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Theorem smflimsup 41457
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 6304 . . . . . . . . 9 (𝑛 = 𝑗 → (ℤ𝑛) = (ℤ𝑗))
76iineq1d 39683 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚))
8 nfcv 2866 . . . . . . . . . 10 𝑞dom (𝐹𝑚)
9 smflimsup.n . . . . . . . . . . . 12 𝑚𝐹
10 nfcv 2866 . . . . . . . . . . . 12 𝑚𝑞
119, 10nffv 6311 . . . . . . . . . . 11 𝑚(𝐹𝑞)
1211nfdm 5474 . . . . . . . . . 10 𝑚dom (𝐹𝑞)
13 fveq2 6304 . . . . . . . . . . 11 (𝑚 = 𝑞 → (𝐹𝑚) = (𝐹𝑞))
1413dmeqd 5433 . . . . . . . . . 10 (𝑚 = 𝑞 → dom (𝐹𝑚) = dom (𝐹𝑞))
158, 12, 14cbviin 4666 . . . . . . . . 9 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1615a1i 11 . . . . . . . 8 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑗)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
177, 16eqtrd 2758 . . . . . . 7 (𝑛 = 𝑗 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
1817cbviunv 4667 . . . . . 6 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
1918eleq2i 2795 . . . . 5 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞))
20 nfcv 2866 . . . . . . . 8 𝑞((𝐹𝑚)‘𝑥)
21 nfcv 2866 . . . . . . . . 9 𝑚𝑥
2211, 21nffv 6311 . . . . . . . 8 𝑚((𝐹𝑞)‘𝑥)
2313fveq1d 6306 . . . . . . . 8 (𝑚 = 𝑞 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑞)‘𝑥))
2420, 22, 23cbvmpt 4857 . . . . . . 7 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))
2524fveq2i 6307 . . . . . 6 (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
2625eleq1i 2794 . . . . 5 ((lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ)
2719, 26anbi12i 735 . . . 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 3291 . . 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 2866 . . . . 5 𝑥𝑍
30 nfcv 2866 . . . . . 6 𝑥(ℤ𝑗)
31 smflimsup.x . . . . . . . 8 𝑥𝐹
32 nfcv 2866 . . . . . . . 8 𝑥𝑞
3331, 32nffv 6311 . . . . . . 7 𝑥(𝐹𝑞)
3433nfdm 5474 . . . . . 6 𝑥dom (𝐹𝑞)
3530, 34nfiin 4657 . . . . 5 𝑥 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
3629, 35nfiun 4656 . . . 4 𝑥 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
37 nfcv 2866 . . . 4 𝑤 𝑗𝑍 𝑞 ∈ (ℤ𝑗)dom (𝐹𝑞)
38 nfv 1956 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ
39 nfcv 2866 . . . . . 6 𝑥lim sup
40 nfcv 2866 . . . . . . . 8 𝑥𝑤
4133, 40nffv 6311 . . . . . . 7 𝑥((𝐹𝑞)‘𝑤)
4229, 41nfmpt 4854 . . . . . 6 𝑥(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))
4339, 42nffv 6311 . . . . 5 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
44 nfcv 2866 . . . . 5 𝑥
4543, 44nfel 2879 . . . 4 𝑥(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ
46 fveq2 6304 . . . . . . 7 (𝑥 = 𝑤 → ((𝐹𝑞)‘𝑥) = ((𝐹𝑞)‘𝑤))
4746mpteq2dv 4853 . . . . . 6 (𝑥 = 𝑤 → (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)) = (𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤)))
4847fveq2d 6308 . . . . 5 (𝑥 = 𝑤 → (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) = (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
4948eleq1d 2788 . . . 4 (𝑥 = 𝑤 → ((lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))) ∈ ℝ))
5036, 37, 38, 45, 49cbvrab 3302 . . 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 2750 . 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 4849 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥))))
54 nfrab1 3225 . . . . 5 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) ∈ ℝ}
555, 54nfcxfr 2864 . . . 4 𝑥𝐷
56 nfcv 2866 . . . 4 𝑤𝐷
57 nfcv 2866 . . . 4 𝑤(lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))
5855, 56, 57, 43, 48cbvmptf 4856 . . 3 (𝑥𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑥)))) = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
5952, 53, 583eqtri 2750 . 2 𝐺 = (𝑤𝐷 ↦ (lim sup‘(𝑞𝑍 ↦ ((𝐹𝑞)‘𝑤))))
60 nfcv 2866 . . . . . . 7 𝑥(ℤ𝑖)
6160, 34nfiin 4657 . . . . . 6 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
62 nfcv 2866 . . . . . 6 𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
63 nfv 1956 . . . . . 6 𝑤sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
6460, 41nfmpt 4854 . . . . . . . . 9 𝑥(𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
6564nfrn 5475 . . . . . . . 8 𝑥ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤))
66 nfcv 2866 . . . . . . . 8 𝑥*
67 nfcv 2866 . . . . . . . 8 𝑥 <
6865, 66, 67nfsup 8473 . . . . . . 7 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
6968, 44nfel 2879 . . . . . 6 𝑥sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
7046mpteq2dv 4853 . . . . . . . . 9 (𝑥 = 𝑤 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7170rneqd 5460 . . . . . . . 8 (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)))
7271supeq1d 8468 . . . . . . 7 (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
7372eleq1d 2788 . . . . . 6 (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
7461, 62, 63, 69, 73cbvrab 3302 . . . . 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 6304 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7776iineq1d 39683 . . . . . . 7 (𝑖 = 𝑘 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) = 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞))
