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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimalegt | 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 left-open, unbounded above intervals, belong to the sigma-algebra. (ii) implies (iii) in Proposition 121B of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimalegt.x | ⊢ Ⅎ𝑥𝜑 |
salpreimalegt.a | ⊢ Ⅎ𝑎𝜑 |
salpreimalegt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimalegt.u | ⊢ 𝐴 = ∪ 𝑆 |
salpreimalegt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimalegt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) |
salpreimalegt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimalegt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimalegt.u | . . . . . 6 ⊢ 𝐴 = ∪ 𝑆 | |
2 | 1 | eqcomi 2660 | . . . . 5 ⊢ ∪ 𝑆 = 𝐴 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴) |
4 | 3 | difeq1d 3760 | . . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) |
5 | salpreimalegt.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
6 | salpreimalegt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
7 | salpreimalegt.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
8 | 7 | rexrd 10127 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
9 | 5, 6, 8 | preimalegt 41234 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) = {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵}) |
10 | 4, 9 | eqtr2d 2686 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶})) |
11 | salpreimalegt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
12 | 7 | ancli 573 | . . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ)) |
13 | nfcv 2793 | . . . . 5 ⊢ Ⅎ𝑎𝐶 | |
14 | salpreimalegt.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
15 | nfv 1883 | . . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ | |
16 | 14, 15 | nfan 1868 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ) |
17 | nfv 1883 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆 | |
18 | 16, 17 | nfim 1865 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆) |
19 | eleq1 2718 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ)) | |
20 | 19 | anbi2d 740 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ))) |
21 | breq2 4689 | . . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝐵 ≤ 𝑎 ↔ 𝐵 ≤ 𝐶)) | |
22 | 21 | rabbidv 3220 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) |
23 | 22 | eleq1d 2715 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)) |
24 | 20, 23 | imbi12d 333 | . . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆))) |
25 | salpreimalegt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝑎} ∈ 𝑆) | |
26 | 13, 18, 24, 25 | vtoclgf 3295 | . . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆)) |
27 | 7, 12, 26 | sylc 65 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶} ∈ 𝑆) |
28 | 11, 27 | saldifcld 40883 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐵 ≤ 𝐶}) ∈ 𝑆) |
29 | 10, 28 | eqeltrd 2730 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 < 𝐵} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 Ⅎwnf 1748 ∈ wcel 2030 {crab 2945 ∖ cdif 3604 ∪ cuni 4468 class class class wbr 4685 ℝcr 9973 ℝ*cxr 10111 < clt 10112 ≤ cle 10113 SAlgcsalg 40846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-xp 5149 df-cnv 5151 df-xr 10116 df-le 10118 df-salg 40847 |
This theorem is referenced by: salpreimalelt 41259 issmfgt 41286 |
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