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Theorem smflimmpt 42961
Description: The limit of a sequence 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 ). 𝐴 can contain 𝑚 as a free variable, in other words it can be thought as an indexed collection 𝐴(𝑚). 𝐵 can be thought as a collection with two indices 𝐵(𝑚, 𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimmpt.p 𝑚𝜑
smflimmpt.x 𝑥𝜑
smflimmpt.n 𝑛𝜑
smflimmpt.m (𝜑𝑀 ∈ ℤ)
smflimmpt.z 𝑍 = (ℤ𝑀)
smflimmpt.a ((𝜑𝑚𝑍) → 𝐴𝑉)
smflimmpt.b ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
smflimmpt.s (𝜑𝑆 ∈ SAlg)
smflimmpt.l ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
smflimmpt.d 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
smflimmpt.g 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵)))
Assertion
Ref Expression
smflimmpt (𝜑𝐺 ∈ (SMblFn‘𝑆))
Distinct variable groups:   𝐴,𝑛,𝑥   𝐵,𝑛   𝑆,𝑚,𝑛   𝑚,𝑍,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐴(𝑚)   𝐵(𝑥,𝑚)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑚,𝑛)   𝑉(𝑥,𝑚,𝑛)   𝑊(𝑥,𝑚,𝑛)

Proof of Theorem smflimmpt
StepHypRef Expression
1 smflimmpt.g . . . 4 𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵)))
21a1i 11 . . 3 (𝜑𝐺 = (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵))))
3 smflimmpt.x . . . 4 𝑥𝜑
4 smflimmpt.d . . . . . 6 𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
54a1i 11 . . . . 5 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
6 smflimmpt.n . . . . . . . . . . . . . 14 𝑛𝜑
7 smflimmpt.p . . . . . . . . . . . . . . . 16 𝑚𝜑
8 nfv 1906 . . . . . . . . . . . . . . . 16 𝑚 𝑛𝑍
97, 8nfan 1891 . . . . . . . . . . . . . . 15 𝑚(𝜑𝑛𝑍)
10 smflimmpt.z . . . . . . . . . . . . . . . . . . . 20 𝑍 = (ℤ𝑀)
1110uztrn2 12250 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1211adantll 710 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 simpll 763 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
14 smflimmpt.a . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → 𝐴𝑉)
1514mptexd 6978 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
1613, 12, 15syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) ∈ V)
17 eqid 2818 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1817fvmpt2 6771 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
1912, 16, 18syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2019dmeqd 5767 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
21 nfv 1906 . . . . . . . . . . . . . . . . . . . 20 𝑥 𝑛𝑍
223, 21nfan 1891 . . . . . . . . . . . . . . . . . . 19 𝑥(𝜑𝑛𝑍)
23 nfv 1906 . . . . . . . . . . . . . . . . . . 19 𝑥 𝑚 ∈ (ℤ𝑛)
2422, 23nfan 1891 . . . . . . . . . . . . . . . . . 18 𝑥((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛))
25 simplll 771 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝜑)
2612adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝑚𝑍)
27 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝑥𝐴)
28 smflimmpt.b . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
2925, 26, 27, 28syl3anc 1363 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝐵𝑊)
30 eqid 2818 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3124, 29, 30fnmptd 6482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) Fn 𝐴)
3231fndmd 6449 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = 𝐴)
3320, 32eqtr2d 2854 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
349, 33iineq2d 4933 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
356, 34iuneq2df 41185 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
36 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → 𝑚𝑍)
3736, 15, 18syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
3837eqcomd 2824 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) = ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3938dmeqd 5767 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4013, 12, 39syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
419, 40iineq2d 4933 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
426, 41iuneq2df 41185 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4335, 42eqtr4d 2856 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4443eleq2d 2895 . . . . . . . . . . 11 (𝜑 → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵)))
4544biimpa 477 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4645adantrr 713 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
47 eliun 4914 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4847biimpi 217 . . . . . . . . . . . 12 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4948adantl 482 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5049adantrr 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
51 nfv 1906 . . . . . . . . . . . . 13 𝑛(𝑚𝑍𝐵) ∈ dom ⇝
526, 51nfan 1891 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )
53 nfv 1906 . . . . . . . . . . . 12 𝑛(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝
54 simpllr 772 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍𝐵) ∈ dom ⇝ )
55 nfcv 2974 . . . . . . . . . . . . . . . . . 18 𝑚𝑥
56 nfii1 4945 . . . . . . . . . . . . . . . . . 18 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
5755, 56nfel 2989 . . . . . . . . . . . . . . . . 17 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
589, 57nfan 1891 . . . . . . . . . . . . . . . 16 𝑚((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5910eluzelz2 41552 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ ℤ)
6059ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
61 eqid 2818 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6210fvexi 6677 . . . . . . . . . . . . . . . . 17 𝑍 ∈ V
6362a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑍 ∈ V)
6410uzssd3 41576 . . . . . . . . . . . . . . . . 17 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
6564ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
66 fvexd 6678 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
67 eliinid 41254 . . . . . . . . . . . . . . . . . 18 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6867adantll 710 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6913adantlr 711 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
7012adantlr 711 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
7169, 70, 68, 28syl3anc 1363 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
7230fvmpt2 6771 . . . . . . . . . . . . . . . . 17 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7368, 71, 72syl2anc 584 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7458, 60, 61, 63, 63, 65, 65, 66, 73climeldmeqmpt3 41846 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7574adantllr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7654, 75mpbird 258 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
