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Theorem smflimmpt 45299
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 1917 . . . . . . . . . . . . . . . 16 𝑚 𝑛𝑍
97, 8nfan 1902 . . . . . . . . . . . . . . 15 𝑚(𝜑𝑛𝑍)
10 smflimmpt.z . . . . . . . . . . . . . . . . . . . 20 𝑍 = (ℤ𝑀)
1110uztrn2 12823 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝑍𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
1211adantll 712 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
14 smflimmpt.a . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑚𝑍) → 𝐴𝑉)
1514mptexd 7210 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
1613, 12, 15syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) ∈ V)
17 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1817fvmpt2 6995 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
1912, 16, 18syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2019dmeqd 5897 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = dom (𝑥𝐴𝐵))
21 nfv 1917 . . . . . . . . . . . . . . . . . . . 20 𝑥 𝑛𝑍
223, 21nfan 1902 . . . . . . . . . . . . . . . . . . 19 𝑥(𝜑𝑛𝑍)
23 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑥 𝑚 ∈ (ℤ𝑛)
2422, 23nfan 1902 . . . . . . . . . . . . . . . . . 18 𝑥((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛))
25 simplll 773 . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) ∧ 𝑥𝐴) → 𝐵𝑊)
30 eqid 2731 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3124, 29, 30fnmptd 6678 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) Fn 𝐴)
3231fndmd 6643 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = 𝐴)
3320, 32eqtr2d 2772 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
349, 33iineq2d 5013 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
356, 34iuneq2df 43504 . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) = ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3938dmeqd 5897 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4013, 12, 39syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
419, 40iineq2d 5013 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
426, 41iuneq2df 43504 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4335, 42eqtr4d 2774 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4443eleq2d 2818 . . . . . . . . . . 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 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → 𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
47 eliun 4994 . . . . . . . . . . . . 13 (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ↔ ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
4847biimpi 215 . . . . . . . . . . . 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 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )) → ∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
51 nfv 1917 . . . . . . . . . . . . 13 𝑛(𝑚𝑍𝐵) ∈ dom ⇝
526, 51nfan 1902 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ )
53 nfv 1917 . . . . . . . . . . . 12 𝑛(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝
54 simpllr 774 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍𝐵) ∈ dom ⇝ )
55 nfcv 2902 . . . . . . . . . . . . . . . . . 18 𝑚𝑥
56 nfii1 5025 . . . . . . . . . . . . . . . . . 18 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
5755, 56nfel 2916 . . . . . . . . . . . . . . . . 17 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
589, 57nfan 1902 . . . . . . . . . . . . . . . 16 𝑚((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5910eluzelz2 43886 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ ℤ)
6059ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
61 eqid 2731 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6210fvexi 6892 . . . . . . . . . . . . . . . . 17 𝑍 ∈ V
6362a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑍 ∈ V)
6410uzssd3 43909 . . . . . . . . . . . . . . . . 17 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
6564ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
66 fvexd 6893 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
67 eliinid 43571 . . . . . . . . . . . . . . . . . 18 ((𝑥 𝑚 ∈ (ℤ𝑛)𝐴𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6867adantll 712 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
6913adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
7012adantlr 713 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
7169, 70, 68, 28syl3anc 1371 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐵𝑊)
7230fvmpt2 6995 . . . . . . . . . . . . . . . . 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 44178 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7574adantllr 717 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
7654, 75mpbird 256 . . . . . . . . . . . . 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 3262 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ))
7978adantrl 714 . . . . . . . . . 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 715 . . . . . . . . 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 1902 . . . . . . . . . . . 12 𝑛(𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
87 simpllr 774 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )
