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Theorem smflim 42615
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 1892 . 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 2949 . . . . . . 7 𝑥𝑍
5 nfcv 2949 . . . . . . . 8 𝑥(ℤ𝑛)
6 smflim.x . . . . . . . . . 10 𝑥𝐹
7 nfcv 2949 . . . . . . . . . 10 𝑥𝑚
86, 7nffv 6548 . . . . . . . . 9 𝑥(𝐹𝑚)
98nfdm 5705 . . . . . . . 8 𝑥dom (𝐹𝑚)
105, 9nfiin 4855 . . . . . . 7 𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
114, 10nfiun 4854 . . . . . 6 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1211ssrab2f 40942 . . . . 5 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
133, 12eqsstri 3922 . . . 4 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1413a1i 11 . . 3 (𝜑𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
15 uzssz 12113 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
16 smflim.z . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
1716eleq2i 2874 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1817biimpi 217 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1915, 18sseldi 3887 . . . . . . . 8 (𝑛𝑍𝑛 ∈ ℤ)
20 uzid 12108 . . . . . . . 8 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
2119, 20syl 17 . . . . . . 7 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
2221adantl 482 . . . . . 6 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑛))
232adantr 481 . . . . . . 7 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
24 smflim.f . . . . . . . 8 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
2524ffvelrnda 6716 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
26 eqid 2795 . . . . . . 7 dom (𝐹𝑛) = dom (𝐹𝑛)
2723, 25, 26smfdmss 42572 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ⊆ 𝑆)
28 smflim.n . . . . . . . . . 10 𝑚𝐹
29 nfcv 2949 . . . . . . . . . 10 𝑚𝑛
3028, 29nffv 6548 . . . . . . . . 9 𝑚(𝐹𝑛)
3130nfdm 5705 . . . . . . . 8 𝑚dom (𝐹𝑛)
32 nfcv 2949 . . . . . . . 8 𝑚 𝑆
3331, 32nfss 3882 . . . . . . 7 𝑚dom (𝐹𝑛) ⊆ 𝑆
34 fveq2 6538 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534dmeqd 5660 . . . . . . . 8 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
3635sseq1d 3919 . . . . . . 7 (𝑚 = 𝑛 → (dom (𝐹𝑚) ⊆ 𝑆 ↔ dom (𝐹𝑛) ⊆ 𝑆))
3733, 36rspce 3554 . . . . . 6 ((𝑛 ∈ (ℤ𝑛) ∧ dom (𝐹𝑛) ⊆ 𝑆) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
3822, 27, 37syl2anc 584 . . . . 5 ((𝜑𝑛𝑍) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
39 iinss 4879 . . . . 5 (∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4038, 39syl 17 . . . 4 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4140iunssd 4873 . . 3 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4214, 41sstrd 3899 . 2 (𝜑𝐷 𝑆)
43 nfv 1892 . . . . 5 𝑚𝜑
44 nfcv 2949 . . . . . 6 𝑚𝑦
45 nfmpt1 5058 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))
46 nfcv 2949 . . . . . . . . 9 𝑚dom ⇝
4745, 46nfel 2961 . . . . . . . 8 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
48 nfcv 2949 . . . . . . . . 9 𝑚𝑍
49 nfii1 4857 . . . . . . . . 9 𝑚 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5048, 49nfiun 4854 . . . . . . . 8 𝑚 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5147, 50nfrab 3345 . . . . . . 7 𝑚{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
523, 51nfcxfr 2947 . . . . . 6 𝑚𝐷
5344, 52nfel 2961 . . . . 5 𝑚 𝑦𝐷
5443, 53nfan 1881 . . . 4 𝑚(𝜑𝑦𝐷)
55 nfcv 2949 . . . 4 𝑤𝐹
562adantr 481 . . . . . 6 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
5724ffvelrnda 6716 . . . . . 6 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
58 eqid 2795 . . . . . 6 dom (𝐹𝑚) = dom (𝐹𝑚)
5956, 57, 58smff 42571 . . . . 5 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
6059adantlr 711 . . . 4 (((𝜑𝑦𝐷) ∧ 𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
61 nfcv 2949 . . . . . . 7 𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
62 nfv 1892 . . . . . . 7 𝑦(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
63 nfcv 2949 . . . . . . . . . 10 𝑥𝑦
648, 63nffv 6548 . . . . . . . . 9 𝑥((𝐹𝑚)‘𝑦)
654, 64nfmpt 5057 . . . . . . . 8 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
6665nfel1 2963 . . . . . . 7 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝
67 fveq2 6538 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6867mpteq2dv 5056 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
6968eleq1d 2867 . . . . . . 7 (𝑥 = 𝑦 → ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ))
7011, 61, 62, 66, 69cbvrab 3433 . . . . . 6 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ }
71 nfcv 2949 . . . . . . . . . . . . 13 𝑙dom (𝐹𝑚)
72 nfcv 2949 . . . . . . . . . . . . . . 15 𝑚𝑙
7328, 72nffv 6548 . . . . . . . . . . . . . 14 𝑚(𝐹𝑙)
7473nfdm 5705 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑙)
75 fveq2 6538 . . . . . . . . . . . . . 14 (𝑚 = 𝑙 → (𝐹𝑚) = (𝐹𝑙))
7675dmeqd 5660 . . . . . . . . . . . . 13 (𝑚 = 𝑙 → dom (𝐹𝑚) = dom (𝐹𝑙))
7771, 74, 76cbviin 4865 . . . . . . . . . . . 12 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙)
