Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagtge | Structured version Visualization version GIF version |
Description: If all the preimages of left-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of left-closed, unbounded above intervals, belong to the sigma-algebra. (iii) implies (iv) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimagtge.x | ⊢ Ⅎ𝑥𝜑 |
salpreimagtge.a | ⊢ Ⅎ𝑎𝜑 |
salpreimagtge.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimagtge.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimagtge.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) |
salpreimagtge.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimagtge | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimagtge.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimagtge.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
3 | salpreimagtge.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 1, 2, 3 | preimageiingt 42875 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
5 | salpreimagtge.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | nnct 13337 | . . . 4 ⊢ ℕ ≼ ω | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
8 | nnn0 41523 | . . . 4 ⊢ ℕ ≠ ∅ | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) |
10 | 3 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ) |
11 | nnrecre 11667 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
12 | 11 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
13 | 10, 12 | resubcld 11056 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 − (1 / 𝑛)) ∈ ℝ) |
14 | salpreimagtge.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
15 | nfv 1906 | . . . . . . 7 ⊢ Ⅎ𝑎(𝐶 − (1 / 𝑛)) ∈ ℝ | |
16 | 14, 15 | nfan 1891 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) |
17 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆 | |
18 | 16, 17 | nfim 1888 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
19 | ovex 7178 | . . . . 5 ⊢ (𝐶 − (1 / 𝑛)) ∈ V | |
20 | eleq1 2897 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 ∈ ℝ ↔ (𝐶 − (1 / 𝑛)) ∈ ℝ)) | |
21 | 20 | anbi2d 628 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ))) |
22 | breq1 5060 | . . . . . . . 8 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 < 𝐵 ↔ (𝐶 − (1 / 𝑛)) < 𝐵)) | |
23 | 22 | rabbidv 3478 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
24 | 23 | eleq1d 2894 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆)) |
25 | 21, 24 | imbi12d 346 | . . . . 5 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆))) |
26 | salpreimagtge.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | |
27 | 18, 19, 25, 26 | vtoclf 3556 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
28 | 13, 27 | syldan 591 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
29 | 5, 7, 9, 28 | saliincl 42487 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
30 | 4, 29 | eqeltrd 2910 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ≠ wne 3013 {crab 3139 ∅c0 4288 ∩ ciin 4911 class class class wbr 5057 (class class class)co 7145 ωcom 7569 ≼ cdom 8495 ℝcr 10524 1c1 10526 ℝ*cxr 10662 < clt 10663 ≤ cle 10664 − cmin 10858 / cdiv 11285 ℕcn 11626 SAlgcsalg 42470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-fl 13150 df-salg 42471 |
This theorem is referenced by: salpreimalelt 42883 salpreimagtlt 42884 issmfge 42923 |
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