Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > decsmflem | Structured version Visualization version GIF version |
Description: A non-increasing 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 9975 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
4 | decsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
5 | ssrab2 3650 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 3598 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
8 | decsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 7, 8 | sstrd 3578 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
10 | 9 | sselda 3568 | . . . . 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 39250 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
14 | 1, 13 | syl5eqel 2692 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
15 | decsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
16 | decsmflem.c | . . . . . . 7 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
17 | nfrab1 3099 | . . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝑅 < (𝐹‘𝑥)} | |
18 | 4, 17 | nfcxfr 2749 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑌 |
19 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥ℝ* | |
20 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥 < | |
21 | 18, 19, 20 | nfsup 8240 | . . . . . . 7 ⊢ Ⅎ𝑥sup(𝑌, ℝ*, < ) |
22 | 16, 21 | nfcxfr 2749 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 |
23 | 22, 18 | nfel 2763 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
24 | 15, 23 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
25 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
26 | decsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
27 | 26 | adantr 480 | . . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
30 | decsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
31 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
32 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
33 | 24, 25, 27, 29, 31, 4, 16, 32, 1 | pimdecfgtioc 39602 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
34 | ineq1 3769 | . . . . 5 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
35 | 34 | eqeq2d 2620 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑌 = (𝑏 ∩ 𝐴) ↔ 𝑌 = (𝐸 ∩ 𝐴))) |
36 | 35 | rspcev 3282 | . . 3 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝑌 = (𝐸 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
37 | 14, 33, 36 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
38 | decsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
39 | 11, 12 | iooborel 39245 | . . . . . 6 ⊢ (-∞(,)𝐶) ∈ 𝐵 |
40 | 38, 39 | eqeltri 2684 | . . . . 5 ⊢ 𝐷 ∈ 𝐵 |
41 | 40 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝐵) |
42 | 41 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐷 ∈ 𝐵) |
43 | 23 | nfn 1768 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
44 | 15, 43 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
45 | decsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
46 | nfv 1830 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
47 | 45, 46 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
48 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
49 | 26 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
50 | 28 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ (𝐹‘𝑥))) |
51 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
52 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
53 | 44, 47, 48, 49, 50, 51, 4, 16, 52, 38 | pimdecfgtioo 39604 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
54 | ineq1 3769 | . . . . 5 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
55 | 54 | eqeq2d 2620 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑌 = (𝑏 ∩ 𝐴) ↔ 𝑌 = (𝐷 ∩ 𝐴))) |
56 | 55 | rspcev 3282 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
57 | 42, 53, 56 | syl2anc 691 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
58 | 37, 57 | pm2.61dan 828 | 1 ⊢ (𝜑 → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supcsup 8229 ℝcr 9814 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 (,]cioc 12047 topGenctg 15921 SalGencsalgen 39208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-ioo 12050 df-ioc 12051 df-fl 12455 df-topgen 15927 df-top 20521 df-bases 20522 df-salg 39205 df-salgen 39209 |
This theorem is referenced by: decsmf 39653 |
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