8874adantllr 717 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) ∧ 𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍𝐵) ∈ dom ⇝ ))
8987, 88mpbid 231 . . . . . . . . . . . . 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 3262 . . . . . . . . . . 11 ((𝜑 ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → (𝑚𝑍𝐵) ∈ dom ⇝ ))
9291adantrl 714 . . . . . . . . . 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 211 . . . . . 6 (𝜑 → ((𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∧ (𝑚𝑍𝐵) ∈ dom ⇝ ) ↔ (𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∧ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ )))
973, 96rabbida3 43595 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
985, 97eqtrd 2771 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
994eleq2i 2824 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
10099biimpi 215 . . . . . . . 8 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
101 rabidim1 3452 . . . . . . . 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 2902 . . . . . . . . 9 𝑛𝑥
105 nfiu1 5024 . . . . . . . . . . 11 𝑛 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴
10651, 105nfrabw 3468 . . . . . . . . . 10 𝑛{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
1074, 106nfcxfr 2900 . . . . . . . . 9 𝑛𝐷
108104, 107nfel 2916 . . . . . . . 8 𝑛 𝑥𝐷
1096, 108nfan 1902 . . . . . . 7 𝑛(𝜑𝑥𝐷)
110 nfv 1917 . . . . . . 7 𝑛( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))
1117, 8, 57nf3an 1904 . . . . . . . . . 10 𝑚(𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
112 simp2 1137 . . . . . . . . . . 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 43900 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
116 fvexd 6893 . . . . . . . . . 10 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
117673ad2antl3 1187 . . . . . . . . . . 11 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑥𝐴)
118 simpl1 1191 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝜑)
119112, 11sylan 580 . . . . . . . . . . . 12 (((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
120118, 119, 117, 28syl3anc 1371 . . . . . . . . . . 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 44171 . . . . . . . . 9 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
1231223exp 1119 . . . . . . . 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 3262 . . . . . 6 ((𝜑𝑥𝐷) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))))
126103, 125mpd 15 . . . . 5 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
127126eqcomd 2737 . . . 4 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍𝐵)) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))))
1283, 98, 127mpteq12da 5226 . . 3 (𝜑 → (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))))
12938eqcomd 2737 . . . . . . . . 9 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
130129fveq1d 6880 . . . . . . . 8 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
1317, 130mpteq2da 5239 . . . . . . 7 (𝜑 → (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))
132131eqcomd 2737 . . . . . 6 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
133132eleq1d 2817 . . . . 5 (𝜑 → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
1343, 42, 133rabbida2 43592 . . . 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 2737 . . . . . 6 ((𝜑𝑚𝑍) → ((𝑥𝐴𝐵)‘𝑥) = (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))
1367, 135mpteq2da 5239 . . . . 5 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
137136fveq2d 6882 . . . 4 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1383, 134, 137mpteq12df 5227 . . 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 2775 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
140 nfmpt1 5249 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
141 nfcv 2902 . . . 4 𝑥𝑍
142 nfmpt1 5249 . . . 4 𝑥(𝑥𝐴𝐵)
143141, 142nfmpt 5248 . . 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 7101 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
148 eqid 2731 . . 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 2731 . . 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 45295 . 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 2832 1 (𝜑𝐺 ∈ (SMblFn‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  wrex 3069  {crab 3431  Vcvv 3473  wss 3944   ciun 4990   ciin 4991  cmpt 5224  dom cdm 5669  cfv 6532  cz 12540  cuz 12804  cli 15410  SAlgcsalg 44797  SMblFncsmblfn 45184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cc 10412  ax-ac2 10440  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-oadd 8452  df-omul 8453  df-er 8686  df-map 8805  df-pm 8806  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-sup 9419  df-inf 9420  df-oi 9487  df-card 9916  df-acn 9919  df-ac 10093  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-n0 12455  df-z 12541  df-uz 12805  df-q 12915  df-rp 12957  df-ioo 13310  df-ico 13312  df-fl 13739  df-seq 13949  df-exp 14010  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-clim 15414  df-rlim 15415  df-rest 17350  df-salg 44798  df-smblfn 45185
This theorem is referenced by:  smflimsuplem3  45311
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