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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmflem | Structured version Visualization version GIF version |
Description: A non decreasing function is Borel measurable. Proposition 121D (c) of [Fremlin1] p. 36 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
incsmflem.x | ⊢ Ⅎ𝑥𝜑 |
incsmflem.y | ⊢ Ⅎ𝑦𝜑 |
incsmflem.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
incsmflem.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
incsmflem.i | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
incsmflem.j | ⊢ 𝐽 = (topGen‘ran (,)) |
incsmflem.b | ⊢ 𝐵 = (SalGen‘𝐽) |
incsmflem.r | ⊢ (𝜑 → 𝑅 ∈ ℝ*) |
incsmflem.l | ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} |
incsmflem.c | ⊢ 𝐶 = sup(𝑌, ℝ*, < ) |
incsmflem.d | ⊢ 𝐷 = (-∞(,)𝐶) |
incsmflem.e | ⊢ 𝐸 = (-∞(,]𝐶) |
Ref | Expression |
---|---|
incsmflem | ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incsmflem.e | . . . 4 ⊢ 𝐸 = (-∞(,]𝐶) | |
2 | mnfxr 10308 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
4 | incsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
5 | ssrab2 3828 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 3776 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
8 | incsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 7, 8 | sstrd 3754 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
10 | 9 | sselda 3744 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ ℝ) |
11 | incsmflem.j | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
12 | incsmflem.b | . . . . 5 ⊢ 𝐵 = (SalGen‘𝐽) | |
13 | 3, 10, 11, 12 | iocborel 41095 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
14 | 1, 13 | syl5eqel 2843 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
15 | incsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
16 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
17 | nfrab1 3261 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
18 | 4, 17 | nfcxfr 2900 | . . . . . 6 ⊢ Ⅎ𝑥𝑌 |
19 | 16, 18 | nfel 2915 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
20 | 15, 19 | nfan 1977 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
21 | incsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
22 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌 | |
23 | 21, 22 | nfan 1977 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐶 ∈ 𝑌) |
24 | 8 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
25 | incsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
26 | 25 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
27 | incsmflem.i | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) | |
28 | 27 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
29 | incsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
30 | 29 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
31 | incsmflem.c | . . . 4 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
32 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
33 | 20, 23, 24, 26, 28, 30, 4, 31, 32, 1 | pimincfltioc 41450 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
34 | ineq1 3950 | . . . . 5 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
35 | 34 | eqeq2d 2770 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑌 = (𝑏 ∩ 𝐴) ↔ 𝑌 = (𝐸 ∩ 𝐴))) |
36 | 35 | rspcev 3449 | . . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
37 | 14, 33, 36 | syl2anc 696 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
38 | incsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
39 | 11, 12 | iooborel 41090 | . . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵 |
40 | 38, 39 | eqeltri 2835 | . . . . 5 ⊢ 𝐷 ∈ 𝐵 |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
42 | 41 | adantr 472 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵) |
43 | 19 | nfn 1933 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
44 | 15, 43 | nfan 1977 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
45 | nfv 1992 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
46 | 21, 45 | nfan 1977 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
47 | 8 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
48 | 25 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
49 | 27 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
50 | 29 | adantr 472 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
51 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
52 | 44, 46, 47, 48, 49, 50, 4, 31, 51, 38 | pimincfltioo 41452 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
53 | ineq1 3950 | . . . . 5 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
54 | 53 | eqeq2d 2770 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑌 = (𝑏 ∩ 𝐴) ↔ 𝑌 = (𝐷 ∩ 𝐴))) |
55 | 54 | rspcev 3449 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
56 | 42, 52, 55 | syl2anc 696 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
57 | 37, 56 | pm2.61dan 867 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 Ⅎwnf 1857 ∈ wcel 2139 ∀wral 3050 ∃wrex 3051 {crab 3054 ∩ cin 3714 ⊆ wss 3715 class class class wbr 4804 ran crn 5267 ⟶wf 6045 ‘cfv 6049 (class class class)co 6814 supcsup 8513 ℝcr 10147 -∞cmnf 10284 ℝ*cxr 10285 < clt 10286 ≤ cle 10287 (,)cioo 12388 (,]cioc 12389 topGenctg 16320 SalGencsalgen 41053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-inf2 8713 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-pre-sup 10226 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-sup 8515 df-inf 8516 df-card 8975 df-acn 8978 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-n0 11505 df-z 11590 df-uz 11900 df-q 12002 df-rp 12046 df-ioo 12392 df-ioc 12393 df-fl 12807 df-topgen 16326 df-top 20921 df-bases 20972 df-salg 41050 df-salgen 41054 |
This theorem is referenced by: incsmf 41475 |
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