smfsupdmmbllem - Mathbox for Glauco Siliprandi

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Theorem smfsupdmmbllem 47672

Description: If a countable set of sigma-measurable functions have domains in the sigma-algebra, then their supremum function has the domain in the sigma-algebra. This is the fourth statement of Proposition 121H of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 24-Jan-2025.)

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

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

Distinct variable groups: 𝐷,𝑚 𝑚,𝐹,𝑦 𝑦,𝐻 𝑆,𝑚,𝑛 𝑚,𝑍,𝑛,𝑥,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 smfsupdmmbllem**
Step	Hyp	Ref	Expression
1		smfsupdmmbllem.1	. . 3 ⊢ Ⅎ𝑛𝜑
2		smfsupdmmbllem.2	. . 3 ⊢ Ⅎ𝑥𝜑
3		smfsupdmmbllem.3	. . 3 ⊢ Ⅎ𝑚𝜑
4		smfsupdmmbllem.4	. . 3 ⊢ Ⅎ𝑥𝐹
5		smfsupdmmbllem.7	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ SAlg)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
7		smfsupdmmbllem.8	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
8	7	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
9		eqid 2760	. . . . 5 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
10	6, 8, 9	smff 47560	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
11	10	frexr 46214	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
12		smfsupdmmbllem.10	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
13		smfsupdmmbllem.12	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ))
14		smfsupdmmbllem.11	. . 3 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
15	1, 2, 3, 4, 11, 12, 13, 14	fsupdm2 47671	. 2 ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
16		nfcv 2922	. . 3 ⊢ Ⅎ𝑚𝑆
17		nfcv 2922	. . 3 ⊢ Ⅎ𝑚ℕ
18		nnct 14045	. . . 4 ⊢ ℕ ≼ ω
19	18	a1i 11	. . 3 ⊢ (𝜑 → ℕ ≼ ω)
20		nfv 1947	. . . . 5 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
21	1, 20	nfan 1932	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
22		nfcv 2922	. . . 4 ⊢ Ⅎ𝑛𝑆
23		nfcv 2922	. . . 4 ⊢ Ⅎ𝑛𝑍
24	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ SAlg)
25		smfsupdmmbllem.6	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
26	25	uzct 45897	. . . . 5 ⊢ 𝑍 ≼ ω
27	26	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≼ ω)
28		smfsupdmmbllem.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
29	28, 25	uzn0d 46253	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅)
30	29	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≠ ∅)
31	24	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
32		smfsupdmmbllem.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
33	32	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
34	31, 33	salrestss 47189	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝑆 ↾_t dom (𝐹‘𝑛)) ⊆ 𝑆)
35		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
36	3, 35	nfan 1932	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
37		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
38	4, 37	nffv 6888	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐹‘𝑛)
39	8	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
40		nnxr 46108	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ^*)
41	40	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℝ^*)
42	38, 31, 39, 9, 41	smfpimltxr 47575	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
43	42	an32s 665	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
44	36, 43	fmptd2f 46064	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
45		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
46		nnex 12263	. . . . . . . . . . 11 ⊢ ℕ ∈ V
47	46	mptex 7222	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V
48	14	fvmpt2 6998	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
49	45, 47, 48	sylancl 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
50	49	feq1d 6684	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)) ↔ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛))))
51	44, 50	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
52	51	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
53		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
54	52, 53	ffvelcdmd 7078	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
55	34, 54	sseldd 3932	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
56	21, 22, 23, 24, 27, 30, 55	saliinclf 47154	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
57	3, 16, 17, 5, 19, 56	saliunclf 47150	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
58	15, 57	eqeltrd 2860	1 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2145 Ⅎwnfc 2907 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ∅c0 4279 ∪ ciun 4951 ∩ ciin 4952 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ωcom 7862 ≼ cdom 8950 supcsup 9410 ℝcr 11123 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 ℕcn 12257 ℤcz 12615 ℤ_≥cuz 12887 ↾_t crest 17505 SAlgcsalg 47136 SMblFncsmblfn 47523

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-oi 9482 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-n0 12529 df-z 12616 df-uz 12888 df-ioo 13402 df-ico 13404 df-rest 17507 df-salg 47137 df-smblfn 47524

This theorem is used by: smfsupdmmbl 47673

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