Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmf | Structured version Visualization version GIF version |
Description: A real-valued, nondecreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
incsmf.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
incsmf.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
incsmf.i | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
incsmf.j | ⊢ 𝐽 = (topGen‘ran (,)) |
incsmf.b | ⊢ 𝐵 = (SalGen‘𝐽) |
Ref | Expression |
---|---|
incsmf | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1911 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | incsmf.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 23369 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2909 | . . . 4 ⊢ 𝐽 ∈ Top |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | incsmf.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐽) | |
7 | 5, 6 | salgencld 42631 | . 2 ⊢ (𝜑 → 𝐵 ∈ SAlg) |
8 | incsmf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 5, 6 | unisalgen2 42636 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 = ∪ 𝐽) |
10 | 2 | unieqi 4850 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 = ∪ (topGen‘ran (,))) |
12 | uniretop 23370 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
13 | 12 | eqcomi 2830 | . . . . 5 ⊢ ∪ (topGen‘ran (,)) = ℝ |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ (topGen‘ran (,)) = ℝ) |
15 | 9, 11, 14 | 3eqtrrd 2861 | . . 3 ⊢ (𝜑 → ℝ = ∪ 𝐵) |
16 | 8, 15 | sseqtrd 4006 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
17 | incsmf.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
18 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑎 ∈ ℝ) | |
19 | nfv 1911 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ ℝ) | |
20 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ⊆ ℝ) |
21 | 17 | frexr 41653 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
22 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝐴⟶ℝ*) |
23 | incsmf.i | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
24 | breq1 5068 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦)) | |
25 | fveq2 6669 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | |
26 | 25 | breq1d 5075 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑦))) |
27 | 24, 26 | imbi12d 347 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)))) |
28 | breq2 5069 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧)) | |
29 | fveq2 6669 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) | |
30 | 29 | breq2d 5077 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
31 | 28, 30 | imbi12d 347 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) |
32 | 27, 31 | cbvral2vw 3461 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
33 | 23, 32 | sylib 220 | . . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
34 | 33 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
35 | rexr 10686 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
37 | 25 | breq1d 5075 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑤) < 𝑎)) |
38 | 37 | cbvrabv 3491 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = {𝑤 ∈ 𝐴 ∣ (𝐹‘𝑤) < 𝑎} |
39 | eqid 2821 | . . . 4 ⊢ sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) = sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) | |
40 | eqid 2821 | . . . 4 ⊢ (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
41 | eqid 2821 | . . . 4 ⊢ (-∞(,]sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,]sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
42 | 18, 19, 20, 22, 34, 2, 6, 36, 38, 39, 40, 41 | incsmflem 43017 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴)) |
43 | reex 10627 | . . . . . . 7 ⊢ ℝ ∈ V | |
44 | 43 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
45 | 44, 8 | ssexd 5227 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
46 | elrest 16700 | . . . . 5 ⊢ ((𝐵 ∈ SAlg ∧ 𝐴 ∈ V) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) | |
47 | 7, 45, 46 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
48 | 47 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
49 | 42, 48 | mpbird 259 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴)) |
50 | 1, 7, 16, 17, 49 | issmfd 43011 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4837 class class class wbr 5065 ran crn 5555 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 supcsup 8903 ℝcr 10535 -∞cmnf 10672 ℝ*cxr 10673 < clt 10674 ≤ cle 10675 (,)cioo 12737 (,]cioc 12738 ↾t crest 16693 topGenctg 16710 Topctop 21500 SAlgcsalg 42592 SalGencsalgen 42596 SMblFncsmblfn 42976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-card 9367 df-acn 9370 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-ioo 12741 df-ioc 12742 df-ico 12743 df-fl 13161 df-rest 16695 df-topgen 16716 df-top 21501 df-bases 21553 df-salg 42593 df-salgen 42597 df-smblfn 42977 |
This theorem is referenced by: smfid 43028 |
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