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Theorem smflim 46733
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 1912 . 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 2903 . . . . . . 7 𝑥𝑍
5 nfcv 2903 . . . . . . . 8 𝑥(ℤ𝑛)
6 smflim.x . . . . . . . . . 10 𝑥𝐹
7 nfcv 2903 . . . . . . . . . 10 𝑥𝑚
86, 7nffv 6917 . . . . . . . . 9 𝑥(𝐹𝑚)
98nfdm 5965 . . . . . . . 8 𝑥dom (𝐹𝑚)
105, 9nfiin 5029 . . . . . . 7 𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
114, 10nfiun 5028 . . . . . 6 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1211ssrab2f 45057 . . . . 5 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
133, 12eqsstri 4030 . . . 4 𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
1413a1i 11 . . 3 (𝜑𝐷 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
15 uzssz 12897 . . . . . . . . 9 (ℤ𝑀) ⊆ ℤ
16 smflim.z . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
1716eleq2i 2831 . . . . . . . . . 10 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1817biimpi 216 . . . . . . . . 9 (𝑛𝑍𝑛 ∈ (ℤ𝑀))
1915, 18sselid 3993 . . . . . . . 8 (𝑛𝑍𝑛 ∈ ℤ)
20 uzid 12891 . . . . . . . 8 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
2119, 20syl 17 . . . . . . 7 (𝑛𝑍𝑛 ∈ (ℤ𝑛))
2221adantl 481 . . . . . 6 ((𝜑𝑛𝑍) → 𝑛 ∈ (ℤ𝑛))
232adantr 480 . . . . . . 7 ((𝜑𝑛𝑍) → 𝑆 ∈ SAlg)
24 smflim.f . . . . . . . 8 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
2524ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑛𝑍) → (𝐹𝑛) ∈ (SMblFn‘𝑆))
26 eqid 2735 . . . . . . 7 dom (𝐹𝑛) = dom (𝐹𝑛)
2723, 25, 26smfdmss 46689 . . . . . 6 ((𝜑𝑛𝑍) → dom (𝐹𝑛) ⊆ 𝑆)
28 smflim.n . . . . . . . . . 10 𝑚𝐹
29 nfcv 2903 . . . . . . . . . 10 𝑚𝑛
3028, 29nffv 6917 . . . . . . . . 9 𝑚(𝐹𝑛)
3130nfdm 5965 . . . . . . . 8 𝑚dom (𝐹𝑛)
32 nfcv 2903 . . . . . . . 8 𝑚 𝑆
3331, 32nfss 3988 . . . . . . 7 𝑚dom (𝐹𝑛) ⊆ 𝑆
34 fveq2 6907 . . . . . . . . 9 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
3534dmeqd 5919 . . . . . . . 8 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
3635sseq1d 4027 . . . . . . 7 (𝑚 = 𝑛 → (dom (𝐹𝑚) ⊆ 𝑆 ↔ dom (𝐹𝑛) ⊆ 𝑆))
3733, 36rspce 3611 . . . . . 6 ((𝑛 ∈ (ℤ𝑛) ∧ dom (𝐹𝑛) ⊆ 𝑆) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
3822, 27, 37syl2anc 584 . . . . 5 ((𝜑𝑛𝑍) → ∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
39 iinss 5061 . . . . 5 (∃𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4038, 39syl 17 . . . 4 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4140iunssd 5055 . . 3 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ 𝑆)
4214, 41sstrd 4006 . 2 (𝜑𝐷 𝑆)
43 nfv 1912 . . . . 5 𝑚𝜑
44 nfcv 2903 . . . . . 6 𝑚𝑦
45 nfmpt1 5256 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))
46 nfcv 2903 . . . . . . . . 9 𝑚dom ⇝
4745, 46nfel 2918 . . . . . . . 8 𝑚(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
48 nfcv 2903 . . . . . . . . 9 𝑚𝑍
49 nfii1 5034 . . . . . . . . 9 𝑚 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5048, 49nfiun 5028 . . . . . . . 8 𝑚 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
5147, 50nfrabw 3473 . . . . . . 7 𝑚{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
523, 51nfcxfr 2901 . . . . . 6 𝑚𝐷
5344, 52nfel 2918 . . . . 5 𝑚 𝑦𝐷
5443, 53nfan 1897 . . . 4 𝑚(𝜑𝑦𝐷)
55 nfcv 2903 . . . 4 𝑤𝐹
562adantr 480 . . . . . 6 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
5724ffvelcdmda 7104 . . . . . 6 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
58 eqid 2735 . . . . . 6 dom (𝐹𝑚) = dom (𝐹𝑚)
5956, 57, 58smff 46688 . . . . 5 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
6059adantlr 715 . . . 4 (((𝜑𝑦𝐷) ∧ 𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
61 nfcv 2903 . . . . . . 7 𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚)
62 nfv 1912 . . . . . . 7 𝑦(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝
63 nfcv 2903 . . . . . . . . . 10 𝑥𝑦
