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Theorem smflimsup 45155
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 6843 . . . . . . . . 9 (𝑛 = 𝑗 → (ℤ𝑛) = (ℤ𝑗))
76iineq1d 43388 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚))
8 nfcv 2904 . . . . . . . . . 10 𝑞dom (𝐹𝑚)
9 smflimsup.n . . . . . . . . . . . 12 𝑚𝐹
10 nfcv 2904 . . . . . . . . . . . 12 𝑚𝑞
119, 10nffv 6853 . . . . . . . . . . 11 𝑚(𝐹𝑞)
1211nfdm 5907 . . . . . . . . . 10 𝑚dom (𝐹𝑞)
13 fveq2 6843 . . . . . . . . . . 11 (𝑚 = 𝑞 → (𝐹𝑚) = (𝐹𝑞))
1413dmeqd 5862 . . . . . . . . . 10 (𝑚 = 𝑞 → dom (𝐹𝑚) = dom (𝐹𝑞))
158, 12, 14cbviin 4998 . . . . . . . . 9 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1615a1i 11 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
177, 16eqtrd 2773 . . . . . . 7 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
1817cbviunv 5001 . . . . . 6 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1918eleq2i 2826 . . . . 5 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
20 nfcv 2904 . . . . . . . 8 𝑞((𝐹𝑚)‘𝑥)
21 nfcv 2904 . . . . . . . . 9 𝑚𝑥
2211, 21nffv 6853 . . . . . . . 8 𝑚((𝐹𝑞)‘𝑥)
2313fveq1d 6845 . . . . . . . 8 (𝑚 = 𝑞 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑞)‘𝑥))
2420, 22, 23cbvmpt 5217 . . . . . . 7 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))
2524fveq2i 6846 . . . . . 6 (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
2625eleq1i 2825 . . . . 5 ((lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ)
2719, 26anbi12i 628 . . . 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 3409 . . 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 2904 . . . . 5 𝑥𝑍
30 nfcv 2904 . . . . . 6 𝑥(ℤ𝑗)
31 smflimsup.x . . . . . . . 8 𝑥𝐹
32 nfcv 2904 . . . . . . . 8 𝑥𝑞
3331, 32nffv 6853 . . . . . . 7 𝑥(𝐹𝑞)
3433nfdm 5907 . . . . . 6 𝑥dom (𝐹𝑞)
3530, 34nfiin 4986 . . . . 5 𝑥 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
3629, 35nfiun 4985 . . . 4 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
37 nfcv 2904 . . . 4 𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
38 nfv 1918 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ
39 nfcv 2904 . . . . . 6 𝑥lim sup
40 nfcv 2904 . . . . . . . 8 𝑥𝑤
4133, 40nffv 6853 . . . . . . 7 𝑥((𝐹𝑞)‘𝑤)
4229, 41nfmpt 5213 . . . . . 6 𝑥(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))
4339, 42nffv 6853 . . . . 5 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
44 nfcv 2904 . . . . 5 𝑥
4543, 44nfel 2918 . . . 4 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ
46 fveq2 6843 . . . . . . 7 (𝑥 = 𝑤 → ((𝐹𝑞)‘𝑥) = ((𝐹𝑞)‘𝑤))
4746mpteq2dv 5208 . . . . . 6 (𝑥 = 𝑤 → (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
4847fveq2d 6847 . . . . 5 (𝑥 = 𝑤 → (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
4948eleq1d 2819 . . . 4 (𝑥 = 𝑤 → ((lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ))
5036, 37, 38, 45, 49cbvrabw 3438 . . 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 2765 . 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 5211 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))))
54 nfrab1 3425 . . . . 5 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
555, 54nfcxfr 2902 . . . 4 𝑥𝐷
56 nfcv 2904 . . . 4 𝑤𝐷
57 nfcv 2904 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
5855, 56, 57, 43, 48cbvmptf 5215 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))) = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
5952, 53, 583eqtri 2765 . 2 𝐺 = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
60 nfcv 2904 . . . . . . 7 𝑥(ℤ𝑖)
6160, 34nfiin 4986 . . . . . 6 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
62 nfcv 2904 . . . . . 6 𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
63 nfv 1918 . . . . . 6 𝑤sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
6460, 41nfmpt 5213 . . . . . . . . 9 𝑥(𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
6564nfrn 5908 . . . . . . . 8 𝑥ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
66 nfcv 2904 . . . . . . . 8 𝑥*
67 nfcv 2904 . . . . . . . 8 𝑥 <
6865, 66, 67nfsup 9392 . . . . . . 7 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
6968, 44nfel 2918 . . . . . 6 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
7046mpteq2dv 5208 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7170rneqd 5894 . . . . . . . 8 (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7271supeq1d 9387 . . . . . . 7 (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
7372eleq1d 2819 . . . . . 6 (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
7461, 62, 63, 69, 73cbvrabw 3438 . . . . 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 6843 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7776iineq1d 43388 . . . . . . 7 (𝑖 = 𝑘 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) = 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞))
