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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 1916 | . 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 4077 | . . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) | |
5 | 3, 4 | eqsstri 4016 | . . . 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 12844 | . . . . . 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 2843 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
12 | fveq2 6891 | . . . . 5 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀)) | |
13 | 12 | dmeqd 5905 | . . . 4 ⊢ (𝑛 = 𝑀 → dom (𝐹‘𝑛) = dom (𝐹‘𝑀)) |
14 | smfsuplem3.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
15 | 14, 11 | ffvelcdmd 7087 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ (SMblFn‘𝑆)) |
16 | eqid 2731 | . . . . 5 ⊢ dom (𝐹‘𝑀) = dom (𝐹‘𝑀) | |
17 | 2, 15, 16 | smfdmss 45907 | . . . 4 ⊢ (𝜑 → dom (𝐹‘𝑀) ⊆ ∪ 𝑆) |
18 | 11, 13, 17 | iinssd 44281 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ ∪ 𝑆) |
19 | 6, 18 | sstrd 3992 | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
20 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) | |
21 | 11 | ne0d 4335 | . . . . 5 ⊢ (𝜑 → 𝑍 ≠ ∅) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑍 ≠ ∅) |
23 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg) |
24 | 14 | ffvelcdmda 7086 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) |
25 | eqid 2731 | . . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
26 | 23, 24, 25 | smff 45906 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
27 | 26 | adantlr 712 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) |
28 | iinss2 5060 | . . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) | |
29 | 28 | adantl 481 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ dom (𝐹‘𝑛)) |
30 | 5 | sseli 3978 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
31 | 30 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) |
32 | 29, 31 | sseldd 3983 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
33 | 32 | adantll 711 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) |
34 | 27, 33 | ffvelcdmd 7087 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) |
35 | 3 | reqabi 3453 | . . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)) |
36 | 35 | simprbi 496 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
37 | 36 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) |
38 | 20, 22, 34, 37 | suprclrnmpt 44413 | . . 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 7115 | . 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ) |
41 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ) |
42 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
43 | 14 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆)) |
44 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
45 | 41, 10, 42, 43, 3, 39, 44 | smfsuplem2 45986 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐺 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) |
46 | 1, 2, 19, 40, 45 | issmfle2d 45983 | 1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3431 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 ∩ ciin 4998 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ⟶wf 6539 ‘cfv 6543 supcsup 9441 ℝcr 11115 < clt 11255 ≤ cle 11256 ℤcz 12565 ℤ≥cuz 12829 SAlgcsalg 45482 SMblFncsmblfn 45869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cc 10436 ax-ac2 10464 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-acn 9943 df-ac 10117 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-ioo 13335 df-ioc 13336 df-ico 13337 df-fl 13764 df-rest 17375 df-topgen 17396 df-top 22715 df-bases 22768 df-salg 45483 df-salgen 45487 df-smblfn 45870 |
This theorem is referenced by: smfsup 45988 |
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