Theorem smflim2 47379

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 47350 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 2927	. 2 ⊢ Ⅎ𝑗𝐹
2		nfcv 2927	. 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 2927	. . . . 5 ⊢ Ⅎ𝑥𝑍
9		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
10		smflim2.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
11		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
12	10, 11	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
13	12	nfdm 5931	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
14	9, 13	nfiin 4984	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
15	8, 14	nfiun 4983	. . . 4 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
16		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
17		nfv 1937	. . . 4 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
18		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑗((𝐹‘𝑚)‘𝑦)
19		smflim2.n	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
20		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑗
21	19, 20	nffv 6881	. . . . . . . 8 ⊢ Ⅎ𝑚(𝐹‘𝑗)
22		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑚𝑦
23	21, 22	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑦)
24		fveq2 6871	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
25	24	fveq1d 6873	. . . . . . 7 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
26	18, 23, 25	cbvmpt 5206	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
27		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑗
28	10, 27	nffv 6881	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑗)
29		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
30	28, 29	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑗)‘𝑦)
31	8, 30	nfmpt 5202	. . . . . 6 ⊢ Ⅎ𝑥(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
32	26, 31	nfcxfr 2925	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
33		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥dom ⇝
34	32, 33	nfel 2941	. . . 4 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
35		fveq2 6871	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
36	35	mpteq2dv 5198	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
37	36	eleq1d 2850	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
38	15, 16, 17, 34, 37	cbvrabw 3452	. . 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 6871	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
40	39	iineq1d 45667	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚))
41		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
42	21	nfdm 5931	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
43	24	dmeqd 5885	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
44	41, 42, 43	cbviin 4995	. . . . . . . . 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 2800	. . . . . . 7 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
47	46	cbviunv 4998	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗)
48	47	eleq2i 2857	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
49	26	eleq1i 2856	. . . . 5 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)) ∈ dom ⇝ )
50	48, 49	anbi12i 639	. . . 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 3420	. . 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 2792	. 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 3437	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
55	7, 54	nfcxfr 2925	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦𝐷
57		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
58		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥 ⇝
59	58, 31	nffv 6881	. . . 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 2800	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
62	61	fveq2d 6875	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
63	55, 56, 57, 59, 62	cbvmptf 5204	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
64	53, 63	eqtri 2788	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 47350	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Ⅎwnfc 2912  {crab 3417  ∪ ciun 4951  ∩ ciin 4952   ↦ cmpt 5185  dom cdm 5651  ⟶wf 6521  ‘cfv 6525  ℤcz 12579  ℤ_≥cuz 12850   ⇝ cli 15523  SAlgcsalg 46881  SMblFncsmblfn 47268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cc 10407  ax-ac2 10435  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-acn 9916  df-ac 10088  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-n0 12493  df-z 12580  df-uz 12851  df-q 12961  df-rp 13005  df-ioo 13364  df-ico 13366  df-fl 13813  df-seq 14026  df-exp 14086  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15527  df-rlim 15528  df-rest 17463  df-salg 46882  df-smblfn 47269

This theorem is referenced by:  smflimmpt  47383