![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimalelt | Structured version Visualization version GIF version |
Description: If all the preimages of right-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (ii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimalelt.x | ⊢ Ⅎ𝑥𝜑 |
salpreimalelt.a | ⊢ Ⅎ𝑎𝜑 |
salpreimalelt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimalelt.u | ⊢ 𝐴 = ∪ 𝑆 |
salpreimalelt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimalelt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) |
salpreimalelt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimalelt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimalelt.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimalelt.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | salpreimalelt.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | salpreimalelt.u | . 2 ⊢ 𝐴 = ∪ 𝑆 | |
5 | salpreimalelt.b | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
6 | nfv 2015 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
7 | 1, 6 | nfan 2004 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
8 | nfv 2015 | . . 3 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑎 ∈ ℝ) | |
9 | 3 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
10 | 5 | adantlr 708 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
11 | nfv 2015 | . . . . . 6 ⊢ Ⅎ𝑥 𝑏 ∈ ℝ | |
12 | 1, 11 | nfan 2004 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑏 ∈ ℝ) |
13 | nfv 2015 | . . . . . 6 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
14 | 2, 13 | nfan 2004 | . . . . 5 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
15 | 3 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑆 ∈ SAlg) |
16 | 5 | adantlr 708 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
17 | salpreimalelt.p | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) | |
18 | 17 | adantlr 708 | . . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ ℝ) ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) |
19 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ) | |
20 | 12, 14, 15, 4, 16, 18, 19 | salpreimalegt 41715 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
21 | 20 | adantlr 708 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
22 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
23 | 7, 8, 9, 10, 21, 22 | salpreimagtge 41729 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) |
24 | salpreimalelt.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
25 | 1, 2, 3, 4, 5, 23, 24 | salpreimagelt 41713 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 {crab 3122 ∪ cuni 4659 class class class wbr 4874 ℝcr 10252 ℝ*cxr 10391 < clt 10392 ≤ cle 10393 SAlgcsalg 41320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-inf2 8816 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 ax-pre-sup 10331 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-iin 4744 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-se 5303 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-isom 6133 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-map 8125 df-en 8224 df-dom 8225 df-sdom 8226 df-sup 8618 df-inf 8619 df-card 9079 df-acn 9082 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-div 11011 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-q 12073 df-rp 12114 df-fl 12889 df-salg 41321 |
This theorem is referenced by: issmflelem 41748 |
Copyright terms: Public domain | W3C validator |