . . . . . . . . . . 12 𝑚𝐹
10 nfcv 2920 . . . . . . . . . . . 12 𝑚𝑞
119, 10nffv 6669 . . . . . . . . . . 11 𝑚(𝐹𝑞)
1211nfdm 5793 . . . . . . . . . 10 𝑚dom (𝐹𝑞)
13 fveq2 6659 . . . . . . . . . . 11 (𝑚 = 𝑞 → (𝐹𝑚) = (𝐹𝑞))
1413dmeqd 5746 . . . . . . . . . 10 (𝑚 = 𝑞 → dom (𝐹𝑚) = dom (𝐹𝑞))
158, 12, 14cbviin 4927 . . . . . . . . 9 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1615a1i 11 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
177, 16eqtrd 2794 . . . . . . 7 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
1817cbviunv 4930 . . . . . 6 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1918eleq2i 2844 . . . . 5 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
20 nfcv 2920 . . . . . . . 8 𝑞((𝐹𝑚)‘𝑥)
21 nfcv 2920 . . . . . . . . 9 𝑚𝑥
2211, 21nffv 6669 . . . . . . . 8 𝑚((𝐹𝑞)‘𝑥)
2313fveq1d 6661 . . . . . . . 8 (𝑚 = 𝑞 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑞)‘𝑥))
2420, 22, 23cbvmpt 5134 . . . . . . 7 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))
2524fveq2i 6662 . . . . . 6 (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
2625eleq1i 2843 . . . . 5 ((lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ)
2719, 26anbi12i 630 . . . 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 3390 . . 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 2920 . . . . 5 𝑥𝑍
30 nfcv 2920 . . . . . 6 𝑥(ℤ𝑗)
31 smflimsup.x . . . . . . . 8 𝑥𝐹
32 nfcv 2920 . . . . . . . 8 𝑥𝑞
3331, 32nffv 6669 . . . . . . 7 𝑥(𝐹𝑞)
3433nfdm 5793 . . . . . 6 𝑥dom (𝐹𝑞)
3530, 34nfiin 4915 . . . . 5 𝑥 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
3629, 35nfiun 4914 . . . 4 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
37 nfcv 2920 . . . 4 𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
38 nfv 1916 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ
39 nfcv 2920 . . . . . 6 𝑥lim sup
40 nfcv 2920 . . . . . . . 8 𝑥𝑤
4133, 40nffv 6669 . . . . . . 7 𝑥((𝐹𝑞)‘𝑤)
4229, 41nfmpt 5130 . . . . . 6 𝑥(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))
4339, 42nffv 6669 . . . . 5 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
44 nfcv 2920 . . . . 5 𝑥
4543, 44nfel 2934 . . . 4 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ
46 fveq2 6659 . . . . . . 7 (𝑥 = 𝑤 → ((𝐹𝑞)‘𝑥) = ((𝐹𝑞)‘𝑤))
4746mpteq2dv 5129 . . . . . 6 (𝑥 = 𝑤 → (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
4847fveq2d 6663 . . . . 5 (𝑥 = 𝑤 → (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
4948eleq1d 2837 . . . 4 (𝑥 = 𝑤 → ((lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ))
5036, 37, 38, 45, 49cbvrabw 3403 . . 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 2786 . 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 5125 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))))
54 nfrab1 3303 . . . . 5 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
555, 54nfcxfr 2918 . . . 4 𝑥𝐷
56 nfcv 2920 . . . 4 𝑤𝐷
57 nfcv 2920 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
5855, 56, 57, 43, 48cbvmptf 5132 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))) = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
5952, 53, 583eqtri 2786 . 2 𝐺 = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
60 nfcv 2920 . . . . . . 7 𝑥(ℤ𝑖)
6160, 34nfiin 4915 . . . . . 6 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
62 nfcv 2920 . . . . . 6 𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
63 nfv 1916 . . . . . 6 𝑤sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
6460, 41nfmpt 5130 . . . . . . . . 9 𝑥(𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
6564nfrn 5794 . . . . . . . 8 𝑥ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
66 nfcv 2920 . . . . . . . 8 𝑥*
67 nfcv 2920 . . . . . . . 8 𝑥 <
6865, 66, 67nfsup 8941 . . . . . . 7 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
6968, 44nfel 2934 . . . . . 6 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
7046mpteq2dv 5129 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7170rneqd 5780 . . . . . . . 8 (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7271supeq1d 8936 . . . . . . 7 (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
7372eleq1d 2837 . . . . . 6 (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
7461, 62, 63, 69, 73cbvrabw 3403 . . . . 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 6659 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7776iineq1d 42092 . . . . . . 7 (𝑖 = 𝑘 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) = 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞))
7877eleq2d 2838 . . . . . 6 (𝑖 = 𝑘 → (𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ↔ 𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞)))
7976mpteq1d 5122 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8079rneqd 5780 . . . . . . . 8 (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8180supeq1d 8936 . . . . . . 7 (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
8281eleq1d 2837 . . . . . 6 (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
