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Theorem smflim 41483
Description: The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
smflim.n 𝑚𝐹
smflim.x 𝑥𝐹
smflim.m (𝜑𝑀 ∈ ℤ)
smflim.z 𝑍 = (ℤ𝑀)
smflim.s (𝜑𝑆 ∈ SAlg)
smflim.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflim.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
smflim.g 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
Assertion
Ref Expression
smflim (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝑛,𝐹   𝑆,𝑚,𝑛   𝑚,𝑍,𝑥,𝑛   𝜑,𝑚,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥)   𝐹(𝑥,𝑚)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑚,𝑛)

Proof of Theorem smflim
Dummy variables 𝑖 𝑗 𝑙 𝑦 𝑘 𝑠 𝑡 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1984 . 2 𝑎𝜑
2 smflim.s . 2 (𝜑𝑆 ∈ SAlg)
3 smflim.d . . . . 5 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
4 nfcv 2894 . . . . . . 7 𝑥𝑍
5 nfcv 2894 . . . . . . . 8 𝑥(ℤ𝑛)
6 smflim.x . . . . . . . . . 10 𝑥𝐹
7 nfcv 2894 . . . . . . . . . 10 𝑥𝑚
86, 7nffv 6351 . . . . . . . . 9 𝑥(𝐹𝑚)
98nfdm 5514 . . . . . . . 8 𝑥dom (𝐹𝑚)
105, 9nfiin 4693 . . . . . . 7 𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
114, 10nfiun 4692 . . . . . 6 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1211ssrab2f 39791 . . . . 5 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
133, 12eqsstri 3768 . . . 4 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1413a1i 11 . . 3 (𝜑𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
15 uzssz 11891 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
16 smflim.z . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
1716eleq2i 2823 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1817biimpi 206 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1915, 18sseldi 3734 . . . . . . . 8 (𝑛𝑍𝑛 ∈ ℤ)
20 uzid 11886 . . . . . . . 8 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
2119, 20syl 17 . . . . . . 7 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
2221adantl 473 . . . . . 6 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑛))
232adantr 472 . . . . . . 7 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
24 smflim.f . . . . . . . 8 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
2524ffvelrnda 6514 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
26 eqid 2752 . . . . . . 7 dom (𝐹𝑛) = dom (𝐹𝑛)
2723, 25, 26smfdmss 41440 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ⊆ 𝑆)
28 smflim.n . . . . . . . . . 10 𝑚𝐹
29 nfcv 2894 . . . . . . . . . 10 𝑚𝑛
3028, 29nffv 6351 . . . . . . . . 9 𝑚(𝐹𝑛)
3130nfdm 5514 . . . . . . . 8 𝑚dom (𝐹𝑛)
32 nfcv 2894 . . . . . . . 8 𝑚 𝑆
3331, 32nfss 3729 . . . . . . 7 𝑚dom (𝐹𝑛) ⊆ 𝑆
34 fveq2 6344 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534dmeqd 5473 . . . . . . . 8 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
3635sseq1d 3765 . . . . . . 7 (𝑚 = 𝑛 → (dom (𝐹𝑚) ⊆ 𝑆 ↔ dom (𝐹𝑛) ⊆ 𝑆))
3733, 36rspce 3436 . . . . . 6 ((𝑛 ∈ (ℤ𝑛) ∧ dom (𝐹𝑛) ⊆ 𝑆) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
3822, 27, 37syl2anc 696 . . . . 5 ((𝜑𝑛𝑍) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
39 iinss 4715 . . . . 5 (∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4038, 39syl 17 . . . 4 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4140iunssd 39762 . . 3 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4214, 41sstrd 3746 . 2 (𝜑𝐷 𝑆)
43 nfv 1984 . . . . 5 𝑚𝜑
44 nfcv 2894 . . . . . 6 𝑚𝑦
45 nfmpt1 4891 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))
46 nfcv 2894 . . . . . . . . 9 𝑚dom ⇝
4745, 46nfel 2907 . . . . . . . 8 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
48 nfcv 2894 . . . . . . . . 9 𝑚𝑍
49 nfii1 4695 . . . . . . . . 9 𝑚 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5048, 49nfiun 4692 . . . . . . . 8 𝑚 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5147, 50nfrab 3254 . . . . . . 7 𝑚{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
523, 51nfcxfr 2892 . . . . . 6 𝑚𝐷
5344, 52nfel 2907 . . . . 5 𝑚 𝑦𝐷
5443, 53nfan 1969 . . . 4 𝑚(𝜑𝑦𝐷)
55 nfcv 2894 . . . 4 𝑤𝐹
562adantr 472 . . . . . 6 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
5724ffvelrnda 6514 . . . . . 6 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
