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Theorem smflim 40313
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 1840 . 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 2761 . . . . . . 7 𝑥𝑍
5 nfcv 2761 . . . . . . . 8 𝑥(ℤ𝑛)
6 smflim.x . . . . . . . . . 10 𝑥𝐹
7 nfcv 2761 . . . . . . . . . 10 𝑥𝑚
86, 7nffv 6160 . . . . . . . . 9 𝑥(𝐹𝑚)
98nfdm 5332 . . . . . . . 8 𝑥dom (𝐹𝑚)
105, 9nfiin 4520 . . . . . . 7 𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
114, 10nfiun 4519 . . . . . 6 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1211ssrab2f 38812 . . . . 5 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
133, 12eqsstri 3619 . . . 4 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1413a1i 11 . . 3 (𝜑𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
15 uzssz 11658 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
16 smflim.z . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
1716eleq2i 2690 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1817biimpi 206 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1915, 18sseldi 3585 . . . . . . . 8 (𝑛𝑍𝑛 ∈ ℤ)
20 uzid 11653 . . . . . . . 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 6320 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
26 eqid 2621 . . . . . . 7 dom (𝐹𝑛) = dom (𝐹𝑛)
2723, 25, 26smfdmss 40270 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ⊆ 𝑆)
28 smflim.n . . . . . . . . . 10 𝑚𝐹
29 nfcv 2761 . . . . . . . . . 10 𝑚𝑛
3028, 29nffv 6160 . . . . . . . . 9 𝑚(𝐹𝑛)
3130nfdm 5332 . . . . . . . 8 𝑚dom (𝐹𝑛)
32 nfcv 2761 . . . . . . . 8 𝑚 𝑆
3331, 32nfss 3580 . . . . . . 7 𝑚dom (𝐹𝑛) ⊆ 𝑆
34 fveq2 6153 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534dmeqd 5291 . . . . . . . 8 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
3635sseq1d 3616 . . . . . . 7 (𝑚 = 𝑛 → (dom (𝐹𝑚) ⊆ 𝑆 ↔ dom (𝐹𝑛) ⊆ 𝑆))
3733, 36rspce 3293 . . . . . 6 ((𝑛 ∈ (ℤ𝑛) ∧ dom (𝐹𝑛) ⊆ 𝑆) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
3822, 27, 37syl2anc 692 . . . . 5 ((𝜑𝑛𝑍) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
39 iinss 4542 . . . . 5 (∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4038, 39syl 17 . . . 4 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4140iunssd 38781 . . 3 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4214, 41sstrd 3597 . 2 (𝜑𝐷 𝑆)
43 nfv 1840 . . . . 5 𝑚𝜑
44 nfcv 2761 . . . . . 6 𝑚𝑦
45 nfmpt1 4712 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))
46 nfcv 2761 . . . . . . . . 9 𝑚dom ⇝
4745, 46nfel 2773 . . . . . . . 8 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
48 nfcv 2761 . . . . . . . . 9 𝑚𝑍
49 nfii1 4522 . . . . . . . . 9 𝑚 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5048, 49nfiun 4519 . . . . . . . 8 𝑚 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5147, 50nfrab 3115 . . . . . . 7 𝑚{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
523, 51nfcxfr 2759 . . . . . 6 𝑚𝐷
5344, 52nfel 2773 . . . . 5 𝑚 𝑦𝐷
5443, 53nfan 1825 . . . 4 𝑚(𝜑𝑦𝐷)
55 nfcv 2761 . . . 4 𝑤𝐹
562adantr 481 . . . . . 6 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
5724ffvelrnda 6320 . . . . . 6 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
58 eqid 2621 . . . . . 6 dom (𝐹𝑚) = dom (𝐹𝑚)
5956, 57, 58smff 40269 . . . . 5 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
6059adantlr 750 . . . 4 (((𝜑𝑦𝐷) ∧ 𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
61 nfcv 2761 . . . . . . 7 𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
62 nfv 1840 . . . . . . 7 𝑦(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
63 nfcv 2761 . . . . . . . . . 10 𝑥𝑦
648, 63nffv 6160 . . . . . . . . 9 𝑥((𝐹𝑚)‘𝑦)
654, 64nfmpt 4711 . . . . . . . 8 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
6665nfel1 2775 . . . . . . 7 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝
67 fveq2 6153 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6867mpteq2dv 4710 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
6968eleq1d 2683 . . . . . . 7 (𝑥 = 𝑦 → ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ))
7011, 61, 62, 66, 69cbvrab 3187 . . . . . 6 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ }
71 nfcv 2761 . . . . . . . . . . . . 13 𝑙dom (𝐹𝑚)
72 nfcv 2761 . . . . . . . . . . . . . . 15 𝑚𝑙
7328, 72nffv 6160 . . . . . . . . . . . . . 14 𝑚(𝐹𝑙)
7473nfdm 5332 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑙)
75 fveq2 6153 . . . . . . . . . . . . . 14 (𝑚 = 𝑙 → (𝐹𝑚) = (𝐹𝑙))
7675dmeqd 5291 . . . . . . . . . . . . 13 (𝑚 = 𝑙 → dom (𝐹𝑚) = dom (𝐹𝑙))
7771, 74, 76cbviin 4529 . . . . . . . . . . . 12 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙)
7877a1i 11 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙))
79 fveq2 6153 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
