Theorem smflim2 45522

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 45493 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 2904	. 2 ⊢ Ⅎ𝑗𝐹
2		nfcv 2904	. 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 2904	. . . . 5 ⊢ Ⅎ𝑥𝑍
9		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
10		smflim2.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
11		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
12	10, 11	nffv 6902	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑚)
13	12	nfdm 5951	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
14	9, 13	nfiin 5029	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
15	8, 14	nfiun 5028	. . . 4 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
16		nfcv 2904	. . . 4 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
17		nfv 1918	. . . 4 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
18		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑗((𝐹‘𝑚)‘𝑦)
19		smflim2.n	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐹
20		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑗
21	19, 20	nffv 6902	. . . . . . . 8 ⊢ Ⅎ𝑚(𝐹‘𝑗)
22		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑚𝑦
23	21, 22	nffv 6902	. . . . . . 7 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑦)
24		fveq2 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
25	24	fveq1d 6894	. . . . . . 7 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑗)‘𝑦))
26	18, 23, 25	cbvmpt 5260	. . . . . 6 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
27		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑗
28	10, 27	nffv 6902	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑗)
29		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
30	28, 29	nffv 6902	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑗)‘𝑦)
31	8, 30	nfmpt 5256	. . . . . 6 ⊢ Ⅎ𝑥(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))
32	26, 31	nfcxfr 2902	. . . . 5 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
33		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥dom ⇝
34	32, 33	nfel 2918	. . . 4 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
35		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
36	35	mpteq2dv 5251	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
37	36	eleq1d 2819	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
38	15, 16, 17, 34, 37	cbvrabw 3468	. . 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 6892	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) = (ℤ≥‘𝑘))
40	39	iineq1d 43779	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑚))
41		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
42	21	nfdm 5951	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
43	24	dmeqd 5906	. . . . . . . . . 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 2773	. . . . . . 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 2826	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑗 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑗))
49	26	eleq1i 2825	. . . . 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 3436	. . 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 2765	. 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 3452	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
55	7, 54	nfcxfr 2902	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2904	. . . 4 ⊢ Ⅎ𝑦𝐷
57		nfcv 2904	. . . 4 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
58		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥 ⇝
59	58, 31	nffv 6902	. . . 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 2773	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦)))
62	61	fveq2d 6896	. . . 4 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
63	55, 56, 57, 59, 62	cbvmptf 5258	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
64	53, 63	eqtri 2761	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑗 ∈ 𝑍 ↦ ((𝐹‘𝑗)‘𝑦))))
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 45493	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  Ⅎwnfc 2884  {crab 3433  ∪ ciun 4998  ∩ ciin 4999   ↦ cmpt 5232  dom cdm 5677  ⟶wf 6540  ‘cfv 6544  ℤcz 12558  ℤ_≥cuz 12822   ⇝ cli 15428  SAlgcsalg 45024  SMblFncsmblfn 45411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-ac2 10458  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-acn 9937  df-ac 10111  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-ioo 13328  df-ico 13330  df-fl 13757  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-rest 17368  df-salg 45025  df-smblfn 45412

This theorem is referenced by:  smflimmpt  45526