58 eqid 2752 . . . . . 6 dom (𝐹𝑚) = dom (𝐹𝑚)
5956, 57, 58smff 41439 . . . . 5 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
6059adantlr 753 . . . 4 (((𝜑𝑦𝐷) ∧ 𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
61 nfcv 2894 . . . . . . 7 𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
62 nfv 1984 . . . . . . 7 𝑦(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
63 nfcv 2894 . . . . . . . . . 10 𝑥𝑦
648, 63nffv 6351 . . . . . . . . 9 𝑥((𝐹𝑚)‘𝑦)
654, 64nfmpt 4890 . . . . . . . 8 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
6665nfel1 2909 . . . . . . 7 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝
67 fveq2 6344 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6867mpteq2dv 4889 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
6968eleq1d 2816 . . . . . . 7 (𝑥 = 𝑦 → ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ))
7011, 61, 62, 66, 69cbvrab 3330 . . . . . 6 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ }
71 nfcv 2894 . . . . . . . . . . . . 13 𝑙dom (𝐹𝑚)
72 nfcv 2894 . . . . . . . . . . . . . . 15 𝑚𝑙
7328, 72nffv 6351 . . . . . . . . . . . . . 14 𝑚(𝐹𝑙)
7473nfdm 5514 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑙)
75 fveq2 6344 . . . . . . . . . . . . . 14 (𝑚 = 𝑙 → (𝐹𝑚) = (𝐹𝑙))
7675dmeqd 5473 . . . . . . . . . . . . 13 (𝑚 = 𝑙 → dom (𝐹𝑚) = dom (𝐹𝑙))
7771, 74, 76cbviin 4702 . . . . . . . . . . . 12 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙)
7877a1i 11 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙))
79 fveq2 6344 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
80 eqidd 2753 . . . . . . . . . . . 12 ((𝑛 = 𝑖𝑙 ∈ (ℤ𝑖)) → dom (𝐹𝑙) = dom (𝐹𝑙))
8179, 80iineq12dv 39780 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8278, 81eqtrd 2786 . . . . . . . . . 10 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8382cbviunv 4703 . . . . . . . . 9 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙)
8483eleq2i 2823 . . . . . . . 8 (𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
85 nfcv 2894 . . . . . . . . . 10 𝑙𝑍
86 nfcv 2894 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑦)
8773, 44nffv 6351 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑦)
8875fveq1d 6346 . . . . . . . . . 10 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑙)‘𝑦))
8948, 85, 86, 87, 88cbvmptf 4892 . . . . . . . . 9 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
9089eleq1i 2822 . . . . . . . 8 ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ )
9184, 90anbi12i 735 . . . . . . 7 ((𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ))
9291rabbia2 3319 . . . . . 6 {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
933, 70, 923eqtri 2778 . . . . 5 𝐷 = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
94 fveq2 6344 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑤))
9594mpteq2dv 4889 . . . . . . . 8 (𝑦 = 𝑤 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)))
9695eleq1d 2816 . . . . . . 7 (𝑦 = 𝑤 → ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ))
9796cbvrabv 3331 . . . . . 6 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ }
98 fveq2 6344 . . . . . . . . . . . . 13 (𝑙 = 𝑚 → (𝐹𝑙) = (𝐹𝑚))
9998dmeqd 5473 . . . . . . . . . . . 12 (𝑙 = 𝑚 → dom (𝐹𝑙) = dom (𝐹𝑚))
10074, 71, 99cbviin 4702 . . . . . . . . . . 11 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
101100a1i 11 . . . . . . . . . 10 (𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
102101iuneq2i 4683 . . . . . . . . 9 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
103102eleq2i 2823 . . . . . . . 8 (𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ↔ 𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
104 nfcv 2894 . . . . . . . . . . 11 𝑚𝑤
10573, 104nffv 6351 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑤)
106 nfcv 2894 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑤)
10798fveq1d 6346 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑤) = ((𝐹𝑚)‘𝑤))
10885, 48, 105, 106, 107cbvmptf 4892 . . . . . . . . 9 (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤))
109108eleq1i 2822 . . . . . . . 8 ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ )
