Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > decsmflem | Structured version Visualization version GIF version |
Description: A nonincreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
decsmflem.x | ⊢ Ⅎ𝑥𝜑 |
decsmflem.y | ⊢ Ⅎ𝑦𝜑 |
decsmflem.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
decsmflem.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
decsmflem.i | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
decsmflem.j | ⊢ 𝐽 = (topGen‘ran (,)) |
decsmflem.b | ⊢ 𝐵 = (SalGen‘𝐽) |
decsmflem.r | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
decsmflem.l | ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} |
decsmflem.c | ⊢ 𝐶 = sup(𝑌, ℝ*, < ) |
decsmflem.d | ⊢ 𝐷 = (-∞(,)𝐶) |
decsmflem.e | ⊢ 𝐸 = (-∞(,]𝐶) |
Ref | Expression |
---|---|
decsmflem | ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | decsmflem.e | . . . 4 ⊢ 𝐸 = (-∞(,]𝐶) | |
2 | mnfxr 10697 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
4 | decsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
5 | ssrab2 4055 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 4000 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
8 | decsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 7, 8 | sstrd 3976 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
10 | 9 | sselda 3966 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ ℝ) |
11 | decsmflem.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
12 | decsmflem.b | . . . . 5 ⊢ 𝐵 = (SalGen‘𝐽) | |
13 | 3, 10, 11, 12 | iocborel 42638 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
14 | 1, 13 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
15 | decsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
16 | decsmflem.c | . . . . . . 7 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
17 | nfrab1 3384 | . . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
18 | 4, 17 | nfcxfr 2975 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑌 |
19 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥ℝ* | |
20 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥 < | |
21 | 18, 19, 20 | nfsup 8914 | . . . . . . 7 ⊢ Ⅎ𝑥sup(𝑌, ℝ*, < ) |
22 | 16, 21 | nfcxfr 2975 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 |
23 | 22, 18 | nfel 2992 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
24 | 15, 23 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
25 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
26 | decsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
27 | 26 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
28 | decsmflem.i | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) | |
29 | 28 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
30 | decsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 30 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
32 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
33 | 24, 25, 27, 29, 31, 4, 16, 32, 1 | pimdecfgtioc 42992 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
34 | ineq1 4180 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
35 | 34 | rspceeqv 3637 | . . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
36 | 14, 33, 35 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
37 | decsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
38 | 11, 12 | iooborel 42633 | . . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵 |
39 | 37, 38 | eqeltri 2909 | . . . . 5 ⊢ 𝐷 ∈ 𝐵 |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
41 | 40 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵) |
42 | 23 | nfn 1853 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
43 | 15, 42 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
44 | decsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
45 | nfv 1911 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
46 | 44, 45 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
47 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
48 | 26 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
49 | 28 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
50 | 30 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
51 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
52 | 43, 46, 47, 48, 49, 50, 4, 16, 51, 37 | pimdecfgtioo 42994 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
53 | ineq1 4180 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
54 | 53 | rspceeqv 3637 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
55 | 41, 52, 54 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
56 | 36, 55 | pm2.61dan 811 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 ∩ cin 3934 ⊆ wss 3935 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 topGenctg 16710 SalGencsalgen 42596 |
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-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-fl 13161 df-topgen 16716 df-top 21501 df-bases 21553 df-salg 42593 df-salgen 42597 |
This theorem is referenced by: decsmf 43042 |
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