80 eqidd 2622 . . . . . . . . . . . 12 ((𝑛 = 𝑖𝑙 ∈ (ℤ𝑖)) → dom (𝐹𝑙) = dom (𝐹𝑙))
8179, 80iineq12dv 38800 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8278, 81eqtrd 2655 . . . . . . . . . 10 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8382cbviunv 4530 . . . . . . . . 9 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙)
8483eleq2i 2690 . . . . . . . 8 (𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
85 nfcv 2761 . . . . . . . . . 10 𝑙𝑍
86 nfcv 2761 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑦)
8773, 44nffv 6160 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑦)
8875fveq1d 6155 . . . . . . . . . 10 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑙)‘𝑦))
8948, 85, 86, 87, 88cbvmptf 4713 . . . . . . . . 9 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
9089eleq1i 2689 . . . . . . . 8 ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ )
9184, 90anbi12i 732 . . . . . . 7 ((𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ))
9291rabbia2 3178 . . . . . 6 {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
933, 70, 923eqtri 2647 . . . . 5 𝐷 = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
94 fveq2 6153 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑤))
9594mpteq2dv 4710 . . . . . . . 8 (𝑦 = 𝑤 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)))
9695eleq1d 2683 . . . . . . 7 (𝑦 = 𝑤 → ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ))
9796cbvrabv 3188 . . . . . 6 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ }
98 fveq2 6153 . . . . . . . . . . . . 13 (𝑙 = 𝑚 → (𝐹𝑙) = (𝐹𝑚))
9998dmeqd 5291 . . . . . . . . . . . 12 (𝑙 = 𝑚 → dom (𝐹𝑙) = dom (𝐹𝑚))
10074, 71, 99cbviin 4529 . . . . . . . . . . 11 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
101100a1i 11 . . . . . . . . . 10 (𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
102101iuneq2i 4510 . . . . . . . . 9 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
103102eleq2i 2690 . . . . . . . 8 (𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ↔ 𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
104 nfcv 2761 . . . . . . . . . . 11 𝑚𝑤
10573, 104nffv 6160 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑤)
106 nfcv 2761 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑤)
10798fveq1d 6155 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑤) = ((𝐹𝑚)‘𝑤))
10885, 48, 105, 106, 107cbvmptf 4713 . . . . . . . . 9 (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤))
109108eleq1i 2689 . . . . . . . 8 ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ )
110103, 109anbi12i 732 . . . . . . 7 ((𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ) ↔ (𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ ))
111110rabbia2 3178 . . . . . 6 {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11297, 111eqtri 2643 . . . . 5 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11393, 112eqtri 2643 . . . 4 𝐷 = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
114 simpr 477 . . . 4 ((𝜑𝑦𝐷) → 𝑦𝐷)
11554, 28, 55, 16, 60, 113, 114fnlimfvre 39333 . . 3 ((𝜑𝑦𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) ∈ ℝ)
116 smflim.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
117 nfrab1 3114 . . . . . 6 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
1183, 117nfcxfr 2759 . . . . 5 𝑥𝐷
119 nfcv 2761 . . . . 5 𝑦𝐷
120 nfcv 2761 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
121 nfcv 2761 . . . . . 6 𝑥
122121, 65nffv 6160 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
12368fveq2d 6157 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
124118, 119, 120, 122, 123cbvmptf 4713 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
125116, 124eqtri 2643 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
126115, 125fmptd 6346 . 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 2761 . . . . . . . . 9 𝑥𝑙
1326, 131nffv 6160 . . . . . . . 8 𝑥(𝐹𝑙)
133132, 63nffv 6160 . . . . . . 7 𝑥((𝐹𝑙)‘𝑦)
1344, 133nfmpt 4711 . . . . . 6 𝑥(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
135121, 134nffv 6160 . . . . 5 𝑥( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
136 nfcv 2761 . . . . . . . . 9 𝑙((𝐹𝑚)‘𝑥)
137 nfcv 2761 . . . . . . . . . 10 𝑚𝑥
13873, 137nffv 6160 . . . . . . . . 9 𝑚((𝐹𝑙)‘𝑥)
13975fveq1d 6155 . . . . . . . . 9 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑙)‘𝑥))
14048, 85, 136, 138, 139cbvmptf 4713 . . . . . . . 8 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥))
141140a1i 11 . . . . . . 7 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)))
142 simpl 473 . . . . . . . . 9 ((𝑥 = 𝑦𝑙𝑍) → 𝑥 = 𝑦)
143142fveq2d 6157 . . . . . . . 8 ((𝑥 = 𝑦𝑙𝑍) → ((𝐹𝑙)‘𝑥) = ((𝐹𝑙)‘𝑦))
144143mpteq2dva 4709 . . . . . . 7 (𝑥 = 𝑦 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
