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Theorem smflim 42930
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 1906 . 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 2974 . . . . . . 7 𝑥𝑍
5 nfcv 2974 . . . . . . . 8 𝑥(ℤ𝑛)
6 smflim.x . . . . . . . . . 10 𝑥𝐹
7 nfcv 2974 . . . . . . . . . 10 𝑥𝑚
86, 7nffv 6673 . . . . . . . . 9 𝑥(𝐹𝑚)
98nfdm 5816 . . . . . . . 8 𝑥dom (𝐹𝑚)
105, 9nfiin 4941 . . . . . . 7 𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
114, 10nfiun 4940 . . . . . 6 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1211ssrab2f 41260 . . . . 5 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
133, 12eqsstri 3998 . . . 4 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1413a1i 11 . . 3 (𝜑𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
15 uzssz 12252 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
16 smflim.z . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
1716eleq2i 2901 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1817biimpi 217 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1915, 18sseldi 3962 . . . . . . . 8 (𝑛𝑍𝑛 ∈ ℤ)
20 uzid 12246 . . . . . . . 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 6843 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
26 eqid 2818 . . . . . . 7 dom (𝐹𝑛) = dom (𝐹𝑛)
2723, 25, 26smfdmss 42887 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ⊆ 𝑆)
28 smflim.n . . . . . . . . . 10 𝑚𝐹
29 nfcv 2974 . . . . . . . . . 10 𝑚𝑛
3028, 29nffv 6673 . . . . . . . . 9 𝑚(𝐹𝑛)
3130nfdm 5816 . . . . . . . 8 𝑚dom (𝐹𝑛)
32 nfcv 2974 . . . . . . . 8 𝑚 𝑆
3331, 32nfss 3957 . . . . . . 7 𝑚dom (𝐹𝑛) ⊆ 𝑆
34 fveq2 6663 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534dmeqd 5767 . . . . . . . 8 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
3635sseq1d 3995 . . . . . . 7 (𝑚 = 𝑛 → (dom (𝐹𝑚) ⊆ 𝑆 ↔ dom (𝐹𝑛) ⊆ 𝑆))
3733, 36rspce 3609 . . . . . 6 ((𝑛 ∈ (ℤ𝑛) ∧ dom (𝐹𝑛) ⊆ 𝑆) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
3822, 27, 37syl2anc 584 . . . . 5 ((𝜑𝑛𝑍) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
39 iinss 4971 . . . . 5 (∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4038, 39syl 17 . . . 4 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4140iunssd 4965 . . 3 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4214, 41sstrd 3974 . 2 (𝜑𝐷 𝑆)
43 nfv 1906 . . . . 5 𝑚𝜑
44 nfcv 2974 . . . . . 6 𝑚𝑦
45 nfmpt1 5155 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))
46 nfcv 2974 . . . . . . . . 9 𝑚dom ⇝
4745, 46nfel 2989 . . . . . . . 8 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
48 nfcv 2974 . . . . . . . . 9 𝑚𝑍
49 nfii1 4945 . . . . . . . . 9 𝑚 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5048, 49nfiun 4940 . . . . . . . 8 𝑚 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5147, 50nfrabw 3383 . . . . . . 7 𝑚{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
523, 51nfcxfr 2972 . . . . . 6 𝑚𝐷
5344, 52nfel 2989 . . . . 5 𝑚 𝑦𝐷
5443, 53nfan 1891 . . . 4 𝑚(𝜑𝑦𝐷)
55 nfcv 2974 . . . 4 𝑤𝐹
562adantr 481 . . . . . 6 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
5724ffvelrnda 6843 . . . . . 6 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
58 eqid 2818 . . . . . 6 dom (𝐹𝑚) = dom (𝐹𝑚)
5956, 57, 58smff 42886 . . . . 5 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
6059adantlr 711 . . . 4 (((𝜑𝑦𝐷) ∧ 𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
61 nfcv 2974 . . . . . . 7 𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
62 nfv 1906 . . . . . . 7 𝑦(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
63 nfcv 2974 . . . . . . . . . 10 𝑥𝑦
648, 63nffv 6673 . . . . . . . . 9 𝑥((𝐹𝑚)‘𝑦)
654, 64nfmpt 5154 . . . . . . . 8 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
6665nfel1 2991 . . . . . . 7 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝
67 fveq2 6663 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6867mpteq2dv 5153 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
6968eleq1d 2894 . . . . . . 7 (𝑥 = 𝑦 → ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ))
7011, 61, 62, 66, 69cbvrabw 3487 . . . . . 6 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ }
71 nfcv 2974 . . . . . . . . . . . . 13 𝑙dom (𝐹𝑚)
72 nfcv 2974 . . . . . . . . . . . . . . 15 𝑚𝑙
7328, 72nffv 6673 . . . . . . . . . . . . . 14 𝑚(𝐹𝑙)
7473nfdm 5816 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑙)
75 fveq2 6663 . . . . . . . . . . . . . 14 (𝑚 = 𝑙 → (𝐹𝑚) = (𝐹𝑙))