7877a1i 11 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙))
79 fveq2 6538 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
80 eqidd 2796 . . . . . . . . . . . 12 ((𝑛 = 𝑖𝑙 ∈ (ℤ𝑖)) → dom (𝐹𝑙) = dom (𝐹𝑙))
8179, 80iineq12dv 40931 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8278, 81eqtrd 2831 . . . . . . . . . 10 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8382cbviunv 4866 . . . . . . . . 9 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙)
8483eleq2i 2874 . . . . . . . 8 (𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
85 nfcv 2949 . . . . . . . . . 10 𝑙𝑍
86 nfcv 2949 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑦)
8773, 44nffv 6548 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑦)
8875fveq1d 6540 . . . . . . . . . 10 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑙)‘𝑦))
8948, 85, 86, 87, 88cbvmptf 5059 . . . . . . . . 9 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
9089eleq1i 2873 . . . . . . . 8 ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ )
9184, 90anbi12i 626 . . . . . . 7 ((𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ))
9291rabbia2 3423 . . . . . 6 {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
933, 70, 923eqtri 2823 . . . . 5 𝐷 = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
94 fveq2 6538 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑤))
9594mpteq2dv 5056 . . . . . . . 8 (𝑦 = 𝑤 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)))
9695eleq1d 2867 . . . . . . 7 (𝑦 = 𝑤 → ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ))
9796cbvrabv 3434 . . . . . 6 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ }
98 fveq2 6538 . . . . . . . . . . . . 13 (𝑙 = 𝑚 → (𝐹𝑙) = (𝐹𝑚))
9998dmeqd 5660 . . . . . . . . . . . 12 (𝑙 = 𝑚 → dom (𝐹𝑙) = dom (𝐹𝑚))
10074, 71, 99cbviin 4865 . . . . . . . . . . 11 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
101100a1i 11 . . . . . . . . . 10 (𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
102101iuneq2i 4845 . . . . . . . . 9 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
103102eleq2i 2874 . . . . . . . 8 (𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ↔ 𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
104 nfcv 2949 . . . . . . . . . . 11 𝑚𝑤
10573, 104nffv 6548 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑤)
106 nfcv 2949 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑤)
10798fveq1d 6540 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑤) = ((𝐹𝑚)‘𝑤))
10885, 48, 105, 106, 107cbvmptf 5059 . . . . . . . . 9 (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤))
109108eleq1i 2873 . . . . . . . 8 ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ )
110103, 109anbi12i 626 . . . . . . 7 ((𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ) ↔ (𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ ))
111110rabbia2 3423 . . . . . 6 {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11297, 111eqtri 2819 . . . . 5 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11393, 112eqtri 2819 . . . 4 𝐷 = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
114 simpr 485 . . . 4 ((𝜑𝑦𝐷) → 𝑦𝐷)
11554, 28, 55, 16, 60, 113, 114fnlimfvre 41516 . . 3 ((𝜑𝑦𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) ∈ ℝ)
116 smflim.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
117 nfrab1 3344 . . . . . 6 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
1183, 117nfcxfr 2947 . . . . 5 𝑥𝐷
119 nfcv 2949 . . . . 5 𝑦𝐷
120 nfcv 2949 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
121 nfcv 2949 . . . . . 6 𝑥
122121, 65nffv 6548 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
12368fveq2d 6542 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
124118, 119, 120, 122, 123cbvmptf 5059 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
125116, 124eqtri 2819 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
126115, 125fmptd 6741 . 2 (𝜑𝐺:𝐷⟶ℝ)
127 smflim.m . . . 4 (𝜑𝑀 ∈ ℤ)
128127adantr 481 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑀 ∈ ℤ)
1292adantr 481 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
13024adantr 481 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆))
131 nfcv 2949 . . . . . . . . 9 𝑥𝑙
1326, 131nffv 6548 . . . . . . . 8 𝑥(𝐹𝑙)
133132, 63nffv 6548 . . . . . . 7 𝑥((𝐹𝑙)‘𝑦)
1344, 133nfmpt 5057 . . . . . 6 𝑥(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
135121, 134nffv 6548 . . . . 5 𝑥( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
136 nfcv 2949 . . . . . . . . 9 𝑙((𝐹𝑚)‘𝑥)
137 nfcv 2949 . . . . . . . . . 10 𝑚𝑥
13873, 137nffv 6548 . . . . . . . . 9 𝑚((𝐹𝑙)‘𝑥)
13975fveq1d 6540 . . . . . . . . 9 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑙)‘𝑥))
14048, 85, 136, 138, 139cbvmptf 5059 . . . . . . . 8 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥))
141140a1i 11 . . . . . . 7 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)))
142 simpl 483 . . . . . . . . 9 ((𝑥 = 𝑦𝑙𝑍) → 𝑥 = 𝑦)
143142fveq2d 6542 . . . . . . . 8 ((𝑥 = 𝑦𝑙𝑍) → ((𝐹𝑙)‘𝑥) = ((𝐹𝑙)‘𝑦))
