smfinfdmmbllem - Mathbox for Glauco Siliprandi

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Theorem smfinfdmmbllem 44983

Description: If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their infimum function has the domain in the sigma-algebra. This is the fifth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 1-Feb-2025.)

**Hypotheses**
Ref	Expression
smfinfdmmbllem.1	⊢ Ⅎ𝑛𝜑
smfinfdmmbllem.2	⊢ Ⅎ𝑥𝜑
smfinfdmmbllem.3	⊢ Ⅎ𝑚𝜑
smfinfdmmbllem.4	⊢ Ⅎ𝑥𝐹
smfinfdmmbllem.5	⊢ (𝜑 → 𝑀 ∈ ℤ)
smfinfdmmbllem.6	⊢ 𝑍 = (ℤ_≥‘𝑀)
smfinfdmmbllem.7	⊢ (𝜑 → 𝑆 ∈ SAlg)
smfinfdmmbllem.8	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smfinfdmmbllem.9	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
smfinfdmmbllem.10	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
smfinfdmmbllem.11	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
smfinfdmmbllem.12	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))

**Assertion**
Ref	Expression
smfinfdmmbllem	⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

Distinct variable groups: 𝐷,𝑚 𝑚,𝐹,𝑦 𝑦,𝐻 𝑆,𝑚,𝑛 𝑚,𝑍,𝑥,𝑛,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑛) 𝑆(𝑥,𝑦) 𝐹(𝑥,𝑛) 𝐺(𝑥,𝑦,𝑚,𝑛) 𝐻(𝑥,𝑚,𝑛) 𝑀(𝑥,𝑦,𝑚,𝑛)

**Proof of Theorem smfinfdmmbllem**
Step	Hyp	Ref	Expression
1		smfinfdmmbllem.1	. . 3 ⊢ Ⅎ𝑛𝜑
2		smfinfdmmbllem.2	. . 3 ⊢ Ⅎ𝑥𝜑
3		smfinfdmmbllem.3	. . 3 ⊢ Ⅎ𝑚𝜑
4		smfinfdmmbllem.4	. . 3 ⊢ Ⅎ𝑥𝐹
5		smfinfdmmbllem.7	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
7		smfinfdmmbllem.8	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
8	7	ffvelcdmda 7031	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
9		eqid 2737	. . . . 5 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
10	6, 8, 9	smff 44867	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
11	10	frexr 43517	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
12		smfinfdmmbllem.10	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)}
13		smfinfdmmbllem.11	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
14		smfinfdmmbllem.12	. . 3 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
15	1, 2, 3, 4, 11, 12, 13, 14	finfdm2 44982	. 2 ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
16		nfcv 2905	. . 3 ⊢ Ⅎ𝑚𝑆
17		nfcv 2905	. . 3 ⊢ Ⅎ𝑚ℕ
18		nnct 13840	. . . 4 ⊢ ℕ ≼ ω
19	18	a1i 11	. . 3 ⊢ (𝜑 → ℕ ≼ ω)
20		nfv 1917	. . . . 5 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
21	1, 20	nfan 1902	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
22		nfcv 2905	. . . 4 ⊢ Ⅎ𝑛𝑆
23		nfcv 2905	. . . 4 ⊢ Ⅎ𝑛𝑍
24	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ SAlg)
25		smfinfdmmbllem.6	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
26	25	uzct 43175	. . . . 5 ⊢ 𝑍 ≼ ω
27	26	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≼ ω)
28		smfinfdmmbllem.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
29	28, 25	uzn0d 43558	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅)
30	29	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≠ ∅)
31	24	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
32		smfinfdmmbllem.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
33	32	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
34	31, 33	salrestss 44496	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝑆 ↾_t dom (𝐹‘𝑛)) ⊆ 𝑆)
35		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
36	3, 35	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
37		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
38	4, 37	nffv 6849	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐹‘𝑛)
39	8	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
40		nnre 12118	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
41	40	renegcld 11540	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ)
42	41	rexrd 11163	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ^*)
43	42	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → -𝑚 ∈ ℝ^*)
44	38, 31, 39, 9, 43	smfpimgtxr 44915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
45	44	an32s 650	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
46	36, 45	fmptd2f 43358	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
47		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
48		nnex 12117	. . . . . . . . . . 11 ⊢ ℕ ∈ V
49	48	mptex 7169	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V
50	14	fvmpt2 6956	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
51	47, 49, 50	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}))
52	51	feq1d 6650	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)) ↔ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛))))
53	46, 52	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
54	53	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
55		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
56	54, 55	ffvelcdmd 7032	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
57	34, 56	sseldd 3943	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
58	21, 22, 23, 24, 27, 30, 57	saliinclf 44461	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
59	3, 16, 17, 5, 19, 58	saliunclf 44457	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
60	15, 59	eqeltrd 2838	1 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2885 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3405 Vcvv 3443 ∅c0 4280 ∪ ciun 4952 ∩ ciin 4953 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5631 ran crn 5632 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ωcom 7794 ≼ cdom 8839 infcinf 9335 ℝcr 11008 ℝ^*cxr 11146 < clt 11147 ≤ cle 11148 -cneg 11344 ℕcn 12111 ℤcz 12457 ℤ_≥cuz 12721 ↾_t crest 17261 SAlgcsalg 44443 SMblFncsmblfn 44830

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cc 10329 ax-ac2 10357 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-oadd 8408 df-omul 8409 df-er 8606 df-map 8725 df-pm 8726 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-acn 9836 df-ac 10010 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-ioo 13222 df-ico 13224 df-fl 13651 df-rest 17263 df-salg 44444 df-smblfn 44831

This theorem is referenced by: smfinfdmmbl 44984

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