648, 63nffv 6917 . . . . . . . . 9 𝑥((𝐹𝑚)‘𝑦)
654, 64nfmpt 5255 . . . . . . . 8 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))
6665nfel1 2920 . . . . . . 7 𝑥(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝
67 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6867mpteq2dv 5250 . . . . . . . 8 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
6968eleq1d 2824 . . . . . . 7 (𝑥 = 𝑦 → ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ))
7011, 61, 62, 66, 69cbvrabw 3471 . . . . . 6 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ }
71 nfcv 2903 . . . . . . . . . . . . 13 𝑙dom (𝐹𝑚)
72 nfcv 2903 . . . . . . . . . . . . . . 15 𝑚𝑙
7328, 72nffv 6917 . . . . . . . . . . . . . 14 𝑚(𝐹𝑙)
7473nfdm 5965 . . . . . . . . . . . . 13 𝑚dom (𝐹𝑙)
75 fveq2 6907 . . . . . . . . . . . . . 14 (𝑚 = 𝑙 → (𝐹𝑚) = (𝐹𝑙))
7675dmeqd 5919 . . . . . . . . . . . . 13 (𝑚 = 𝑙 → dom (𝐹𝑚) = dom (𝐹𝑙))
7771, 74, 76cbviin 5042 . . . . . . . . . . . 12 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙)
7877a1i 11 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙))
79 fveq2 6907 . . . . . . . . . . . 12 (𝑛 = 𝑖 → (ℤ𝑛) = (ℤ𝑖))
80 eqidd 2736 . . . . . . . . . . . 12 ((𝑛 = 𝑖𝑙 ∈ (ℤ𝑖)) → dom (𝐹𝑙) = dom (𝐹𝑙))
8179, 80iineq12dv 45046 . . . . . . . . . . 11 (𝑛 = 𝑖 𝑙 ∈ (ℤ𝑛)dom (𝐹𝑙) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8278, 81eqtrd 2775 . . . . . . . . . 10 (𝑛 = 𝑖 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
8382cbviunv 5045 . . . . . . . . 9 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) = 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙)
8483eleq2i 2831 . . . . . . . 8 (𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ↔ 𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙))
85 nfcv 2903 . . . . . . . . . 10 𝑙𝑍
86 nfcv 2903 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑦)
8773, 44nffv 6917 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑦)
8875fveq1d 6909 . . . . . . . . . 10 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑙)‘𝑦))
8948, 85, 86, 87, 88cbvmptf 5257 . . . . . . . . 9 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
9089eleq1i 2830 . . . . . . . 8 ((𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ )
9184, 90anbi12i 628 . . . . . . 7 ((𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ))
9291rabbia2 3436 . . . . . 6 {𝑦 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
933, 70, 923eqtri 2767 . . . . 5 𝐷 = {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ }
94 fveq2 6907 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑤))
9594mpteq2dv 5250 . . . . . . . 8 (𝑦 = 𝑤 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)))
9695eleq1d 2824 . . . . . . 7 (𝑦 = 𝑤 → ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ))
9796cbvrabv 3444 . . . . . 6 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ }
98 fveq2 6907 . . . . . . . . . . . . 13 (𝑙 = 𝑚 → (𝐹𝑙) = (𝐹𝑚))
9998dmeqd 5919 . . . . . . . . . . . 12 (𝑙 = 𝑚 → dom (𝐹𝑙) = dom (𝐹𝑚))
10074, 71, 99cbviin 5042 . . . . . . . . . . 11 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
101100a1i 11 . . . . . . . . . 10 (𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
102101iuneq2i 5018 . . . . . . . . 9 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) = 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚)
103102eleq2i 2831 . . . . . . . 8 (𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ↔ 𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚))
104 nfcv 2903 . . . . . . . . . . 11 𝑚𝑤
10573, 104nffv 6917 . . . . . . . . . 10 𝑚((𝐹𝑙)‘𝑤)
106 nfcv 2903 . . . . . . . . . 10 𝑙((𝐹𝑚)‘𝑤)
10798fveq1d 6909 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑤) = ((𝐹𝑚)‘𝑤))
10885, 48, 105, 106, 107cbvmptf 5257 . . . . . . . . 9 (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤))
