Theorem smflim2 44226

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 44199 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 2906	. 2 ⊢ Ⅎ𝑗𝐹
2		nfcv 2906	. 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 2906	. . . . 5 ⊢ Ⅎ𝑥𝑍
9		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
10		smflim2.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
11		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
12	10, 11	nffv 6766	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
13	12	nfdm 5849	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
14	9, 13	nfiin 4952	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
15	8, 14	nfiun 4951	. . . 4 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
16		nfcv 2906	. . . 4 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
17		nfv 1918	. . . 4 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
18		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑗((𝐹‘𝑚)‘𝑦)
19		smflim2.n	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
20		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑗
21	19, 20	nffv 6766	. . . . . . . 8 ⊢ Ⅎ𝑚(𝐹‘𝑗)
22		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑚𝑦
23	21, 22	nffv 6766	. . . . . . 7 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑦)
24		fveq2 6756	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
25	24	fveq1d 6758	. . . . . . 7 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
26	18, 23, 25	cbvmpt 5181	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
27		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑗
28	10, 27	nffv 6766	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑗)
29		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
30	28, 29	nffv 6766	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑗)‘𝑦)
31	8, 30	nfmpt 5177	. . . . . 6 ⊢ Ⅎ𝑥(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
32	26, 31	nfcxfr 2904	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
33		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥dom ⇝
34	32, 33	nfel 2920	. . . 4 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
35		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
36	35	mpteq2dv 5172	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
37	36	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
38	15, 16, 17, 34, 37	cbvrabw 3414	. . 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 6756	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
40	39	iineq1d 42529	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚))
41		nfcv 2906	. . . . . . . . . 10 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
42	21	nfdm 5849	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
43	24	dmeqd 5803	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
44	41, 42, 43	cbviin 4963	. . . . . . . . 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 2778	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
47	46	cbviunv 4966	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗)
48	47	eleq2i 2830	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
49	26	eleq1i 2829	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ )
50	48, 49	anbi12i 626	. . . 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 3401	. . 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 2770	. 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 3310	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
55	7, 54	nfcxfr 2904	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2906	. . . 4 ⊢ Ⅎ𝑦𝐷
57		nfcv 2906	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
58		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥 ⇝
59	58, 31	nffv 6766	. . . 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 2778	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
62	61	fveq2d 6760	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
63	55, 56, 57, 59, 62	cbvmptf 5179	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
64	53, 63	eqtri 2766	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 44199	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  {crab 3067  ∪ ciun 4921  ∩ ciin 4922   ↦ cmpt 5153  dom cdm 5580  ⟶wf 6414  ‘cfv 6418  ℤcz 12249  ℤ_≥cuz 12511   ⇝ cli 15121  SAlgcsalg 43739  SMblFncsmblfn 44123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cc 10122  ax-ac2 10150  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-acn 9631  df-ac 9803  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-ioo 13012  df-ico 13014  df-fl 13440  df-seq 13650  df-exp 13711  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-rest 17050  df-salg 43740  df-smblfn 44124

This theorem is referenced by:  smflimmpt  44230