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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfsupdmmbllem | Structured version Visualization version GIF version |
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.) |
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, < )) |
Ref | Expression |
---|---|
smfsupdmmbllem | ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) |
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 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
7 | smfsupdmmbllem.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 | 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 44978 | . 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 | smfsupdmmbllem.6 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
26 | 25 | uzct 43175 | . . . . 5 ⊢ 𝑍 ≼ ω |
27 | 26 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≼ ω) |
28 | smfsupdmmbllem.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 | smfsupdmmbllem.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 | nnxr 43406 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ*) | |
41 | 40 | ad2antlr 725 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℝ*) |
42 | 38, 31, 39, 9, 41 | smfpimltxr 44882 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
43 | 42 | an32s 650 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
44 | 36, 43 | fmptd2f 43358 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
45 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
46 | nnex 12117 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
47 | 46 | mptex 7169 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V |
48 | 14 | fvmpt2 6956 | . . . . . . . . . 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 6650 | . . . . . . . 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 256 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
52 | 51 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
53 | simplr 767 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ) | |
54 | 52, 53 | ffvelcdmd 7032 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
55 | 34, 54 | sseldd 3943 | . . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
56 | 21, 22, 23, 24, 27, 30, 55 | saliinclf 44461 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
57 | 3, 16, 17, 5, 19, 56 | saliunclf 44457 | . 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
58 | 15, 57 | 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 supcsup 9334 ℝcr 11008 ℝ*cxr 11146 < clt 11147 ≤ cle 11148 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-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-nn 12112 df-n0 12372 df-z 12458 df-uz 12722 df-ioo 13222 df-ico 13224 df-rest 17263 df-salg 44444 df-smblfn 44831 |
This theorem is referenced by: smfsupdmmbl 44980 |
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