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Theorem smflimmpt 41337
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 indexes 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 1883 . . . . . . . . . . . . . . . 16 𝑚 𝑛𝑍
97, 8nfan 1868 . . . . . . . . . . . . . . 15 𝑚(𝜑𝑛𝑍)
10 smflimmpt.z . . . . . . . . . . . . . . . . . . . 20 𝑍 = (ℤ𝑀)
1110uztrn2 11743 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1211adantll 750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 simpll 805 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
14 smflimmpt.a . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → 𝐴𝑉)
1514mptexd 6528 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
1613, 12, 15syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) ∈ V)
17 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1817fvmpt2 6330 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
1912, 16, 18syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2019dmeqd 5358 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
21 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑥 𝑛𝑍
223, 21nfan 1868 . . . . . . . . . . . . . . . . . . 19 𝑥(𝜑𝑛𝑍)
23 nfv 1883 . . . . . . . . . . . . . . . . . . 19 𝑥 𝑚 ∈ (ℤ𝑛)
2422, 23nfan 1868 . . . . . . . . . . . . . . . . . 18 𝑥((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛))
25 simplll 813 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝜑)
2612adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝑚𝑍)
27 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝑥𝐴)
28 smflimmpt.b . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍𝑥𝐴) → 𝐵𝑊)
2925, 26, 27, 28syl3anc 1366 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝐵𝑊)
30 eqid 2651 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3124, 29, 30fnmptd 39748 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) Fn 𝐴)
3231fndmd 39755 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = 𝐴)
3320, 32eqtr2d 2686 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
349, 33iineq2d 4573 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
356, 34iuneq2df 39526 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
36 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → 𝑚𝑍)
3736, 15, 18syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
3837eqcomd 2657 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) = ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3938dmeqd 5358 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4013, 12, 39syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
419, 40iineq2d 4573 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
426, 41iuneq2df 39526 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4335, 42eqtr4d 2688 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4443eleq2d 2716 . . . . . . . . . . 11 (𝜑 → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵)))
4544biimpa 500 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4645adantrr 753 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
47 eliun 4556 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4847biimpi 206 . . . . . . . . . . . 12 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4948adantl 481 . . . . . . . . . . 11 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5049adantrr 753 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
51 nfv 1883 . . . . . . . . . . . . 13 𝑛(𝑚𝑍𝐵) ∈ dom ⇝
526, 51nfan 1868 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )
53 nfv 1883 . . . . . . . . . . . 12 𝑛(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝
54 simpllr 815 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍𝐵) ∈ dom ⇝ )
55 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑚𝑥
56 nfii1 4583 . . . . . . . . . . . . . . . . . 18 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
5755, 56nfel 2806 . . . . . . . . . . . . . . . . 17 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
589, 57nfan 1868 . . . . . . . . . . . . . . . 16 𝑚((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5910eluzelz2 39940 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ ℤ)
6059ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
61 eqid 2651 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6210fvexi 6240 . . . . . . . . . . . . . . . . 17 𝑍 ∈ V
6362a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑍 ∈ V)
6410uzssd3 39966 . . . . . . . . . . . . . . . . 17 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
6564ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
66 fvexd 6241 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
67 eliinid 39608 . . . . . . . . . . . . . . . . . 18 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6867adantll 750 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6913adantlr 751 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
7012adantlr 751 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
7169, 70, 68, 28syl3anc 1366 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
7230fvmpt2 6330 . . . . . . . . . . . . . . . . 17 ((𝑥𝐴𝐵𝑊) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7368, 71, 72syl2anc 694 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
7458, 60, 61, 63, 63, 65, 65, 66, 73climeldmeqmpt3 40239 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7574adantllr 755 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7654, 75mpbird 247 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
7776exp31 629 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
7852, 53, 77rexlimd 3055 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
