Theorem smflim2 47634

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 ). TODO: this has fewer distinct variable conditions than smflim 47605 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflim2.n	⊢ Ⅎ𝑚𝐹
smflim2.x	⊢ Ⅎ𝑥𝐹
smflim2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflim2.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflim2.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflim2.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflim2.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflim2.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))

Assertion

Ref	Expression
smflim2	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups:   𝑛,𝐹   𝑚,𝑍,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑚, 𝑛)   𝐷(𝑥, 𝑚, 𝑛)   𝑆(𝑥, 𝑚, 𝑛)   𝐹(𝑥, 𝑚)   𝐺(𝑥, 𝑚, 𝑛)   𝑀(𝑥, 𝑚, 𝑛)

Proof of Theorem smflim2

Dummy variables 𝑦 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2922	. 2 ⊢ Ⅎ𝑗𝐹
2		nfcv 2922	. 2 ⊢ Ⅎ𝑦𝐹
3		smflim2.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
4		smflim2.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
5		smflim2.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6		smflim2.f	. 2 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
7		smflim2.d	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
8		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑍
9		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
10		smflim2.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
11		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
12	10, 11	nffv 6888	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
13	12	nfdm 5935	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
14	9, 13	nfiin 4983	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
15	8, 14	nfiun 4982	. . . 4 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
16		nfcv 2922	. . . 4 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
17		nfv 1947	. . . 4 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
18		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑗((𝐹‘𝑚)‘𝑦)
19		smflim2.n	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
20		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑗
21	19, 20	nffv 6888	. . . . . . . 8 ⊢ Ⅎ𝑚(𝐹‘𝑗)
22		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑚𝑦
23	21, 22	nffv 6888	. . . . . . 7 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑦)
24		fveq2 6878	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
25	24	fveq1d 6880	. . . . . . 7 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
26	18, 23, 25	cbvmpt 5207	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
27		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑗
28	10, 27	nffv 6888	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑗)
29		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
30	28, 29	nffv 6888	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑗)‘𝑦)
31	8, 30	nfmpt 5203	. . . . . 6 ⊢ Ⅎ𝑥(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
32	26, 31	nfcxfr 2920	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
33		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥dom ⇝
34	32, 33	nfel 2936	. . . 4 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
35		fveq2 6878	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
36	35	mpteq2dv 5199	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
37	36	eleq1d 2845	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
38	15, 16, 17, 34, 37	cbvrabw 3446	. . 3 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ }
39		fveq2 6878	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
40	39	iineq1d 45922	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚))
41		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
42	21	nfdm 5935	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
43	24	dmeqd 5889	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
44	41, 42, 43	cbviin 4994	. . . . . . . . 9 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗)
45	44	a1i 11	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
46	40, 45	eqtrd 2795	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
47	46	cbviunv 4997	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗)
48	47	eleq2i 2852	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
49	26	eleq1i 2851	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ )
50	48, 49	anbi12i 640	. . . 4 ⊢ ((𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗) ∧ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ ))
51	50	rabbia2 3415	. . 3 ⊢ {𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗) ∣ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ }
52	7, 38, 51	3eqtri 2787	. 2 ⊢ 𝐷 = {𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗) ∣ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ }
53		smflim2.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
54		nfrab1 3431	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
55	7, 54	nfcxfr 2920	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2922	. . . 4 ⊢ Ⅎ𝑦𝐷
57		nfcv 2922	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
58		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥 ⇝
59	58, 31	nffv 6888	. . . 4 ⊢ Ⅎ𝑥( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
60	26	a1i 11	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
61	36, 60	eqtrd 2795	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
62	61	fveq2d 6882	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
63	55, 56, 57, 59, 62	cbvmptf 5205	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
64	53, 63	eqtri 2783	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 47605	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  {crab 3412  ∪ ciun 4951  ∩ ciin 4952   ↦ cmpt 5186  dom cdm 5655  ⟶wf 6529  ‘cfv 6533  ℤcz 12615  ℤ_≥cuz 12887   ⇝ cli 15571  SAlgcsalg 47136  SMblFncsmblfn 47523

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cc 10437  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-oadd 8459  df-omul 8460  df-er 8696  df-map 8828  df-pm 8829  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9944  df-acn 9947  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-z 12616  df-uz 12888  df-q 12998  df-rp 13043  df-ioo 13402  df-ico 13404  df-fl 13853  df-seq 14066  df-exp 14126  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-rlim 15576  df-rest 17507  df-salg 47137  df-smblfn 47524

This theorem is used by:  smflimmpt  47638