Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagtlt | Structured version Visualization version GIF version |
Description: If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimagtlt.x | ⊢ Ⅎ𝑥𝜑 |
salpreimagtlt.a | ⊢ Ⅎ𝑎𝜑 |
salpreimagtlt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimagtlt.u | ⊢ 𝐴 = ∪ 𝑆 |
salpreimagtlt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimagtlt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) |
salpreimagtlt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimagtlt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimagtlt.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | salpreimagtlt.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | salpreimagtlt.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | salpreimagtlt.u | . 2 ⊢ 𝐴 = ∪ 𝑆 | |
5 | salpreimagtlt.b | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
6 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
7 | 1, 6 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
8 | nfv 1915 | . . 3 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑎 ∈ ℝ) | |
9 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
10 | 5 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
11 | nfv 1915 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
12 | 2, 11 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
13 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆 | |
14 | 12, 13 | nfim 1897 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
15 | eleq1w 2895 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℝ ↔ 𝑏 ∈ ℝ)) | |
16 | 15 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝑏 ∈ ℝ))) |
17 | breq1 5069 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 < 𝐵 ↔ 𝑏 < 𝐵)) | |
18 | 17 | rabbidv 3480 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵}) |
19 | 18 | eleq1d 2897 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆)) |
20 | 16, 19 | imbi12d 347 | . . . . 5 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆))) |
21 | salpreimagtlt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | |
22 | 14, 20, 21 | chvarfv 2242 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
23 | 22 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
24 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
25 | 7, 8, 9, 10, 23, 24 | salpreimagtge 43022 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) |
26 | salpreimagtlt.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
27 | 1, 2, 3, 4, 5, 25, 26 | salpreimagelt 43006 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 {crab 3142 ∪ cuni 4838 class class class wbr 5066 ℝcr 10536 ℝ*cxr 10674 < clt 10675 SAlgcsalg 42613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-card 9368 df-acn 9371 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-fl 13163 df-salg 42614 |
This theorem is referenced by: issmfgtlem 43052 |
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