109108eleq1i 2830 . . . . . . . 8 ((𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ )
110103, 109anbi12i 628 . . . . . . 7 ((𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∧ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ ) ↔ (𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∧ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ ))
111110rabbia2 3436 . . . . . 6 {𝑤 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11297, 111eqtri 2763 . . . . 5 {𝑦 𝑖𝑍 𝑙 ∈ (ℤ𝑖)dom (𝐹𝑙) ∣ (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
11393, 112eqtri 2763 . . . 4 𝐷 = {𝑤 𝑖𝑍 𝑚 ∈ (ℤ𝑖)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑤)) ∈ dom ⇝ }
114 simpr 484 . . . 4 ((𝜑𝑦𝐷) → 𝑦𝐷)
11554, 28, 55, 16, 60, 113, 114fnlimfvre 45630 . . 3 ((𝜑𝑦𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))) ∈ ℝ)
116 smflim.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))))
117 nfrab1 3454 . . . . . 6 𝑥{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) ∈ dom ⇝ }
1183, 117nfcxfr 2901 . . . . 5 𝑥𝐷
119 nfcv 2903 . . . . 5 𝑦𝐷
120 nfcv 2903 . . . . 5 𝑦( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))
121 nfcv 2903 . . . . . 6 𝑥
122121, 65nffv 6917 . . . . 5 𝑥( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦)))
12368fveq2d 6911 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
124118, 119, 120, 122, 123cbvmptf 5257 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
125116, 124eqtri 2763 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑦))))
126115, 125fmptd 7134 . 2 (𝜑𝐺:𝐷⟶ℝ)
127 smflim.m . . . 4 (𝜑𝑀 ∈ ℤ)
128127adantr 480 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑀 ∈ ℤ)
1292adantr 480 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
13024adantr 480 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆))
131 nfcv 2903 . . . . . . . . 9 𝑥𝑙
1326, 131nffv 6917 . . . . . . . 8 𝑥(𝐹𝑙)
133132, 63nffv 6917 . . . . . . 7 𝑥((𝐹𝑙)‘𝑦)
1344, 133nfmpt 5255 . . . . . 6 𝑥(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))
135121, 134nffv 6917 . . . . 5 𝑥( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
136 nfcv 2903 . . . . . . . . 9 𝑙((𝐹𝑚)‘𝑥)
137 nfcv 2903 . . . . . . . . . 10 𝑚𝑥
13873, 137nffv 6917 . . . . . . . . 9 𝑚((𝐹𝑙)‘𝑥)
13975fveq1d 6909 . . . . . . . . 9 (𝑚 = 𝑙 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑙)‘𝑥))
14048, 85, 136, 138, 139cbvmptf 5257 . . . . . . . 8 (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥))
141140a1i 11 . . . . . . 7 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)))
142 simpl 482 . . . . . . . . 9 ((𝑥 = 𝑦𝑙𝑍) → 𝑥 = 𝑦)
143142fveq2d 6911 . . . . . . . 8 ((𝑥 = 𝑦𝑙𝑍) → ((𝐹𝑙)‘𝑥) = ((𝐹𝑙)‘𝑦))
144143mpteq2dva 5248 . . . . . . 7 (𝑥 = 𝑦 → (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
145141, 144eqtrd 2775 . . . . . 6 (𝑥 = 𝑦 → (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)) = (𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦)))
146145fveq2d 6911 . . . . 5 (𝑥 = 𝑦 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥))) = ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
147118, 119, 120, 135, 146cbvmptf 5257 . . . 4 (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑥)))) = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
148116, 147eqtri 2763 . . 3 𝐺 = (𝑦𝐷 ↦ ( ⇝ ‘(𝑙𝑍 ↦ ((𝐹𝑙)‘𝑦))))
149 simpr 484 . . 3 ((𝜑𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150 nfcv 2903 . . . . . . . . 9 𝑚 <
151 nfcv 2903 . . . . . . . . 9 𝑚(𝑎 + (1 / 𝑗))
15287, 150, 151nfbr 5195 . . . . . . . 8 𝑚((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153152, 74nfrabw 3473 . . . . . . 7 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154 nfcv 2903 . . . . . . . 8 𝑚𝑡
155154, 74nfin 4232 . . . . . . 7 𝑚(𝑡 ∩ dom (𝐹𝑙))
156153, 155nfeq 2917 . . . . . 6 𝑚{𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))
157 nfcv 2903 . . . . . 6 𝑚𝑆
158156, 157nfrabw 3473 . . . . 5 𝑚{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
