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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimaltle | Structured version Visualization version GIF version |
Description: If all the preimages of right-open, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-closed, unbounded below intervals, belong to the sigma-algebra. (i) implies (ii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimaltle.x | ⊢ Ⅎ𝑥𝜑 |
salpreimaltle.a | ⊢ Ⅎ𝑎𝜑 |
salpreimaltle.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimaltle.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimaltle.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ 𝑆) |
salpreimaltle.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimaltle | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimaltle.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimaltle.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
3 | salpreimaltle.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
4 | 1, 2, 3 | preimaleiinlt 46032 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))}) |
5 | salpreimaltle.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | nnct 13970 | . . . 4 ⊢ ℕ ≼ ω | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) |
8 | nnn0 44683 | . . . 4 ⊢ ℕ ≠ ∅ | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) |
10 | simpl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝜑) | |
11 | simpl 482 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ) | |
12 | nnrecre 12276 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
13 | 12 | adantl 481 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) |
14 | 11, 13 | readdcld 11265 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝐶 + (1 / 𝑛)) ∈ ℝ) |
15 | 3, 14 | sylan 579 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 + (1 / 𝑛)) ∈ ℝ) |
16 | salpreimaltle.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
17 | nfv 1910 | . . . . . . 7 ⊢ Ⅎ𝑎(𝐶 + (1 / 𝑛)) ∈ ℝ | |
18 | 16, 17 | nfan 1895 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ (𝐶 + (1 / 𝑛)) ∈ ℝ) |
19 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆 | |
20 | 18, 19 | nfim 1892 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ (𝐶 + (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆) |
21 | ovex 7447 | . . . . 5 ⊢ (𝐶 + (1 / 𝑛)) ∈ V | |
22 | eleq1 2816 | . . . . . . 7 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → (𝑎 ∈ ℝ ↔ (𝐶 + (1 / 𝑛)) ∈ ℝ)) | |
23 | 22 | anbi2d 628 | . . . . . 6 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ (𝐶 + (1 / 𝑛)) ∈ ℝ))) |
24 | breq2 5146 | . . . . . . . 8 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → (𝐵 < 𝑎 ↔ 𝐵 < (𝐶 + (1 / 𝑛)))) | |
25 | 24 | rabbidv 3435 | . . . . . . 7 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))}) |
26 | 25 | eleq1d 2813 | . . . . . 6 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → ({𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆)) |
27 | 23, 26 | imbi12d 344 | . . . . 5 ⊢ (𝑎 = (𝐶 + (1 / 𝑛)) → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ 𝑆) ↔ ((𝜑 ∧ (𝐶 + (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆))) |
28 | salpreimaltle.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ 𝑆) | |
29 | 20, 21, 27, 28 | vtoclf 3547 | . . . 4 ⊢ ((𝜑 ∧ (𝐶 + (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆) |
30 | 10, 15, 29 | syl2anc 583 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆) |
31 | 5, 7, 9, 30 | saliincl 45638 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ 𝐵 < (𝐶 + (1 / 𝑛))} ∈ 𝑆) |
32 | 4, 31 | eqeltrd 2828 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ≠ wne 2935 {crab 3427 ∅c0 4318 ∩ ciin 4992 class class class wbr 5142 (class class class)co 7414 ωcom 7864 ≼ cdom 8953 ℝcr 11129 1c1 11131 + caddc 11133 ℝ*cxr 11269 < clt 11270 ≤ cle 11271 / cdiv 11893 ℕcn 12234 SAlgcsalg 45619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-card 9954 df-acn 9957 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-q 12955 df-rp 12999 df-fl 13781 df-salg 45620 |
This theorem is referenced by: issmfle 46056 |
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