Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfsuplem3 | Structured version Visualization version GIF version |
Description: The supremum of a countable set of sigma-measurable functions is sigma-measurable. Proposition 121F (b) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
smfsuplem3.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
smfsuplem3.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
smfsuplem3.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfsuplem3.f | ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
smfsuplem3.d | ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} |
smfsuplem3.g | ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) |
Ref | Expression |
---|---|
smfsuplem3 | ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | smfsuplem3.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfsuplem3.d | . . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} | |
4 | ssrab2 4053 | . . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) | |
5 | 3, 4 | eqsstri 3998 | . . . 4 ⊢ 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
7 | smfsuplem3.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | uzid 12246 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
10 | smfsuplem3.z | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
11 | 9, 10 | eleqtrrdi 2921 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
12 | fveq2 6663 | . . . . 5 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀)) | |
13 | 12 | dmeqd 5767 | . . . 4 ⊢ (𝑛 = 𝑀 → dom (𝐹‘𝑛) = dom (𝐹‘𝑀)) |
14 | smfsuplem3.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
15 | 14, 11 | ffvelrnd 6844 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ (SMblFn‘𝑆)) |
16 | eqid 2818 | . . . . 5 ⊢ dom (𝐹‘𝑀) = dom (𝐹‘𝑀) | |
17 | 2, 15, 16 | smfdmss 42887 | . . . 4 ⊢ (𝜑 → dom (𝐹‘𝑀) ⊆ ∪ 𝑆) |
18 | 11, 13, 17 | iinssd 41273 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ ∪ 𝑆) |
19 | 6, 18 | sstrd 3974 | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
20 | nfv 1906 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) | |
21 | 11 | ne0d 4298 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅) |
23 | 2 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
24 | 14 | ffvelrnda 6843 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
25 | eqid 2818 | . . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
26 | 23, 24, 25 | smff 42886 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
27 | 26 | adantlr 711 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
28 | iinss2 4972 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) | |
29 | 28 | adantl 482 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) |
30 | 5 | sseli 3960 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
31 | 30 | adantr 481 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
32 | 29, 31 | sseldd 3965 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
33 | 32 | adantll 710 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
34 | 27, 33 | ffvelrnd 6844 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
35 | 3 | rabeq2i 3485 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
36 | 35 | simprbi 497 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
37 | 36 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
38 | 20, 22, 34, 37 | suprclrnmpt 41399 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < ) ∈ ℝ) |
39 | smfsuplem3.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ sup(ran (𝑛 ∈ 𝑍 ↦ ((𝐹‘𝑛)‘𝑥)), ℝ, < )) | |
40 | 38, 39 | fmptd 6870 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
41 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ) |
42 | 2 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
43 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
44 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
45 | 41, 10, 42, 43, 3, 39, 44 | smfsuplem2 42963 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐺 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) |
46 | 1, 2, 19, 40, 45 | issmfle2d 42960 | 1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 {crab 3139 ⊆ wss 3933 ∅c0 4288 ∪ cuni 4830 ∩ ciin 4911 class class class wbr 5057 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 ⟶wf 6344 ‘cfv 6348 supcsup 8892 ℝcr 10524 < clt 10663 ≤ cle 10664 ℤcz 11969 ℤ≥cuz 12231 SAlgcsalg 42470 SMblFncsmblfn 42854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ioc 12731 df-ico 12732 df-fl 13150 df-rest 16684 df-topgen 16705 df-top 21430 df-bases 21482 df-salg 42471 df-salgen 42475 df-smblfn 42855 |
This theorem is referenced by: smfsup 42965 |
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