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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 1915 | . 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 4007 | . . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) | |
5 | 3, 4 | eqsstri 3949 | . . . 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 2901 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
12 | fveq2 6645 | . . . . 5 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀)) | |
13 | 12 | dmeqd 5738 | . . . 4 ⊢ (𝑛 = 𝑀 → dom (𝐹‘𝑛) = dom (𝐹‘𝑀)) |
14 | smfsuplem3.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
15 | 14, 11 | ffvelrnd 6829 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ (SMblFn‘𝑆)) |
16 | eqid 2798 | . . . . 5 ⊢ dom (𝐹‘𝑀) = dom (𝐹‘𝑀) | |
17 | 2, 15, 16 | smfdmss 43367 | . . . 4 ⊢ (𝜑 → dom (𝐹‘𝑀) ⊆ ∪ 𝑆) |
18 | 11, 13, 17 | iinssd 41766 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ ∪ 𝑆) |
19 | 6, 18 | sstrd 3925 | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
20 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) | |
21 | 11 | ne0d 4251 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
22 | 21 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅) |
23 | 2 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
24 | 14 | ffvelrnda 6828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
25 | eqid 2798 | . . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
26 | 23, 24, 25 | smff 43366 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
27 | 26 | adantlr 714 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
28 | iinss2 4944 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) | |
29 | 28 | adantl 485 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) |
30 | 5 | sseli 3911 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
31 | 30 | adantr 484 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
32 | 29, 31 | sseldd 3916 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
33 | 32 | adantll 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
34 | 27, 33 | ffvelrnd 6829 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
35 | 3 | rabeq2i 3435 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
36 | 35 | simprbi 500 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
37 | 36 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
38 | 20, 22, 34, 37 | suprclrnmpt 41889 | . . 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 6855 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
41 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ) |
42 | 2 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
43 | 14 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
44 | simpr 488 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
45 | 41, 10, 42, 43, 3, 39, 44 | smfsuplem2 43443 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐺 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) |
46 | 1, 2, 19, 40, 45 | issmfle2d 43440 | 1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ∩ ciin 4882 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ⟶wf 6320 ‘cfv 6324 supcsup 8888 ℝcr 10525 < clt 10664 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 SAlgcsalg 42950 SMblFncsmblfn 43334 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-acn 9355 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 13157 df-rest 16688 df-topgen 16709 df-top 21499 df-bases 21551 df-salg 42951 df-salgen 42955 df-smblfn 43335 |
This theorem is referenced by: smfsup 43445 |
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