7978adantrl 752 . . . . . . . . . 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 553 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
8281ex 449 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
8344biimpar 501 . . . . . . . . . 10 ((𝜑𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵)) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴)
8483adantrr 753 . . . . . . . . 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 1868 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
87 simpllr 815 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
8874adantllr 755 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
8987, 88mpbid 222 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍𝐵) ∈ dom ⇝ )
9089exp31 629 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ )))
9186, 51, 90rexlimd 3055 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ ))
9291adantrl 752 . . . . . . . . . 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 553 . . . . . . . 8 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ))
9594ex 449 . . . . . . 7 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )))
9682, 95impbid 202 . . . . . 6 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ↔ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
973, 96rabbida3 39634 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
985, 97eqtrd 2685 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
994eleq2i 2722 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
10099biimpi 206 . . . . . . . 8 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
101 rabidim1 3147 . . . . . . . 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 481 . . . . . 6 ((𝜑𝑥𝐷) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
104 nfcv 2793 . . . . . . . . 9 𝑛𝑥
105 nfiu1 4582 . . . . . . . . . . 11 𝑛 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴
10651, 105nfrab 3153 . . . . . . . . . 10 𝑛{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
1074, 106nfcxfr 2791 . . . . . . . . 9 𝑛𝐷
108104, 107nfel 2806 . . . . . . . 8 𝑛 𝑥𝐷
1096, 108nfan 1868 . . . . . . 7 𝑛(𝜑𝑥𝐷)
110 nfv 1883 . . . . . . 7 𝑛( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))
1117, 8, 57nf3an 1871 . . . . . . . . . 10 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
112 simp2 1082 . . . . . . . . . . 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 39957 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
116 fvexd 6241 . . . . . . . . . 10 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
117673ad2antl3 1245 . . . . . . . . . . 11 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
118 simpl1 1084 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
119112, 11sylan 487 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
120118, 119, 117, 28syl3anc 1366 . . . . . . . . . . 11 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
121117, 120, 72syl2anc 694 . . . . . . . . . 10 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
122111, 113, 61, 114, 114, 115, 115, 116, 121climfveqmpt3 40232 . . . . . . . . 9 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
1231223exp 1283 . . . . . . . 8 (𝜑 → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))))
124123adantr 480 . . . . . . 7 ((𝜑𝑥𝐷) → (𝑛𝑍 → (𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))))
125109, 110, 124rexlimd 3055 . . . . . 6 ((𝜑𝑥𝐷) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))))
126103, 125mpd 15 . . . . 5 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
127126eqcomd 2657 . . . 4 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍𝐵)) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))))
1283, 98, 127mpteq12da 39766 . . 3 (𝜑 → (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))))
12938eqcomd 2657 . . . . . . . . 9 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
130129fveq1d 6231 . . . . . . . 8 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
1317, 130mpteq2da 4776 . . . . . . 7 (𝜑 → (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))
132131eqcomd 2657 . . . . . 6 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
133132eleq1d 2715 . . . . 5 (𝜑 → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
1343, 42, 133rabbida2 39631 . . . 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 2657 . . . . . 6 ((𝜑𝑚𝑍) → ((𝑥𝐴𝐵)‘𝑥) = (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))
1367, 135mpteq2da 4776 . . . . 5 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
137136fveq2d 6233 . . . 4 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1383, 134, 137mpteq12d 4767 . . 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 2689 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
140 nfmpt1 4780 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
141 nfcv 2793 . . . 4 𝑥𝑍
142 nfmpt1 4780 . . . 4 𝑥(𝑥𝐴𝐵)
143141, 142nfmpt 4779 . . 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 6427 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
148 eqid 2651 . . 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 2651 . . 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 41333 . 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 2730 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wnf 1748  wcel 2030  wrex 2942  {crab 2945  Vcvv 3231  wss 3607   ciun 4552   ciin 4553  cmpt 4762  dom cdm 5143  cfv 5926  cz 11415  cuz 11725  cli 14259  SAlgcsalg 40846  SMblFncsmblfn 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-fl 12633  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-rest 16130  df-salg 40847  df-smblfn 41231
This theorem is referenced by:  smflimsuplem3  41349
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