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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfinfdmmbllem | 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 |
---|---|
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)})) |
Ref | Expression |
---|---|
smfinfdmmbllem | ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) |
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 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
7 | smfinfdmmbllem.8 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
8 | 7 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
9 | eqid 2735 | . . . . 5 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
10 | 6, 8, 9 | smff 46688 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
11 | 10 | frexr 45335 | . . 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 46803 | . 2 ⊢ (𝜑 → dom 𝐺 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) |
16 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑚𝑆 | |
17 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑚ℕ | |
18 | nnct 14019 | . . . 4 ⊢ ℕ ≼ ω | |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
20 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ | |
21 | 1, 20 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ) |
22 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑛𝑆 | |
23 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑛𝑍 | |
24 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑆 ∈ SAlg) |
25 | smfinfdmmbllem.6 | . . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
26 | 25 | uzct 45003 | . . . . 5 ⊢ 𝑍 ≼ ω |
27 | 26 | a1i 11 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑍 ≼ ω) |
28 | smfinfdmmbllem.5 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
29 | 28, 25 | uzn0d 45375 | . . . . 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 | smfinfdmmbllem.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 46317 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝑆 ↾t dom (𝐹‘𝑛)) ⊆ 𝑆) |
35 | nfv 1912 | . . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍 | |
36 | 3, 35 | nfan 1897 | . . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
37 | nfcv 2903 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛 | |
38 | 4, 37 | nffv 6917 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝐹‘𝑛) |
39 | 8 | adantlr 715 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
40 | nnre 12271 | . . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ) | |
41 | 40 | renegcld 11688 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ) |
42 | 41 | rexrd 11309 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ ℕ → -𝑚 ∈ ℝ*) |
43 | 42 | ad2antlr 727 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → -𝑚 ∈ ℝ*) |
44 | 38, 31, 39, 9, 43 | smfpimgtxr 46736 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
45 | 44 | an32s 652 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)} ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
46 | 36, 45 | fmptd2f 45178 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
47 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) | |
48 | nnex 12270 | . . . . . . . . . . 11 ⊢ ℕ ∈ V | |
49 | 48 | mptex 7243 | . . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ -𝑚 < ((𝐹‘𝑛)‘𝑥)}) ∈ V |
50 | 14 | fvmpt2 7027 | . . . . . . . . . 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 6721 | . . . . . . . 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 257 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
54 | 53 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):ℕ⟶(𝑆 ↾t dom (𝐹‘𝑛))) |
55 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ) | |
56 | 54, 55 | ffvelcdmd 7105 | . . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ (𝑆 ↾t dom (𝐹‘𝑛))) |
57 | 34, 56 | sseldd 3996 | . . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
58 | 21, 22, 23, 24, 27, 30, 57 | saliinclf 46282 | . . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
59 | 3, 16, 17, 5, 19, 58 | saliunclf 46278 | . 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∈ 𝑆) |
60 | 15, 59 | eqeltrd 2839 | 1 ⊢ (𝜑 → dom 𝐺 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 ∅c0 4339 ∪ ciun 4996 ∩ ciin 4997 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ωcom 7887 ≼ cdom 8982 infcinf 9479 ℝcr 11152 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 -cneg 11491 ℕcn 12264 ℤcz 12611 ℤ≥cuz 12876 ↾t crest 17467 SAlgcsalg 46264 SMblFncsmblfn 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-ac2 10501 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-acn 9980 df-ac 10154 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-ioo 13388 df-ico 13390 df-fl 13829 df-rest 17469 df-salg 46265 df-smblfn 46652 |
This theorem is referenced by: smfinfdmmbl 46805 |
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