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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfinfdmmbl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
smfinfdmmbl.1 | ⊢ Ⅎ𝑛𝜑 |
smfinfdmmbl.2 | ⊢ Ⅎ𝑥𝜑 |
smfinfdmmbl.3 | ⊢ Ⅎ𝑥𝐹 |
smfinfdmmbl.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
smfinfdmmbl.5 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
smfinfdmmbl.6 | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfinfdmmbl.7 | ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
smfinfdmmbl.8 | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) |
smfinfdmmbl.9 | ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} |
smfinfdmmbl.10 | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
Ref | Expression |
---|---|
smfinfdmmbl | ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfinfdmmbl.1 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | smfinfdmmbl.2 | . 2 ⊢ Ⅎ𝑥𝜑 | |
3 | nfv 1917 | . 2 ⊢ Ⅎ𝑚𝜑 | |
4 | smfinfdmmbl.3 | . 2 ⊢ Ⅎ𝑥𝐹 | |
5 | smfinfdmmbl.4 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | smfinfdmmbl.5 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | smfinfdmmbl.6 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
8 | smfinfdmmbl.7 | . 2 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
9 | smfinfdmmbl.8 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ∈ 𝑆) | |
10 | smfinfdmmbl.9 | . 2 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑦 ≤ ((𝐹‘𝑛)‘𝑥)} | |
11 | smfinfdmmbl.10 | . 2 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ inf(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) | |
12 | eqid 2737 | . 2 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)})) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | smfinfdmmbllem 44983 | 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 ∀wral 3062 ∃wrex 3071 {crab 3405 ∩ ciin 4953 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5631 ran crn 5632 ⟶wf 6489 ‘cfv 6493 infcinf 9335 ℝcr 11008 < clt 11147 ≤ cle 11148 -cneg 11344 ℕcn 12111 ℤcz 12457 ℤ≥cuz 12721 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: (None) |
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