159 nfcv 2903 . . . . 5 𝑘{𝑡𝑆 ∣ {𝑦 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹𝑙))}
160 nfcv 2903 . . . . 5 𝑙{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
161 nfcv 2903 . . . . 5 𝑗{𝑠𝑆 ∣ {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹𝑚))}
162 nfcv 2903 . . . . . . . . . . . 12 𝑦dom (𝐹𝑙)
163132nfdm 5965 . . . . . . . . . . . 12 𝑥dom (𝐹𝑙)
164 nfcv 2903 . . . . . . . . . . . . 13 𝑥 <
165 nfcv 2903 . . . . . . . . . . . . 13 𝑥(𝑎 + (1 / 𝑗))
166133, 164, 165nfbr 5195 . . . . . . . . . . . 12 𝑥((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167 nfv 1912 . . . . . . . . . . . 12 𝑦((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168 fveq2 6907 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → ((𝐹𝑙)‘𝑦) = ((𝐹𝑙)‘𝑥))
169168breq1d 5158 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (((𝐹𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170162, 163, 166, 167, 169cbvrabw 3471 . . . . . . . . . . 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 4221 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑙)))
173171, 172eqeq12d 2751 . . . . . . . . 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 3444 . . . . . . . 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 2825 . . . . . . . . . . 11 (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹𝑙) ↔ 𝑥 ∈ dom (𝐹𝑚)))
17798fveq1d 6909 . . . . . . . . . . . 12 (𝑙 = 𝑚 → ((𝐹𝑙)‘𝑥) = ((𝐹𝑚)‘𝑥))
178177breq1d 5158 . . . . . . . . . . 11 (𝑙 = 𝑚 → (((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179176, 178anbi12d 632 . . . . . . . . . 10 (𝑙 = 𝑚 → ((𝑥 ∈ dom (𝐹𝑙) ∧ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))) ↔ (𝑥 ∈ dom (𝐹𝑚) ∧ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)))))
180179rabbidva2 3435 . . . . . . . . 9 (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹𝑙) ∣ ((𝐹𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
18199ineq2d 4228 . . . . . . . . 9 (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹𝑙)) = (𝑠 ∩ dom (𝐹𝑚)))
182180, 181eqeq12d 2751 . . . . . . . 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 3441 . . . . . . 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 2775 . . . . . 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 7439 . . . . . . . . . . 11 (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186185oveq2d 7447 . . . . . . . . . 10 (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187186breq2d 5160 . . . . . . . . 9 (𝑗 = 𝑘 → (((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188187rabbidv 3441 . . . . . . . 8 (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹𝑚) ∣ ((𝐹𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189188eqeq1d 2737 . . . . . . 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 3441 . . . . . 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 2795 . . . . 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 7527 . . . 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 2744 . . 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 46732 . 2 ((𝜑𝑎 ∈ ℝ) → {𝑦𝐷 ∣ (𝐺𝑦) ≤ 𝑎} ∈ (𝑆t 𝐷))
1951, 2, 42, 126, 194issmfled 46713 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wnfc 2888  wrex 3068  {crab 3433  cin 3962  wss 3963   cuni 4912   ciun 4996   ciin 4997   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  cr 11152  1c1 11154   + caddc 11156   < clt 11293   / cdiv 11918  cn 12264  cz 12611  cuz 12876  cli 15517  SAlgcsalg 46264  SMblFncsmblfn 46651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cc 10473  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-ioo 13388  df-ico 13390  df-fl 13829  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-rlim 15522  df-rest 17469  df-salg 46265  df-smblfn 46652
This theorem is referenced by:  smflim2  46762
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