Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > incsmflem | Structured version Visualization version GIF version |
Description: A nondecreasing 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 10698 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → -∞ ∈ ℝ*) |
4 | incsmflem.l | . . . . . . . . 9 ⊢ 𝑌 = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
5 | ssrab2 4056 | . . . . . . . . 9 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} ⊆ 𝐴 | |
6 | 4, 5 | eqsstri 4001 | . . . . . . . 8 ⊢ 𝑌 ⊆ 𝐴 |
7 | 6 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝐴) |
8 | incsmflem.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
9 | 7, 8 | sstrd 3977 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ ℝ) |
10 | 9 | sselda 3967 | . . . . 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 42659 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → (-∞(,]𝐶) ∈ 𝐵) |
14 | 1, 13 | eqeltrid 2917 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐸 ∈ 𝐵) |
15 | incsmflem.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
16 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
17 | nfrab1 3384 | . . . . . . 7 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < 𝑅} | |
18 | 4, 17 | nfcxfr 2975 | . . . . . 6 ⊢ Ⅎ𝑥𝑌 |
19 | 16, 18 | nfel 2992 | . . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝑌 |
20 | 15, 19 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐶 ∈ 𝑌) |
21 | incsmflem.y | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
22 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 𝐶 ∈ 𝑌 | |
23 | 21, 22 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ 𝐶 ∈ 𝑌) |
24 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
25 | incsmflem.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
26 | 25 | adantr 483 | . . . 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
29 | incsmflem.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ ℝ*) | |
30 | 29 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
31 | incsmflem.c | . . . 4 ⊢ 𝐶 = sup(𝑌, ℝ*, < ) | |
32 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝐶 ∈ 𝑌) | |
33 | 20, 23, 24, 26, 28, 30, 4, 31, 32, 1 | pimincfltioc 43014 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑌) → 𝑌 = (𝐸 ∩ 𝐴)) |
34 | ineq1 4181 | . . . 4 ⊢ (𝑏 = 𝐸 → (𝑏 ∩ 𝐴) = (𝐸 ∩ 𝐴)) | |
35 | 34 | rspceeqv 3638 | . . 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 | incsmflem.d | . . . . . 6 ⊢ 𝐷 = (-∞(,)𝐶) | |
38 | 11, 12 | iooborel 42654 | . . . . . 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 | 19 | nfn 1857 | . . . . 5 ⊢ Ⅎ𝑥 ¬ 𝐶 ∈ 𝑌 |
43 | 15, 42 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
44 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦 ¬ 𝐶 ∈ 𝑌 | |
45 | 21, 44 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑦(𝜑 ∧ ¬ 𝐶 ∈ 𝑌) |
46 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐴 ⊆ ℝ) |
47 | 25 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝐹:𝐴⟶ℝ*) |
48 | 27 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
49 | 29 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑅 ∈ ℝ*) |
50 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ¬ 𝐶 ∈ 𝑌) | |
51 | 43, 45, 46, 47, 48, 49, 4, 31, 50, 37 | pimincfltioo 43016 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → 𝑌 = (𝐷 ∩ 𝐴)) |
52 | ineq1 4181 | . . . 4 ⊢ (𝑏 = 𝐷 → (𝑏 ∩ 𝐴) = (𝐷 ∩ 𝐴)) | |
53 | 52 | rspceeqv 3638 | . . 3 ⊢ ((𝐷 ∈ 𝐵 ∧ 𝑌 = (𝐷 ∩ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
54 | 41, 51, 53 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐶 ∈ 𝑌) → ∃𝑏 ∈ 𝐵 𝑌 = (𝑏 ∩ 𝐴)) |
55 | 36, 54 | 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 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 {crab 3142 ∩ cin 3935 ⊆ wss 3936 class class class wbr 5066 ran crn 5556 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 supcsup 8904 ℝcr 10536 -∞cmnf 10673 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 (,)cioo 12739 (,]cioc 12740 topGenctg 16711 SalGencsalgen 42617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-ioo 12743 df-ioc 12744 df-fl 13163 df-topgen 16717 df-top 21502 df-bases 21554 df-salg 42614 df-salgen 42618 |
This theorem is referenced by: incsmf 43039 |
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