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, non-decreasing 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 1843 | . 2 ⊢ Ⅎ𝑎𝜑 | |
2 | incsmf.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
3 | retop 22565 | . . . . 5 ⊢ (topGen‘ran (,)) ∈ Top | |
4 | 2, 3 | eqeltri 2697 | . . . 4 ⊢ 𝐽 ∈ Top |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐽 ∈ Top) |
6 | incsmf.b | . . 3 ⊢ 𝐵 = (SalGen‘𝐽) | |
7 | 5, 6 | salgencld 40567 | . 2 ⊢ (𝜑 → 𝐵 ∈ SAlg) |
8 | incsmf.a | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 5, 6 | unisalgen2 40572 | . . . 4 ⊢ (𝜑 → ∪ 𝐵 = ∪ 𝐽) |
10 | 2 | unieqi 4445 | . . . . 5 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝐽 = ∪ (topGen‘ran (,))) |
12 | uniretop 22566 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
13 | 12 | eqcomi 2631 | . . . . 5 ⊢ ∪ (topGen‘ran (,)) = ℝ |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ (topGen‘ran (,)) = ℝ) |
15 | 9, 11, 14 | 3eqtrrd 2661 | . . 3 ⊢ (𝜑 → ℝ = ∪ 𝐵) |
16 | 8, 15 | sseqtrd 3641 | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝐵) |
17 | incsmf.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
18 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑎 ∈ ℝ) | |
19 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑎 ∈ ℝ) | |
20 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐴 ⊆ ℝ) |
21 | 17 | frexr 39604 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
22 | 21 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝐴⟶ℝ*) |
23 | incsmf.i | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
24 | breq1 4656 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 ≤ 𝑦 ↔ 𝑤 ≤ 𝑦)) | |
25 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) | |
26 | 25 | breq1d 4663 | . . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑦))) |
27 | 24, 26 | imbi12d 334 | . . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)))) |
28 | breq2 4657 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝑤 ≤ 𝑦 ↔ 𝑤 ≤ 𝑧)) | |
29 | fveq2 6191 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) | |
30 | 29 | breq2d 4665 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
31 | 28, 30 | imbi12d 334 | . . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝑤 ≤ 𝑦 → (𝐹‘𝑤) ≤ (𝐹‘𝑦)) ↔ (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧)))) |
32 | 27, 31 | cbvral2v 3179 | . . . . . 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 208 | . . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
34 | 33 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑤 ≤ 𝑧 → (𝐹‘𝑤) ≤ (𝐹‘𝑧))) |
35 | rexr 10085 | . . . . 5 ⊢ (𝑎 ∈ ℝ → 𝑎 ∈ ℝ*) | |
36 | 35 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ*) |
37 | 25 | breq1d 4663 | . . . . 5 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑤) < 𝑎)) |
38 | 37 | cbvrabv 3199 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = {𝑤 ∈ 𝐴 ∣ (𝐹‘𝑤) < 𝑎} |
39 | eqid 2622 | . . . 4 ⊢ sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) = sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < ) | |
40 | eqid 2622 | . . . 4 ⊢ (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) = (-∞(,)sup({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎}, ℝ*, < )) | |
41 | eqid 2622 | . . . 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 40950 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴)) |
43 | reex 10027 | . . . . . . 7 ⊢ ℝ ∈ V | |
44 | 43 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ∈ V) |
45 | 44, 8 | ssexd 4805 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
46 | elrest 16088 | . . . . 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 693 | . . . 4 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴) ↔ ∃𝑏 ∈ 𝐵 {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} = (𝑏 ∩ 𝐴))) |
48 | 47 | adantr 481 | . . 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 247 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝐵 ↾t 𝐴)) |
50 | 1, 7, 16, 17, 49 | issmfd 40944 | 1 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ∪ cuni 4436 class class class wbr 4653 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 ℝcr 9935 -∞cmnf 10072 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 (,]cioc 12176 ↾t crest 16081 topGenctg 16098 Topctop 20698 SAlgcsalg 40528 SalGencsalgen 40532 SMblFncsmblfn 40909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-card 8765 df-acn 8768 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ioc 12180 df-ico 12181 df-fl 12593 df-rest 16083 df-topgen 16104 df-top 20699 df-bases 20750 df-salg 40529 df-salgen 40533 df-smblfn 40910 |
This theorem is referenced by: smfid 40961 |
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