7675dmeqd 5767 . . . . . . . . . . . . 13 (𝑚 = 𝑙 → dom (𝐹𝑚) = dom (𝐹𝑙))
7771, 74, 76cbviin 4953 . . . . . . . . . . . 12 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙)
7877a1i 11 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙))
79 fveq2 6663 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
80 eqidd 2819 . . . . . . . . . . . 12 ((𝑛 = 𝑖𝑙 ∈ (ℤ𝑖)) → dom (𝐹𝑙) = dom (𝐹𝑙))
8179, 80iineq12dv 41249 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8278, 81eqtrd 2853 . . . . . . . . . 10 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8382cbviunv 4956 . . . . . . . . 9 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙)
8483eleq2i 2901 . . . . . . . 8 (𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
85 nfcv 2974 . . . . . . . . . 10 𝑙𝑍
86 nfcv 2974 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑦)
8773, 44nffv 6673 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑦)
8875fveq1d 6665 . . . . . . . . . 10 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑙)‘𝑦))
8948, 85, 86, 87, 88cbvmptf 5156 . . . . . . . . 9 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
9089eleq1i 2900 . . . . . . . 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 3475 . . . . . 6 {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
933, 70, 923eqtri 2845 . . . . 5 𝐷 = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
94 fveq2 6663 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑤))
9594mpteq2dv 5153 . . . . . . . 8 (𝑦 = 𝑤 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)))
9695eleq1d 2894 . . . . . . 7 (𝑦 = 𝑤 → ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ))
9796cbvrabv 3489 . . . . . 6 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ }
98 fveq2 6663 . . . . . . . . . . . . 13 (𝑙 = 𝑚 → (𝐹𝑙) = (𝐹𝑚))
9998dmeqd 5767 . . . . . . . . . . . 12 (𝑙 = 𝑚 → dom (𝐹𝑙) = dom (𝐹𝑚))
10074, 71, 99cbviin 4953 . . . . . . . . . . 11 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
101100a1i 11 . . . . . . . . . 10 (𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
102101iuneq2i 4931 . . . . . . . . 9 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
103102eleq2i 2901 . . . . . . . 8 (𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ↔ 𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
104 nfcv 2974 . . . . . . . . . . 11 𝑚𝑤
10573, 104nffv 6673 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑤)
106 nfcv 2974 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑤)
10798fveq1d 6665 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑤) = ((𝐹𝑚)‘𝑤))
10885, 48, 105, 106, 107cbvmptf 5156 . . . . . . . . 9 (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤))
109108eleq1i 2900 . . . . . . . 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 3475 . . . . . 6 {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11297, 111eqtri 2841 . . . . 5 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11393, 112eqtri 2841 . . . 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 41831 . . 3 ((𝜑𝑦𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) ∈ ℝ)
116 smflim.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
117 nfrab1 3382 . . . . . 6 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
1183, 117nfcxfr 2972 . . . . 5 𝑥𝐷
119 nfcv 2974 . . . . 5 𝑦𝐷
120 nfcv 2974 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
121 nfcv 2974 . . . . . 6 𝑥
122121, 65nffv 6673 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
12368fveq2d 6667 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
124118, 119, 120, 122, 123cbvmptf 5156 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
125116, 124eqtri 2841 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
126115, 125fmptd 6870 . 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 2974 . . . . . . . . 9 𝑥𝑙
1326, 131nffv 6673 . . . . . . . 8 𝑥(𝐹𝑙)
133132, 63nffv 6673 . . . . . . 7 𝑥((𝐹𝑙)‘𝑦)
1344, 133nfmpt 5154 . . . . . 6 𝑥(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
135121, 134nffv 6673 . . . . 5 𝑥( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
136 nfcv 2974 . . . . . . . . 9 𝑙((𝐹𝑚)‘𝑥)
137 nfcv 2974 . . . . . . . . . 10 𝑚𝑥
13873, 137nffv 6673 . . . . . . . . 9 𝑚((𝐹𝑙)‘𝑥)
13975fveq1d 6665 . . . . . . . . 9 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑙)‘𝑥))
14048, 85, 136, 138, 139cbvmptf 5156 . . . . . . . 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 6667 . . . . . . . 8 ((𝑥 = 𝑦𝑙𝑍) → ((𝐹𝑙)‘𝑥) = ((𝐹𝑙)‘𝑦))