7776exp31 420 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
7852, 53, 77rexlimd 3314 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
7978adantrl 712 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
8050, 79mpd 15 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
8146, 80jca 512 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
8281ex 413 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
8344biimpar 478 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8483adantrr 713 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8584, 48syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
866, 53nfan 1891 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
87 simpllr 772 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
8874adantllr 715 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
8987, 88mpbid 233 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍𝐵) ∈ dom ⇝ )
9089exp31 420 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ )))
9186, 51, 90rexlimd 3314 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ ))
9291adantrl 712 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ ))
9385, 92mpd 15 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → (𝑚𝑍𝐵) ∈ dom ⇝ )
9484, 93jca 512 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ))
9594ex 413 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )))
9682, 95impbid 213 . . . . . 6 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ↔ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
973, 96rabbida3 41278 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
985, 97eqtrd 2853 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
994eleq2i 2901 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
10099biimpi 217 . . . . . . . 8 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
101 rabidim1 3378 . . . . . . . 8 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ } → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
102100, 101, 483syl 18 . . . . . . 7 (𝑥𝐷 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
103102adantl 482 . . . . . 6 ((𝜑𝑥𝐷) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
104 nfcv 2974 . . . . . . . . 9 𝑛𝑥
105 nfiu1 4944 . . . . . . . . . . 11 𝑛 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴
10651, 105nfrabw 3383 . . . . . . . . . 10 𝑛{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
1074, 106nfcxfr 2972 . . . . . . . . 9 𝑛𝐷
108104, 107nfel 2989 . . . . . . . 8 𝑛 𝑥𝐷
1096, 108nfan 1891 . . . . . . 7 𝑛(𝜑𝑥𝐷)
110 nfv 1906 . . . . . . 7 𝑛( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))
1117, 8, 57nf3an 1893 . . . . . . . . . 10 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
112 simp2 1129 . . . . . . . . . . 11 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛𝑍)
113112, 59syl 17 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
11462a1i 11 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑍 ∈ V)
11510, 112uzssd2 41567 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
116 fvexd 6678 . . . . . . . . . 10 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
117673ad2antl3 1179 . . . . . . . . . . 11 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
118 simpl1 1183 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
119112, 11sylan 580 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
120118, 119, 117, 28syl3anc 1363 . . . . . . . . . . 11 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
121117, 120, 72syl2anc 584 . . . . . . . . . 10 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
122111, 113, 61, 114, 114, 115, 115, 116, 121climfveqmpt3 41839 . . . . . . . . 9 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
1231223exp 1111 . . . . . . . 8 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))))
124123adantr 481 . . . . . . 7 ((𝜑𝑥𝐷) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))))
125109, 110, 124rexlimd 3314 . . . . . 6 ((𝜑𝑥𝐷) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))))
126103, 125mpd 15 . . . . 5 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
127126eqcomd 2824 . . . 4 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍𝐵)) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))))
1283, 98, 127mpteq12da 41389 . . 3 (𝜑 → (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))))
12938eqcomd 2824 . . . . . . . . 9 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
130129fveq1d 6665 . . . . . . . 8 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
1317, 130mpteq2da 5151 . . . . . . 7 (𝜑 → (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))
132131eqcomd 2824 . . . . . 6 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
133132eleq1d 2894 . . . . 5 (𝜑 → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
1343, 42, 133rabbida2 41275 . . . 4 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ })
135130eqcomd 2824 . . . . . 6 ((𝜑𝑚𝑍) → ((𝑥𝐴𝐵)‘𝑥) = (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))
1367, 135mpteq2da 5151 . . . . 5 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
137136fveq2d 6667 . . . 4 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1383, 134, 137mpteq12df 5139 . . 3 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
1392, 128, 1383eqtrd 2857 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
140 nfmpt1 5155 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
141 nfcv 2974 . . . 4 𝑥𝑍
142 nfmpt1 5155 . . . 4 𝑥(𝑥𝐴𝐵)
143141, 142nfmpt 5154 . . 3 𝑥(𝑚𝑍 ↦ (𝑥𝐴𝐵))
144 smflimmpt.m . . 3 (𝜑𝑀 ∈ ℤ)
145 smflimmpt.s . . 3 (𝜑𝑆 ∈ SAlg)
146 smflimmpt.l . . . 4 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ (SMblFn‘𝑆))
1477, 146, 17fmptdf 6873 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
148 eqid 2818 . . 3 {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ }
149 eqid 2818 . . 3 (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
150140, 143, 144, 10, 145, 147, 148, 149smflim2 42957 . 2 (𝜑 → (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))) ∈ (SMblFn‘𝑆))
151139, 150eqeltrd 2910 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wnf 1775  wcel 2105  wrex 3136  {crab 3139  Vcvv 3492  wss 3933   ciun 4910   ciin 4911  cmpt 5137  dom cdm 5548  cfv 6348  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:  smflimsuplem3  42973
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