144143mpteq2dva 5055 . . . . . . 7 (𝑥 = 𝑦 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
145141, 144eqtrd 2831 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
146145fveq2d 6542 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
147118, 119, 120, 135, 146cbvmptf 5059 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
148116, 147eqtri 2819 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
149 simpr 485 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150 nfcv 2949 . . . . . . . . 9 𝑚 <
151 nfcv 2949 . . . . . . . . 9 𝑚(𝑎 + (1 / 𝑗))
15287, 150, 151nfbr 5009 . . . . . . . 8 𝑚((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153152, 74nfrab 3345 . . . . . . 7 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154 nfcv 2949 . . . . . . . 8 𝑚𝑡
155154, 74nfin 4113 . . . . . . 7 𝑚(𝑡 ∩ dom (𝐹𝑙))
156153, 155nfeq 2960 . . . . . 6 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))
157 nfcv 2949 . . . . . 6 𝑚𝑆
158156, 157nfrab 3345 . . . . 5 𝑚{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
159 nfcv 2949 . . . . 5 𝑘{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
160 nfcv 2949 . . . . 5 𝑙{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
161 nfcv 2949 . . . . 5 𝑗{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
162 nfcv 2949 . . . . . . . . . . . 12 𝑦dom (𝐹𝑙)
163132nfdm 5705 . . . . . . . . . . . 12 𝑥dom (𝐹𝑙)
164 nfcv 2949 . . . . . . . . . . . . 13 𝑥 <
165 nfcv 2949 . . . . . . . . . . . . 13 𝑥(𝑎 + (1 / 𝑗))
166133, 164, 165nfbr 5009 . . . . . . . . . . . 12 𝑥((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167 nfv 1892 . . . . . . . . . . . 12 𝑦((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168 fveq2 6538 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑥))
169168breq1d 4972 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170162, 163, 166, 167, 169cbvrab 3433 . . . . . . . . . . 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 4101 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑙)))
173171, 172eqeq12d 2810 . . . . . . . . 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 3434 . . . . . . . 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 2868 . . . . . . . . . . 11 (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹𝑙) ↔ 𝑥 ∈ dom (𝐹𝑚)))
17798fveq1d 6540 . . . . . . . . . . . 12 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑥) = ((𝐹𝑚)‘𝑥))
178177breq1d 4972 . . . . . . . . . . 11 (𝑙 = 𝑚 → (((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179176, 178anbi12d 630 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝑥 ∈ dom (𝐹𝑙) ∧ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))) ↔ (𝑥 ∈ dom (𝐹𝑚) ∧ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)))))
180179rabbidva2 3422 . . . . . . . . 9 (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
18199ineq2d 4109 . . . . . . . . 9 (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑚)))
182180, 181eqeq12d 2810 . . . . . . . 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 3425 . . . . . . 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 2831 . . . . . 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 7024 . . . . . . . . . . 11 (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186185oveq2d 7032 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187186breq2d 4974 . . . . . . . . 9 (𝑗 = 𝑘 → (((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188187rabbidv 3425 . . . . . . . 8 (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189188eqeq1d 2797 . . . . . . 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 3425 . . . . . 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 2851 . . . . 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, 191cbvmpo 7104 . . . 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 2804 . . 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 42614 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑦𝐷 ∣ (𝐺𝑦) ≤ 𝑎} ∈ (𝑆t 𝐷))
1951, 2, 42, 126, 194issmfled 42596 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wnfc 2933  wrex 3106  {crab 3109  cin 3858  wss 3859   cuni 4745   ciun 4825   ciin 4826   class class class wbr 4962  cmpt 5041  dom cdm 5443  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  cr 10382  1c1 10384   + caddc 10386   < clt 10521   / cdiv 11145  cn 11486  cz 11829  cuz 12093  cli 14675  SAlgcsalg 42155  SMblFncsmblfn 42539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cc 9703  ax-ac2 9731  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-omul 7958  df-er 8139  df-map 8258  df-pm 8259  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-acn 9217  df-ac 9388  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-ioo 12592  df-ico 12594  df-fl 13012  df-seq 13220  df-exp 13280  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-rest 16525  df-salg 42156  df-smblfn 42540
This theorem is referenced by:  smflim2  42642
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