110103, 109anbi12i 735 . . . . . . 7 ((𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ) ↔ (𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ ))
111110rabbia2 3319 . . . . . 6 {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11297, 111eqtri 2774 . . . . 5 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11393, 112eqtri 2774 . . . 4 𝐷 = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
114 simpr 479 . . . 4 ((𝜑𝑦𝐷) → 𝑦𝐷)
11554, 28, 55, 16, 60, 113, 114fnlimfvre 40401 . . 3 ((𝜑𝑦𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) ∈ ℝ)
116 smflim.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
117 nfrab1 3253 . . . . . 6 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
1183, 117nfcxfr 2892 . . . . 5 𝑥𝐷
119 nfcv 2894 . . . . 5 𝑦𝐷
120 nfcv 2894 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
121 nfcv 2894 . . . . . 6 𝑥
122121, 65nffv 6351 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
12368fveq2d 6348 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
124118, 119, 120, 122, 123cbvmptf 4892 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
125116, 124eqtri 2774 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
126115, 125fmptd 6540 . 2 (𝜑𝐺:𝐷⟶ℝ)
127 smflim.m . . . 4 (𝜑𝑀 ∈ ℤ)
128127adantr 472 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑀 ∈ ℤ)
1292adantr 472 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
13024adantr 472 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆))
131 nfcv 2894 . . . . . . . . 9 𝑥𝑙
1326, 131nffv 6351 . . . . . . . 8 𝑥(𝐹𝑙)
133132, 63nffv 6351 . . . . . . 7 𝑥((𝐹𝑙)‘𝑦)
1344, 133nfmpt 4890 . . . . . 6 𝑥(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
135121, 134nffv 6351 . . . . 5 𝑥( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
136 nfcv 2894 . . . . . . . . 9 𝑙((𝐹𝑚)‘𝑥)
137 nfcv 2894 . . . . . . . . . 10 𝑚𝑥
13873, 137nffv 6351 . . . . . . . . 9 𝑚((𝐹𝑙)‘𝑥)
13975fveq1d 6346 . . . . . . . . 9 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑙)‘𝑥))
14048, 85, 136, 138, 139cbvmptf 4892 . . . . . . . 8 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥))
141140a1i 11 . . . . . . 7 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)))
142 simpl 474 . . . . . . . . 9 ((𝑥 = 𝑦𝑙𝑍) → 𝑥 = 𝑦)
143142fveq2d 6348 . . . . . . . 8 ((𝑥 = 𝑦𝑙𝑍) → ((𝐹𝑙)‘𝑥) = ((𝐹𝑙)‘𝑦))
144143mpteq2dva 4888 . . . . . . 7 (𝑥 = 𝑦 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
145141, 144eqtrd 2786 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
146145fveq2d 6348 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
147118, 119, 120, 135, 146cbvmptf 4892 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
148116, 147eqtri 2774 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
149 simpr 479 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150 nfcv 2894 . . . . . . . . 9 𝑚 <
151 nfcv 2894 . . . . . . . . 9 𝑚(𝑎 + (1 / 𝑗))
15287, 150, 151nfbr 4843 . . . . . . . 8 𝑚((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153152, 74nfrab 3254 . . . . . . 7 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154 nfcv 2894 . . . . . . . 8 𝑚𝑡
155154, 74nfin 3955 . . . . . . 7 𝑚(𝑡 ∩ dom (𝐹𝑙))
156153, 155nfeq 2906 . . . . . 6 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))
157 nfcv 2894 . . . . . 6 𝑚𝑆
158156, 157nfrab 3254 . . . . 5 𝑚{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
159 nfcv 2894 . . . . 5 𝑘{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
160 nfcv 2894 . . . . 5 𝑙{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
161 nfcv 2894 . . . . 5 𝑗{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
162 nfcv 2894 . . . . . . . . . . . 12 𝑦dom (𝐹𝑙)
163132nfdm 5514 . . . . . . . . . . . 12 𝑥dom (𝐹𝑙)
164 nfcv 2894 . . . . . . . . . . . . 13 𝑥 <
165 nfcv 2894 . . . . . . . . . . . . 13 𝑥(𝑎 + (1 / 𝑗))
166133, 164, 165nfbr 4843 . . . . . . . . . . . 12 𝑥((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167 nfv 1984 . . . . . . . . . . . 12 𝑦((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168 fveq2 6344 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑥))
