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| Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfgtlem | Structured version Visualization version GIF version | ||
| Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) | 
| Ref | Expression | 
|---|---|
| issmfgtlem.x | ⊢ Ⅎ𝑥𝜑 | 
| issmfgtlem.a | ⊢ Ⅎ𝑎𝜑 | 
| issmfgtlem.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| issmfgtlem.d | ⊢ 𝐷 = dom 𝐹 | 
| issmfgtlem.i | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | 
| issmfgtlem.f | ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | 
| issmfgtlem.p | ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | 
| Ref | Expression | 
|---|---|
| issmfgtlem | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | issmfgtlem.i | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) | |
| 2 | issmfgtlem.f | . . 3 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ) | |
| 3 | issmfgtlem.s | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 4 | 3, 1 | restuni4 45126 | . . . . . . . 8 ⊢ (𝜑 → ∪ (𝑆 ↾t 𝐷) = 𝐷) | 
| 5 | 4 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ (𝑆 ↾t 𝐷)) | 
| 6 | 5 | rabeqdv 3452 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) | 
| 7 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} = {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏}) | 
| 8 | issmfgtlem.x | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
| 9 | nfv 1914 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ | |
| 10 | 8, 9 | nfan 1899 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ) | 
| 11 | issmfgtlem.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
| 12 | nfv 1914 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
| 13 | 11, 12 | nfan 1899 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) | 
| 14 | 3 | uniexd 7762 | . . . . . . . . . . 11 ⊢ (𝜑 → ∪ 𝑆 ∈ V) | 
| 15 | 14 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → ∪ 𝑆 ∈ V) | 
| 16 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ⊆ ∪ 𝑆) | |
| 17 | 15, 16 | ssexd 5324 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐷 ⊆ ∪ 𝑆) → 𝐷 ∈ V) | 
| 18 | 1, 17 | mpdan 687 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V) | 
| 19 | eqid 2737 | . . . . . . . 8 ⊢ (𝑆 ↾t 𝐷) = (𝑆 ↾t 𝐷) | |
| 20 | 3, 18, 19 | subsalsal 46374 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ↾t 𝐷) ∈ SAlg) | 
| 21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → (𝑆 ↾t 𝐷) ∈ SAlg) | 
| 22 | eqid 2737 | . . . . . 6 ⊢ ∪ (𝑆 ↾t 𝐷) = ∪ (𝑆 ↾t 𝐷) | |
| 23 | 2 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝐹:𝐷⟶ℝ) | 
| 24 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) | |
| 25 | 4 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → ∪ (𝑆 ↾t 𝐷) = 𝐷) | 
| 26 | 24, 25 | eleqtrd 2843 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → 𝑥 ∈ 𝐷) | 
| 27 | 23, 26 | ffvelcdmd 7105 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ) | 
| 28 | 27 | rexrd 11311 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) | 
| 29 | 28 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ ∪ (𝑆 ↾t 𝐷)) → (𝐹‘𝑥) ∈ ℝ*) | 
| 30 | 4 | rabeqdv 3452 | . . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}) | 
| 31 | 30 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} = {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)}) | 
| 32 | issmfgtlem.p | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | |
| 33 | 32 | r19.21bi 3251 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | 
| 34 | 31, 33 | eqeltrd 2841 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | 
| 35 | 34 | adantlr 715 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ 𝑎 < (𝐹‘𝑥)} ∈ (𝑆 ↾t 𝐷)) | 
| 36 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) | |
| 37 | 10, 13, 21, 22, 29, 35, 36 | salpreimagtlt 46745 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ ∪ (𝑆 ↾t 𝐷) ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) | 
| 38 | 7, 37 | eqeltrd 2841 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) | 
| 39 | 38 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)) | 
| 40 | 1, 2, 39 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷))) | 
| 41 | issmfgtlem.d | . . 3 ⊢ 𝐷 = dom 𝐹 | |
| 42 | 3, 41 | issmf 46743 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑏} ∈ (𝑆 ↾t 𝐷)))) | 
| 43 | 40, 42 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 ℝ*cxr 11294 < clt 11295 ↾t crest 17465 SAlgcsalg 46323 SMblFncsmblfn 46710 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-ac2 10503 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-ac 10156 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-ioo 13391 df-ico 13393 df-fl 13832 df-rest 17467 df-salg 46324 df-smblfn 46711 | 
| This theorem is referenced by: issmfgt 46771 | 
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