144143mpteq2dva 5152 . . . . . . 7 (𝑥 = 𝑦 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
145141, 144eqtrd 2853 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
146145fveq2d 6667 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
147118, 119, 120, 135, 146cbvmptf 5156 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
148116, 147eqtri 2841 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
149 simpr 485 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150 nfcv 2974 . . . . . . . . 9 𝑚 <
151 nfcv 2974 . . . . . . . . 9 𝑚(𝑎 + (1 / 𝑗))
15287, 150, 151nfbr 5104 . . . . . . . 8 𝑚((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153152, 74nfrabw 3383 . . . . . . 7 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154 nfcv 2974 . . . . . . . 8 𝑚𝑡
155154, 74nfin 4190 . . . . . . 7 𝑚(𝑡 ∩ dom (𝐹𝑙))
156153, 155nfeq 2988 . . . . . 6 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))
157 nfcv 2974 . . . . . 6 𝑚𝑆
158156, 157nfrabw 3383 . . . . 5 𝑚{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
159 nfcv 2974 . . . . 5 𝑘{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
160 nfcv 2974 . . . . 5 𝑙{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
161 nfcv 2974 . . . . 5 𝑗{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
162 nfcv 2974 . . . . . . . . . . . 12 𝑦dom (𝐹𝑙)
163132nfdm 5816 . . . . . . . . . . . 12 𝑥dom (𝐹𝑙)
164 nfcv 2974 . . . . . . . . . . . . 13 𝑥 <
165 nfcv 2974 . . . . . . . . . . . . 13 𝑥(𝑎 + (1 / 𝑗))
166133, 164, 165nfbr 5104 . . . . . . . . . . . 12 𝑥((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167 nfv 1906 . . . . . . . . . . . 12 𝑦((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168 fveq2 6663 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑥))
169168breq1d 5067 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170162, 163, 166, 167, 169cbvrabw 3487 . . . . . . . . . . 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 4178 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑙)))
173171, 172eqeq12d 2834 . . . . . . . . 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 3489 . . . . . . . 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 2895 . . . . . . . . . . 11 (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹𝑙) ↔ 𝑥 ∈ dom (𝐹𝑚)))
17798fveq1d 6665 . . . . . . . . . . . 12 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑥) = ((𝐹𝑚)‘𝑥))
178177breq1d 5067 . . . . . . . . . . 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 3474 . . . . . . . . 9 (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
18199ineq2d 4186 . . . . . . . . 9 (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑚)))
182180, 181eqeq12d 2834 . . . . . . . 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 3478 . . . . . . 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 2853 . . . . . 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 7153 . . . . . . . . . . 11 (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186185oveq2d 7161 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187186breq2d 5069 . . . . . . . . 9 (𝑗 = 𝑘 → (((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188187rabbidv 3478 . . . . . . . 8 (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189188eqeq1d 2820 . . . . . . 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 3478 . . . . . 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 2873 . . . . 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 7237 . . . 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 2827 . . 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 42929 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑦𝐷 ∣ (𝐺𝑦) ≤ 𝑎} ∈ (𝑆t 𝐷))
1951, 2, 42, 126, 194issmfled 42911 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wrex 3136  {crab 3139  cin 3932  wss 3933   cuni 4830   ciun 4910   ciin 4911   class class class wbr 5057  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  cr 10524  1c1 10526   + caddc 10528   < clt 10663   / cdiv 11285  cn 11626  cz 11969  cuz 12231  cli 14829  SAlgcsalg 42470  SMblFncsmblfn 42854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cc 9845  ax-ac2 9873  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-acn 9359  df-ac 9530  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-ioo 12730  df-ico 12732  df-fl 13150  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-rest 16684  df-salg 42471  df-smblfn 42855
This theorem is referenced by:  smflim2  42957
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