Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smflimmpt Structured version   Visualization version   GIF version

Theorem smflimmpt 45605
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 12843 . . . . . . . . . . . . . . . . . . 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 7228 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) ∈ V)
1613, 12, 15syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) ∈ V)
17 eqid 2732 . . . . . . . . . . . . . . . . . . 19 (𝑚𝑍 ↦ (𝑥𝐴𝐵)) = (𝑚𝑍 ↦ (𝑥𝐴𝐵))
1817fvmpt2 7009 . . . . . . . . . . . . . . . . . 18 ((𝑚𝑍 ∧ (𝑥𝐴𝐵) ∈ V) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
1912, 16, 18syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
2019dmeqd 5905 . . . . . . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
3124, 29, 30fnmptd 6691 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → (𝑥𝐴𝐵) Fn 𝐴)
3231fndmd 6654 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = 𝐴)
3320, 32eqtr2d 2773 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → 𝐴 = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
349, 33iineq2d 5020 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
356, 34iuneq2df 43815 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚𝑍) → (𝑥𝐴𝐵) = ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
3938dmeqd 5905 . . . . . . . . . . . . . . . 16 ((𝜑𝑚𝑍) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4013, 12, 39syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑛𝑍) ∧ 𝑚 ∈ (ℤ𝑛)) → dom (𝑥𝐴𝐵) = dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
419, 40iineq2d 5020 . . . . . . . . . . . . . 14 ((𝜑𝑛𝑍) → 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
426, 41iuneq2df 43815 . . . . . . . . . . . . 13 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚))
4335, 42eqtr4d 2775 . . . . . . . . . . . 12 (𝜑 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 = 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵))
4443eleq2d 2819 . . . . . . . . . . 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 5001 . . . . . . . . . . . . 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 2903 . . . . . . . . . . . . . . . . . 18 𝑚𝑥
56 nfii1 5032 . . . . . . . . . . . . . . . . . 18 𝑚 𝑚 ∈ (ℤ𝑛)𝐴
5755, 56nfel 2917 . . . . . . . . . . . . . . . . 17 𝑚 𝑥 𝑚 ∈ (ℤ𝑛)𝐴
589, 57nfan 1902 . . . . . . . . . . . . . . . 16 𝑚((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴)
5910eluzelz2 44192 . . . . . . . . . . . . . . . . 17 (𝑛𝑍𝑛 ∈ ℤ)
6059ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑛 ∈ ℤ)
61 eqid 2732 . . . . . . . . . . . . . . . 16 (ℤ𝑛) = (ℤ𝑛)
6210fvexi 6905 . . . . . . . . . . . . . . . . 17 𝑍 ∈ V
6362a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → 𝑍 ∈ V)
6410uzssd3 44215 . . . . . . . . . . . . . . . . 17 (𝑛𝑍 → (ℤ𝑛) ⊆ 𝑍)
6564ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
66 fvexd 6906 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛𝑍) ∧ 𝑥 𝑚 ∈ (ℤ𝑛)𝐴) ∧ 𝑚 ∈ (ℤ𝑛)) → ((𝑥𝐴𝐵)‘𝑥) ∈ V)
67 eliinid 43882 . . . . . . . . . . . . . . . . . 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 7009 . . . . . . . . . . . . . . . . 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 44484 . . . . . . . . . . . . . . 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 3263 . . . . . . . . . . 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 3263 . . . . . . . . . . 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 43906 . . . . 5 (𝜑 → {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ } = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
985, 97eqtrd 2772 . . . 4 (𝜑𝐷 = {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ })
994eleq2i 2825 . . . . . . . . 9 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