7877eleq2d 2820 . . . . . 6 (𝑖 = 𝑘 → (𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ↔ 𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞)))
7976mpteq1d 5201 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8079rneqd 5894 . . . . . . . 8 (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8180supeq1d 9387 . . . . . . 7 (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
8281eleq1d 2819 . . . . . 6 (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
8378, 82anbi12d 632 . . . . 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 3408 . . . 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 2773 . . 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 5219 . 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 6843 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝐹𝑝)‘𝑦) = ((𝐹𝑝)‘𝑤))
8887mpteq2dv 5208 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
8988rneqd 5894 . . . . . . . 8 (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
9089supeq1d 9387 . . . . . . 7 (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
9190cbvmptv 5219 . . . . . 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 6843 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
9392dmeqd 5862 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → dom (𝐹𝑝) = dom (𝐹𝑞))
9493cbviinv 5002 . . . . . . . . . . . 12 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) = 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
9594eleq2i 2826 . . . . . . . . . . 11 (𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ↔ 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞))
96 nfcv 2904 . . . . . . . . . . . . . . 15 𝑞((𝐹𝑝)‘𝑥)
97 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑝(𝐹𝑞)
98 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑝𝑥
9997, 98nffv 6853 . . . . . . . . . . . . . . 15 𝑝((𝐹𝑞)‘𝑥)
10092fveq1d 6845 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑥) = ((𝐹𝑞)‘𝑥))
10196, 99, 100cbvmpt 5217 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
102101rneqi 5893 . . . . . . . . . . . . 13 ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
103102supeq1i 9388 . . . . . . . . . . . 12 sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < )
104103eleq1i 2825 . . . . . . . . . . 11 (sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
10595, 104anbi12i 628 . . . . . . . . . 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 3409 . . . . . . . . 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 5211 . . . . . . . 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 6844 . . . . . . 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 6845 . . . . . . . . . 10 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑤) = ((𝐹𝑞)‘𝑤))
110109cbvmptv 5219 . . . . . . . . 9 (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
111110rneqi 5893 . . . . . . . 8 ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
112111supeq1i 9388 . . . . . . 7 sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
113108, 112mpteq12i 5212 . . . . . 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 2761 . . . . 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 6843 . . . . 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 6843 . . . . . . . 8 (𝑙 = 𝑘 → (ℤ𝑙) = (ℤ𝑘))
118117mpteq1d 5201 . . . . . . 7 (𝑙 = 𝑘 → (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
119118rneqd 5894 . . . . . 6 (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
120119supeq1d 9387 . . . . 5 (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
121116, 120mpteq12dv 5197 . . . 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 2773 . . 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 5219 . 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 45154 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wnfc 2884  {crab 3406   ciun 4955   ciin 4956  cmpt 5189  dom cdm 5634  ran crn 5635  wf 6493  cfv 6497  supcsup 9381  cr 11055  *cxr 11193   < clt 11194  cz 12504  cuz 12768  lim supclsp 15358  SAlgcsalg 44635  SMblFncsmblfn 45022
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cc 10376  ax-ac2 10404  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-omul 8418  df-er 8651  df-map 8770  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-acn 9883  df-ac 10057  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-ioo 13274  df-ioc 13275  df-ico 13276  df-fz 13431  df-fl 13703  df-ceil 13704  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-limsup 15359  df-clim 15376  df-rlim 15377  df-rest 17309  df-topgen 17330  df-top 22259  df-bases 22312  df-salg 44636  df-salgen 44640  df-smblfn 45023
This theorem is referenced by:  smflimsupmpt  45156
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