169168breq1d 4806 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170162, 163, 166, 167, 169cbvrab 3330 . . . . . . . . . . 11 {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))}
171170a1i 11 . . . . . . . . . 10 (𝑡 = 𝑠 → {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))})
172 ineq1 3942 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑙)))
173171, 172eqeq12d 2767 . . . . . . . . 9 (𝑡 = 𝑠 → ({𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙)) ↔ {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑙))))
174173cbvrabv 3331 . . . . . . . 8 {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑙))}
175174a1i 11 . . . . . . 7 (𝑙 = 𝑚 → {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑙))})
17699eleq2d 2817 . . . . . . . . . . 11 (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹𝑙) ↔ 𝑥 ∈ dom (𝐹𝑚)))
17798fveq1d 6346 . . . . . . . . . . . 12 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑥) = ((𝐹𝑚)‘𝑥))
178177breq1d 4806 . . . . . . . . . . 11 (𝑙 = 𝑚 → (((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179176, 178anbi12d 749 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝑥 ∈ dom (𝐹𝑙) ∧ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))) ↔ (𝑥 ∈ dom (𝐹𝑚) ∧ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)))))
180179rabbidva2 3318 . . . . . . . . 9 (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
18199ineq2d 3949 . . . . . . . . 9 (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑚)))
182180, 181eqeq12d 2767 . . . . . . . 8 (𝑙 = 𝑚 → ({𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑙)) ↔ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑚))))
183182rabbidv 3321 . . . . . . 7 (𝑙 = 𝑚 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑙))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑚))})
184175, 183eqtrd 2786 . . . . . 6 (𝑙 = 𝑚 → {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑚))})
185 oveq2 6813 . . . . . . . . . . 11 (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186185oveq2d 6821 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187186breq2d 4808 . . . . . . . . 9 (𝑗 = 𝑘 → (((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188187rabbidv 3321 . . . . . . . 8 (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189188eqeq1d 2754 . . . . . . 7 (𝑗 = 𝑘 → ({𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑚)) ↔ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))))
190189rabbidv 3321 . . . . . 6 (𝑗 = 𝑘 → {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹𝑚))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
191184, 190sylan9eq 2806 . . . . 5 ((𝑙 = 𝑚𝑗 = 𝑘) → {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))} = {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
192158, 159, 160, 161, 191cbvmpt2 6891 . . . 4 (𝑙𝑍, 𝑗 ∈ ℕ ↦ {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}) = (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))})
193192eqcomi 2761 . . 3 (𝑚𝑍, 𝑘 ∈ ℕ ↦ {𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}) = (𝑙𝑍, 𝑗 ∈ ℕ ↦ {𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))})
194128, 16, 129, 130, 93, 148, 149, 193smflimlem6 41482 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑦𝐷 ∣ (𝐺𝑦) ≤ 𝑎} ∈ (𝑆t 𝐷))
1951, 2, 42, 126, 194issmfled 41464 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wnfc 2881  wrex 3043  {crab 3046  cin 3706  wss 3707   cuni 4580   ciun 4664   ciin 4665   class class class wbr 4796  cmpt 4873  dom cdm 5258  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807  cr 10119  1c1 10121   + caddc 10123   < clt 10258   / cdiv 10868  cn 11204  cz 11561  cuz 11871  cli 14406  SAlgcsalg 41023  SMblFncsmblfn 41407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cc 9441  ax-ac2 9469  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-omul 7726  df-er 7903  df-map 8017  df-pm 8018  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-acn 8950  df-ac 9121  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-ioo 12364  df-ico 12366  df-fl 12779  df-seq 12988  df-exp 13047  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-rest 16277  df-salg 41024  df-smblfn 41408
This theorem is referenced by:  smflim2  41510
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