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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 46735 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} = ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) | 
| 5 | salpreimagtge.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 6 | nnct 14022 | . . . 4 ⊢ ℕ ≼ ω | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≼ ω) | 
| 8 | nnn0 45389 | . . . 4 ⊢ ℕ ≠ ∅ | |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → ℕ ≠ ∅) | 
| 10 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 ∈ ℝ) | 
| 11 | nnrecre 12308 | . . . . . 6 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ) | |
| 12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ) | 
| 13 | 10, 12 | resubcld 11691 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶 − (1 / 𝑛)) ∈ ℝ) | 
| 14 | salpreimagtge.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
| 15 | nfv 1914 | . . . . . . 7 ⊢ Ⅎ𝑎(𝐶 − (1 / 𝑛)) ∈ ℝ | |
| 16 | 14, 15 | nfan 1899 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) | 
| 17 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆 | |
| 18 | 16, 17 | nfim 1896 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ (𝐶 − (1 / 𝑛)) ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) | 
| 19 | ovex 7464 | . . . . 5 ⊢ (𝐶 − (1 / 𝑛)) ∈ V | |
| 20 | eleq1 2829 | . . . . . . 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 5146 | . . . . . . . 8 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → (𝑎 < 𝐵 ↔ (𝐶 − (1 / 𝑛)) < 𝐵)) | |
| 23 | 22 | rabbidv 3444 | . . . . . . 7 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵}) | 
| 24 | 23 | eleq1d 2826 | . . . . . 6 ⊢ (𝑎 = (𝐶 − (1 / 𝑛)) → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆)) | 
| 25 | 21, 24 | imbi12d 344 | . . . . 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 3564 | . . . 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 46342 | . 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ {𝑥 ∈ 𝐴 ∣ (𝐶 − (1 / 𝑛)) < 𝐵} ∈ 𝑆) | 
| 30 | 4, 29 | eqeltrd 2841 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 {crab 3436 ∅c0 4333 ∩ ciin 4992 class class class wbr 5143 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 ℝcr 11154 1c1 11156 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 SAlgcsalg 46323 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-fl 13832 df-salg 46324 | 
| This theorem is referenced by: salpreimalelt 46744 salpreimagtlt 46745 issmfge 46785 | 
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