10099biimpi 215 . . . . . . . 8 (𝑥𝐷𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ })
101 rabidim1 3453 . . . . . . . 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 2903 . . . . . . . . 9 𝑛𝑥
105 nfiu1 5031 . . . . . . . . . . 11 𝑛 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴
10651, 105nfrabw 3468 . . . . . . . . . 10 𝑛{𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)𝐴 ∣ (𝑚𝑍𝐵) ∈ dom ⇝ }
1074, 106nfcxfr 2901 . . . . . . . . 9 𝑛𝐷
108104, 107nfel 2917 . . . . . . . 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 44206 . . . . . . . . . 10 ((𝜑𝑛𝑍𝑥 𝑚 ∈ (ℤ𝑛)𝐴) → (ℤ𝑛) ⊆ 𝑍)
116 fvexd 6906 . . . . . . . . . 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 44477 . . . . . . . . 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 3263 . . . . . 6 ((𝜑𝑥𝐷) → (∃𝑛𝑍 𝑥 𝑚 ∈ (ℤ𝑛)𝐴 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵))))
126103, 125mpd 15 . . . . 5 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍𝐵)))
127126eqcomd 2738 . . . 4 ((𝜑𝑥𝐷) → ( ⇝ ‘(𝑚𝑍𝐵)) = ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))))
1283, 98, 127mpteq12da 5233 . . 3 (𝜑 → (𝑥𝐷 ↦ ( ⇝ ‘(𝑚𝑍𝐵))) = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom (𝑥𝐴𝐵) ∣ (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))))
12938eqcomd 2738 . . . . . . . . 9 ((𝜑𝑚𝑍) → ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) = (𝑥𝐴𝐵))
130129fveq1d 6893 . . . . . . . 8 ((𝜑𝑚𝑍) → (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥) = ((𝑥𝐴𝐵)‘𝑥))
1317, 130mpteq2da 5246 . . . . . . 7 (𝜑 → (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) = (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)))
132131eqcomd 2738 . . . . . 6 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
133132eleq1d 2818 . . . . 5 (𝜑 → ((𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) ∈ dom ⇝ ↔ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ ))
1343, 42, 133rabbida2 43903 . . . 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 2738 . . . . . 6 ((𝜑𝑚𝑍) → ((𝑥𝐴𝐵)‘𝑥) = (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))
1367, 135mpteq2da 5246 . . . . 5 (𝜑 → (𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥)) = (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))
137136fveq2d 6895 . . . 4 (𝜑 → ( ⇝ ‘(𝑚𝑍 ↦ ((𝑥𝐴𝐵)‘𝑥))) = ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥))))
1383, 134, 137mpteq12df 5234 . . 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 2776 . 2 (𝜑𝐺 = (𝑥 ∈ {𝑥 𝑛𝑍 𝑚 ∈ (ℤ𝑛)dom ((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚) ∣ (𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑚𝑍 ↦ (((𝑚𝑍 ↦ (𝑥𝐴𝐵))‘𝑚)‘𝑥)))))
140 nfmpt1 5256 . . 3 𝑚(𝑚𝑍 ↦ (𝑥𝐴𝐵))
141 nfcv 2903 . . . 4 𝑥𝑍
142 nfmpt1 5256 . . . 4 𝑥(𝑥𝐴𝐵)
143141, 142nfmpt 5255 . . 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 7118 . . 3 (𝜑 → (𝑚𝑍 ↦ (𝑥𝐴𝐵)):𝑍⟶(SMblFn‘𝑆))
148 eqid 2732 . . 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 2732 . . 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 45601 . 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 2833 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 3070  {crab 3432  Vcvv 3474  wss 3948   ciun 4997   ciin 4998  cmpt 5231  dom cdm 5676  cfv 6543  cz 12560  cuz 12824  cli 15430  SAlgcsalg 45103  SMblFncsmblfn 45490
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-acn 9939  df-ac 10113  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-ioo 13330  df-ico 13332  df-fl 13759  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-rlim 15435  df-rest 17370  df-salg 45104  df-smblfn 45491
This theorem is referenced by:  smflimsuplem3  45617
  Copyright terms: Public domain W3C validator