smfsupdmmbllem - Mathbox for Glauco Siliprandi

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Theorem smfsupdmmbllem 46839

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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
7		smfsupdmmbllem.8	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
8	7	ffvelcdmda 7011	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
9		eqid 2729	. . . . 5 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
10	6, 8, 9	smff 46727	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ)
11	10	frexr 45380	. . 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 46838	. 2 ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
16		nfcv 2891	. . 3 ⊢ Ⅎ𝑚𝑆
17		nfcv 2891	. . 3 ⊢ Ⅎ𝑚ℕ
18		nnct 13876	. . . 4 ⊢ ℕ ≼ ω
19	18	a1i 11	. . 3 ⊢ (𝜑 → ℕ ≼ ω)
20		nfv 1914	. . . . 5 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
21	1, 20	nfan 1899	. . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
22		nfcv 2891	. . . 4 ⊢ Ⅎ𝑛𝑆
23		nfcv 2891	. . . 4 ⊢ Ⅎ𝑛𝑍
24	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ SAlg)
25		smfsupdmmbllem.6	. . . . . 6 ⊢ 𝑍 = (ℤ_≥‘𝑀)
26	25	uzct 45057	. . . . 5 ⊢ 𝑍 ≼ ω
27	26	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≼ ω)
28		smfsupdmmbllem.5	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
29	28, 25	uzn0d 45420	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅)
30	29	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≠ ∅)
31	24	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
32		smfsupdmmbllem.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
33	32	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆)
34	31, 33	salrestss 46356	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝑆 ↾_t dom (𝐹‘𝑛)) ⊆ 𝑆)
35		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
36	3, 35	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍)
37		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
38	4, 37	nffv 6826	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐹‘𝑛)
39	8	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
40		nnxr 45273	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ^*)
41	40	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℝ^*)
42	38, 31, 39, 9, 41	smfpimltxr 46742	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
43	42	an32s 652	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾_t dom (𝐹‘𝑛)))
44	36, 43	fmptd2f 45229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
45		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
46		nnex 12122	. . . . . . . . . . 11 ⊢ ℕ ∈ V
47	46	mptex 7151	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V
48	14	fvmpt2 6934	. . . . . . . . . 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 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
50	49	feq1d 6628	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
52	51	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾_t dom (𝐹‘𝑛)))
53		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
54	52, 53	ffvelcdmd 7012	. . . . 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 46321	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
57	3, 16, 17, 5, 19, 56	saliunclf 46317	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆)
58	15, 57	eqeltrd 2828	1 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3392 Vcvv 3433 ∅c0 4280 ∪ ciun 4938 ∩ ciin 4939 class class class wbr 5088 ↦ cmpt 5169 dom cdm 5613 ran crn 5614 ⟶wf 6472 ‘cfv 6476 (class class class)co 7340 ωcom 7790 ≼ cdom 8861 supcsup 9318 ℝcr 10996 ℝ^*cxr 11136 < clt 11137 ≤ cle 11138 ℕcn 12116 ℤcz 12459 ℤ_≥cuz 12723 ↾_t crest 17311 SAlgcsalg 46303 SMblFncsmblfn 46690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-inf2 9525 ax-cc 10317 ax-ac2 10345 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4895 df-iun 4940 df-iin 4941 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-se 5567 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-om 7791 df-1st 7915 df-2nd 7916 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-2o 8380 df-oadd 8383 df-omul 8384 df-er 8616 df-map 8746 df-pm 8747 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-sup 9320 df-oi 9390 df-card 9823 df-acn 9826 df-ac 9998 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-nn 12117 df-n0 12373 df-z 12460 df-uz 12724 df-ioo 13240 df-ico 13242 df-rest 17313 df-salg 46304 df-smblfn 46691

This theorem is referenced by: smfsupdmmbl 46840

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