Theorem smflim2 46761

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 46732 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 2902	. 2 ⊢ Ⅎ𝑗𝐹
2		nfcv 2902	. 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 2902	. . . . 5 ⊢ Ⅎ𝑥𝑍
9		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
10		smflim2.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
11		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
12	10, 11	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
13	12	nfdm 5964	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
14	9, 13	nfiin 5028	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
15	8, 14	nfiun 5027	. . . 4 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
16		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
17		nfv 1911	. . . 4 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
18		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑗((𝐹‘𝑚)‘𝑦)
19		smflim2.n	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
20		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑗
21	19, 20	nffv 6916	. . . . . . . 8 ⊢ Ⅎ𝑚(𝐹‘𝑗)
22		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑚𝑦
23	21, 22	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑦)
24		fveq2 6906	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
25	24	fveq1d 6908	. . . . . . 7 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
26	18, 23, 25	cbvmpt 5258	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
27		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑗
28	10, 27	nffv 6916	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑗)
29		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
30	28, 29	nffv 6916	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑗)‘𝑦)
31	8, 30	nfmpt 5254	. . . . . 6 ⊢ Ⅎ𝑥(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
32	26, 31	nfcxfr 2900	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
33		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥dom ⇝
34	32, 33	nfel 2917	. . . 4 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
35		fveq2 6906	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
36	35	mpteq2dv 5249	. . . . 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 3470	. . 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 6906	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
40	39	iineq1d 45029	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚))
41		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
42	21	nfdm 5964	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
43	24	dmeqd 5918	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
44	41, 42, 43	cbviin 5041	. . . . . . . . 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 2774	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
47	46	cbviunv 5044	. . . . . 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 628	. . . 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 3435	. . 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 2766	. 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 3453	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
55	7, 54	nfcxfr 2900	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦𝐷
57		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
58		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥 ⇝
59	58, 31	nffv 6916	. . . 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 2774	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
62	61	fveq2d 6910	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
63	55, 56, 57, 59, 62	cbvmptf 5256	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
64	53, 63	eqtri 2762	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 46732	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105  Ⅎwnfc 2887  {crab 3432  ∪ ciun 4995  ∩ ciin 4996   ↦ cmpt 5230  dom cdm 5688  ⟶wf 6558  ‘cfv 6562  ℤcz 12610  ℤ_≥cuz 12875   ⇝ cli 15516  SAlgcsalg 46263  SMblFncsmblfn 46650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-ac2 10500  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-ac 10153  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-ioo 13387  df-ico 13389  df-fl 13828  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-rest 17468  df-salg 46264  df-smblfn 46651

This theorem is referenced by:  smflimmpt  46765