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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 45035 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
5 | salpreimagtge.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | nnct 13893 | . . . 4 ⊢ ℕ ≼ ω | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
8 | nnn0 43686 | . . . 4 ⊢ ℕ ≠ ∅ | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) |
10 | 3 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ) |
11 | nnrecre 12202 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
12 | 11 | adantl 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
13 | 10, 12 | resubcld 11590 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 − (1 / 𝑛)) ∈ ℝ) |
14 | salpreimagtge.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
15 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑎(𝐶 − (1 / 𝑛)) ∈ ℝ | |
16 | 14, 15 | nfan 1903 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) |
17 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆 | |
18 | 16, 17 | nfim 1900 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
19 | ovex 7395 | . . . . 5 ⊢ (𝐶 − (1 / 𝑛)) ∈ V | |
20 | eleq1 2826 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 ∈ ℝ ↔ (𝐶 − (1 / 𝑛)) ∈ ℝ)) | |
21 | 20 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ))) |
22 | breq1 5113 | . . . . . . . 8 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 < 𝐵 ↔ (𝐶 − (1 / 𝑛)) < 𝐵)) | |
23 | 22 | rabbidv 3418 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) |
24 | 23 | eleq1d 2823 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆)) |
25 | 21, 24 | imbi12d 345 | . . . . 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 3519 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
28 | 13, 27 | syldan 592 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
29 | 5, 7, 9, 28 | saliincl 44642 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) |
30 | 4, 29 | eqeltrd 2838 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ≠ wne 2944 {crab 3410 ∅c0 4287 ∩ ciin 4960 class class class wbr 5110 (class class class)co 7362 ωcom 7807 ≼ cdom 8888 ℝcr 11057 1c1 11059 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 − cmin 11392 / cdiv 11819 ℕcn 12160 SAlgcsalg 44623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-fl 13704 df-salg 44624 |
This theorem is referenced by: salpreimalelt 45044 salpreimagtlt 45045 issmfge 45085 |
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