7877eleq2d 2789 . . . . . 6 (𝑖 = 𝑘 → (𝑤 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞) ↔ 𝑤 𝑞 ∈ (ℤ𝑘)dom (𝐹𝑞)))
7976mpteq1d 4846 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8079rneqd 5460 . . . . . . . 8 (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
8180supeq1d 8468 . . . . . . 7 (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
8281eleq1d 2788 . . . . . 6 (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
8378, 82anbi12d 749 . . . . 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 3290 . . . 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 2758 . . 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 4858 . 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 6304 . . . . . . . . . 10 (𝑦 = 𝑤 → ((𝐹𝑝)‘𝑦) = ((𝐹𝑝)‘𝑤))
8887mpteq2dv 4853 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
8988rneqd 5460 . . . . . . . 8 (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)))
9089supeq1d 8468 . . . . . . 7 (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ))
9190cbvmptv 4858 . . . . . 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 6304 . . . . . . . . . . . . . 14 (𝑝 = 𝑞 → (𝐹𝑝) = (𝐹𝑞))
9392dmeqd 5433 . . . . . . . . . . . . 13 (𝑝 = 𝑞 → dom (𝐹𝑝) = dom (𝐹𝑞))
9493cbviinv 4668 . . . . . . . . . . . 12 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) = 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞)
9594eleq2i 2795 . . . . . . . . . . 11 (𝑥 𝑝 ∈ (ℤ𝑖)dom (𝐹𝑝) ↔ 𝑥 𝑞 ∈ (ℤ𝑖)dom (𝐹𝑞))
96 nfcv 2866 . . . . . . . . . . . . . . 15 𝑞((𝐹𝑝)‘𝑥)
97 nfcv 2866 . . . . . . . . . . . . . . . 16 𝑝(𝐹𝑞)
98 nfcv 2866 . . . . . . . . . . . . . . . 16 𝑝𝑥
9997, 98nffv 6311 . . . . . . . . . . . . . . 15 𝑝((𝐹𝑞)‘𝑥)
10092fveq1d 6306 . . . . . . . . . . . . . . 15 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑥) = ((𝐹𝑞)‘𝑥))
10196, 99, 100cbvmpt 4857 . . . . . . . . . . . . . 14 (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
102101rneqi 5459 . . . . . . . . . . . . 13 ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥))
103102supeq1i 8469 . . . . . . . . . . . 12 sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < )
104103eleq1i 2794 . . . . . . . . . . 11 (sup(ran (𝑝 ∈ (ℤ𝑖) ↦ ((𝐹𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ𝑖) ↦ ((𝐹𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
10595, 104anbi12i 735 . . . . . . . . . 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 3291 . . . . . . . . 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 4849 . . . . . . . 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 6305 . . . . . . 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 6306 . . . . . . . . . 10 (𝑝 = 𝑞 → ((𝐹𝑝)‘𝑤) = ((𝐹𝑞)‘𝑤))
110109cbvmptv 4858 . . . . . . . . 9 (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
111110rneqi 5459 . . . . . . . 8 ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤))
112111supeq1i 8469 . . . . . . 7 sup(ran (𝑝 ∈ (ℤ𝑙) ↦ ((𝐹𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < )
113108, 112mpteq12i 4850 . . . . . 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 2746 . . . . 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 6304 . . . . 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 6304 . . . . . . . 8 (𝑙 = 𝑘 → (ℤ𝑙) = (ℤ𝑘))
118117mpteq1d 4846 . . . . . . 7 (𝑙 = 𝑘 → (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
119118rneqd 5460 . . . . . 6 (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)))
120119supeq1d 8468 . . . . 5 (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ𝑙) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ𝑘) ↦ ((𝐹𝑞)‘𝑤)), ℝ*, < ))
121116, 120mpteq12dv 4841 . . . 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 2758 . . 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 4858 . 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 41456 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103  wnfc 2853  {crab 3018   ciun 4628   ciin 4629  cmpt 4837  dom cdm 5218  ran crn 5219  wf 5997  cfv 6001  supcsup 8462  cr 10048  *cxr 10186   < clt 10187  cz 11490  cuz 11800  lim supclsp 14321  SAlgcsalg 40948  SMblFncsmblfn 41332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cc 9370  ax-ac2 9398  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126  ax-pre-sup 10127
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-iin 4631  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-omul 7685  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-sup 8464  df-inf 8465  df-oi 8531  df-card 8878  df-acn 8881  df-ac 9052  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-div 10798  df-nn 11134  df-2 11192  df-3 11193  df-n0 11406  df-z 11491  df-uz 11801  df-q 11903  df-rp 11947  df-ioo 12293  df-ioc 12294  df-ico 12295  df-fz 12441  df-fl 12708  df-ceil 12709  df-seq 12917  df-exp 12976  df-cj 13959  df-re 13960  df-im 13961  df-sqrt 14095  df-abs 14096  df-limsup 14322  df-clim 14339  df-rlim 14340  df-rest 16206  df-topgen 16227  df-top 20822  df-bases 20873  df-salg 40949  df-salgen 40953  df-smblfn 41333
This theorem is referenced by:  smflimsupmpt  41458
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