145141, 144eqtrd 2655 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
146145fveq2d 6157 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
147118, 119, 120, 135, 146cbvmptf 4713 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
148116, 147eqtri 2643 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
149 simpr 477 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150 nfcv 2761 . . . . . . . . 9 𝑚 <
151 nfcv 2761 . . . . . . . . 9 𝑚(𝑎 + (1 / 𝑗))
15287, 150, 151nfbr 4664 . . . . . . . 8 𝑚((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153152, 74nfrab 3115 . . . . . . 7 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154 nfcv 2761 . . . . . . . 8 𝑚𝑡
155154, 74nfin 3803 . . . . . . 7 𝑚(𝑡 ∩ dom (𝐹𝑙))
156153, 155nfeq 2772 . . . . . 6 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))
157 nfcv 2761 . . . . . 6 𝑚𝑆
158156, 157nfrab 3115 . . . . 5 𝑚{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
159 nfcv 2761 . . . . 5 𝑘{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
160 nfcv 2761 . . . . 5 𝑙{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
161 nfcv 2761 . . . . 5 𝑗{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
162 nfcv 2761 . . . . . . . . . . . 12 𝑦dom (𝐹𝑙)
163132nfdm 5332 . . . . . . . . . . . 12 𝑥dom (𝐹𝑙)
164 nfcv 2761 . . . . . . . . . . . . 13 𝑥 <
165 nfcv 2761 . . . . . . . . . . . . 13 𝑥(𝑎 + (1 / 𝑗))
166133, 164, 165nfbr 4664 . . . . . . . . . . . 12 𝑥((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167 nfv 1840 . . . . . . . . . . . 12 𝑦((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168 fveq2 6153 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑥))
169168breq1d 4628 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170162, 163, 166, 167, 169cbvrab 3187 . . . . . . . . . . 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 3790 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑙)))
173171, 172eqeq12d 2636 . . . . . . . . 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 3188 . . . . . . . 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 2684 . . . . . . . . . . 11 (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹𝑙) ↔ 𝑥 ∈ dom (𝐹𝑚)))
17798fveq1d 6155 . . . . . . . . . . . 12 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑥) = ((𝐹𝑚)‘𝑥))
178177breq1d 4628 . . . . . . . . . . 11 (𝑙 = 𝑚 → (((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179176, 178anbi12d 746 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝑥 ∈ dom (𝐹𝑙) ∧ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))) ↔ (𝑥 ∈ dom (𝐹𝑚) ∧ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)))))
180179rabbidva2 3177 . . . . . . . . 9 (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
18199ineq2d 3797 . . . . . . . . 9 (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑚)))
182180, 181eqeq12d 2636 . . . . . . . 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 3180 . . . . . . 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 2655 . . . . . 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 6618 . . . . . . . . . . 11 (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186185oveq2d 6626 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187186breq2d 4630 . . . . . . . . 9 (𝑗 = 𝑘 → (((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188187rabbidv 3180 . . . . . . . 8 (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189188eqeq1d 2623 . . . . . . 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 3180 . . . . . 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 2675 . . . . 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 6694 . . . 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 2630 . . 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 40312 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑦𝐷 ∣ (𝐺𝑦) ≤ 𝑎} ∈ (𝑆t 𝐷))
1951, 2, 42, 126, 194issmfled 40294 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wnfc 2748  wrex 2908  {crab 2911  cin 3558  wss 3559   cuni 4407   ciun 4490   ciin 4491   class class class wbr 4618  cmpt 4678  dom cdm 5079  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  cr 9886  1c1 9888   + caddc 9890   < clt 10025   / cdiv 10635  cn 10971  cz 11328  cuz 11638  cli 14156  SAlgcsalg 39856  SMblFncsmblfn 40237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cc 9208  ax-ac2 9236  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-acn 8719  df-ac 8890  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-ioo 12128  df-ico 12130  df-fl 12540  df-seq 12749  df-exp 12808  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-rlim 14161  df-rest 16011  df-salg 39857  df-smblfn 40238
This theorem is referenced by:  smflim2  40340
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