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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 10687 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
4 | decsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
5 | ssrab2 4007 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 3949 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
8 | decsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 7, 8 | sstrd 3925 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
10 | 9 | sselda 3915 | . . . . 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 42996 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
14 | 1, 13 | eqeltrid 2894 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
15 | decsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
16 | decsmflem.c | . . . . . . 7 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
17 | nfrab1 3337 | . . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
18 | 4, 17 | nfcxfr 2953 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑌 |
19 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑥ℝ* | |
20 | nfcv 2955 | . . . . . . . 8 ⊢ Ⅎ𝑥 < | |
21 | 18, 19, 20 | nfsup 8899 | . . . . . . 7 ⊢ Ⅎ𝑥sup(𝑌, ℝ*, < ) |
22 | 16, 21 | nfcxfr 2953 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 |
23 | 22, 18 | nfel 2969 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
24 | 15, 23 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
25 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
26 | decsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
27 | 26 | adantr 484 | . . . 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 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
30 | decsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 30 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
32 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
33 | 24, 25, 27, 29, 31, 4, 16, 32, 1 | pimdecfgtioc 43350 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
34 | ineq1 4131 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
35 | 34 | rspceeqv 3586 | . . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
36 | 14, 33, 35 | syl2anc 587 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
37 | decsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
38 | 11, 12 | iooborel 42991 | . . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵 |
39 | 37, 38 | eqeltri 2886 | . . . . 5 ⊢ 𝐷 ∈ 𝐵 |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
41 | 40 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵) |
42 | 23 | nfn 1858 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
43 | 15, 42 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
44 | decsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
45 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
46 | 44, 45 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
47 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
48 | 26 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
49 | 28 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
50 | 30 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
51 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
52 | 43, 46, 47, 48, 49, 50, 4, 16, 51, 37 | pimdecfgtioo 43352 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
53 | ineq1 4131 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
54 | 53 | rspceeqv 3586 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
55 | 41, 52, 54 | syl2anc 587 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
56 | 36, 55 | pm2.61dan 812 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {crab 3110 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 supcsup 8888 ℝcr 10525 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 (,)cioo 12726 (,]cioc 12727 topGenctg 16703 SalGencsalgen 42954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ioc 12731 df-fl 13157 df-topgen 16709 df-top 21499 df-bases 21551 df-salg 42951 df-salgen 42955 |
This theorem is referenced by: decsmf 43400 |
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