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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 1913 | . 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 4079 | . . . . 5 ⊢ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) | |
| 5 | 3, 4 | eqsstri 4029 | . . . 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 12894 | . . . . . 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 2851 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) | 
| 12 | fveq2 6905 | . . . . 5 ⊢ (𝑛 = 𝑀 → (𝐹‘𝑛) = (𝐹‘𝑀)) | |
| 13 | 12 | dmeqd 5915 | . . . 4 ⊢ (𝑛 = 𝑀 → dom (𝐹‘𝑛) = dom (𝐹‘𝑀)) | 
| 14 | smfsuplem3.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆)) | |
| 15 | 14, 11 | ffvelcdmd 7104 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ (SMblFn‘𝑆)) | 
| 16 | eqid 2736 | . . . . 5 ⊢ dom (𝐹‘𝑀) = dom (𝐹‘𝑀) | |
| 17 | 2, 15, 16 | smfdmss 46753 | . . . 4 ⊢ (𝜑 → dom (𝐹‘𝑀) ⊆ ∪ 𝑆) | 
| 18 | 11, 13, 17 | iinssd 45141 | . . 3 ⊢ (𝜑 → ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ⊆ ∪ 𝑆) | 
| 19 | 6, 18 | sstrd 3993 | . 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | 
| 20 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ 𝐷) | |
| 21 | 11 | ne0d 4341 | . . . . 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 7103 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆)) | 
| 25 | eqid 2736 | . . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛) | |
| 26 | 23, 24, 25 | smff 46752 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) | 
| 27 | 26 | adantlr 715 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ) | 
| 28 | iinss2 5056 | . . . . . . . 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 714 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛)) | 
| 34 | 27, 33 | ffvelcdmd 7104 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ) | 
| 35 | 3 | reqabi 3459 | . . . . . 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 45263 | . . 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 7133 | . 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 46832 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (◡𝐺 “ (-∞(,]𝑎)) ∈ (𝑆 ↾t 𝐷)) | 
| 46 | 1, 2, 19, 40, 45 | issmfle2d 46829 | 1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 ⊆ wss 3950 ∅c0 4332 ∪ cuni 4906 ∩ ciin 4991 class class class wbr 5142 ↦ cmpt 5224 dom cdm 5684 ran crn 5685 ⟶wf 6556 ‘cfv 6560 supcsup 9481 ℝcr 11155 < clt 11296 ≤ cle 11297 ℤcz 12615 ℤ≥cuz 12879 SAlgcsalg 46328 SMblFncsmblfn 46715 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-ioo 13392 df-ioc 13393 df-ico 13394 df-fl 13833 df-rest 17468 df-topgen 17489 df-top 22901 df-bases 22954 df-salg 46329 df-salgen 46333 df-smblfn 46716 | 
| This theorem is referenced by: smfsup 46834 | 
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