8378, 82anbi12d 634 . . . . 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 3389 . . . 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 2794 . . 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 5136 . 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 6659 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝐹𝑝)‘𝑦) = ((𝐹𝑝)‘𝑤))
8887mpteq2dv 5129 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
8988rneqd 5780 . . . . . . . 8 (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
9089supeq1d 8936 . . . . . . 7 (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
9190cbvmptv 5136 . . . . . 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 6659 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
9392dmeqd 5746 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → dom (𝐹𝑝) = dom (𝐹𝑞))
9493cbviinv 4931 . . . . . . . . . . . 12 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) = 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
9594eleq2i 2844 . . . . . . . . . . 11 (𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ↔ 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞))
96 nfcv 2920 . . . . . . . . . . . . . . 15 𝑞((𝐹𝑝)‘𝑥)
97 nfcv 2920 . . . . . . . . . . . . . . . 16 𝑝(𝐹𝑞)
98 nfcv 2920 . . . . . . . . . . . . . . . 16 𝑝𝑥
9997, 98nffv 6669 . . . . . . . . . . . . . . 15 𝑝((𝐹𝑞)‘𝑥)
10092fveq1d 6661 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑥) = ((𝐹𝑞)‘𝑥))
10196, 99, 100cbvmpt 5134 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
102101rneqi 5779 . . . . . . . . . . . . 13 ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
103102supeq1i 8937 . . . . . . . . . . . 12 sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < )
104103eleq1i 2843 . . . . . . . . . . 11 (sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
10595, 104anbi12i 630 . . . . . . . . . 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 3390 . . . . . . . . 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 5125 . . . . . . . 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 6660 . . . . . . 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 6661 . . . . . . . . . 10 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑤) = ((𝐹𝑞)‘𝑤))
110109cbvmptv 5136 . . . . . . . . 9 (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
111110rneqi 5779 . . . . . . . 8 ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
112111supeq1i 8937 . . . . . . 7 sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
113108, 112mpteq12i 5126 . . . . . 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 2782 . . . . 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 6659 . . . . 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 6659 . . . . . . . 8 (𝑙 = 𝑘 → (ℤ𝑙) = (ℤ𝑘))
118117mpteq1d 5122 . . . . . . 7 (𝑙 = 𝑘 → (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
119118rneqd 5780 . . . . . 6 (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
120119supeq1d 8936 . . . . 5 (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
121116, 120mpteq12dv 5118 . . . 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 2794 . . 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 5136 . 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 43817 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  Ⅎwnfc 2900  {crab 3075  ∪ ciun 4884  ∩ ciin 4885   ↦ cmpt 5113  dom cdm 5525  ran crn 5526  ⟶wf 6332  ‘cfv 6336  supcsup 8930  ℝcr 10567  ℝ*cxr 10705   < clt 10706  ℤcz 12013  ℤ≥cuz 12275  lim supclsp 14868  SAlgcsalg 43309  SMblFncsmblfn 43693 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-inf2 9130  ax-cc 9888  ax-ac2 9916  ax-cnex 10624  ax-resscn 10625  ax-1cn 10626  ax-icn 10627  ax-addcl 10628  ax-addrcl 10629  ax-mulcl 10630  ax-mulrcl 10631  ax-mulcom 10632  ax-addass 10633  ax-mulass 10634  ax-distr 10635  ax-i2m1 10636  ax-1ne0 10637  ax-1rid 10638  ax-rnegex 10639  ax-rrecex 10640  ax-cnre 10641  ax-pre-lttri 10642  ax-pre-lttrn 10643  ax-pre-ltadd 10644  ax-pre-mulgt0 10645  ax-pre-sup 10646 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-se 5485  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-oadd 8117  df-omul 8118  df-er 8300  df-map 8419  df-pm 8420  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-sup 8932  df-inf 8933  df-oi 9000  df-card 9394  df-acn 9397  df-ac 9569  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-sub 10903  df-neg 10904  df-div 11329  df-nn 11668  df-2 11730  df-3 11731  df-n0 11928  df-z 12014  df-uz 12276  df-q 12382  df-rp 12424  df-ioo 12776  df-ioc 12777  df-ico 12778  df-fz 12933  df-fl 13204  df-ceil 13205  df-seq 13412  df-exp 13473  df-cj 14499  df-re 14500  df-im 14501  df-sqrt 14635  df-abs 14636  df-limsup 14869  df-clim 14886  df-rlim 14887  df-rest 16747  df-topgen 16768  df-top 21587  df-bases 21639  df-salg 43310  df-salgen 43314  df-smblfn 43694 This theorem is referenced by:  smflimsupmpt  43819
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