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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagtlt | Structured version Visualization version GIF version |
Description: If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimagtlt.x | ⊢ Ⅎ𝑥𝜑 |
salpreimagtlt.a | ⊢ Ⅎ𝑎𝜑 |
salpreimagtlt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimagtlt.u | ⊢ 𝐴 = ∪ 𝑆 |
salpreimagtlt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimagtlt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) |
salpreimagtlt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimagtlt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimagtlt.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimagtlt.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | salpreimagtlt.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | salpreimagtlt.u | . 2 ⊢ 𝐴 = ∪ 𝑆 | |
5 | salpreimagtlt.b | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
6 | nfv 2013 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
7 | 1, 6 | nfan 2002 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
8 | nfv 2013 | . . 3 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑎 ∈ ℝ) | |
9 | 3 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
10 | 5 | adantlr 706 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
11 | nfv 2013 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
12 | 2, 11 | nfan 2002 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
13 | nfv 2013 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆 | |
14 | 12, 13 | nfim 1999 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
15 | eleq1w 2889 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℝ ↔ 𝑏 ∈ ℝ)) | |
16 | 15 | anbi2d 622 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝑏 ∈ ℝ))) |
17 | breq1 4878 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 < 𝐵 ↔ 𝑏 < 𝐵)) | |
18 | 17 | rabbidv 3402 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵}) |
19 | 18 | eleq1d 2891 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆)) |
20 | 16, 19 | imbi12d 336 | . . . . 5 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆))) |
21 | salpreimagtlt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | |
22 | 14, 20, 21 | chvar 2415 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
23 | 22 | adantlr 706 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
24 | simpr 479 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
25 | 7, 8, 9, 10, 23, 24 | salpreimagtge 41726 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) |
26 | salpreimagtlt.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
27 | 1, 2, 3, 4, 5, 25, 26 | salpreimagelt 41710 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 Ⅎwnf 1882 ∈ wcel 2164 {crab 3121 ∪ cuni 4660 class class class wbr 4875 ℝcr 10258 ℝ*cxr 10397 < clt 10398 SAlgcsalg 41317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-card 9085 df-acn 9088 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-fl 12895 df-salg 41318 |
This theorem is referenced by: issmfgtlem 41756 |
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