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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 1912 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
7 | 1, 6 | nfan 1897 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) |
8 | nfv 1912 | . . 3 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑎 ∈ ℝ) | |
9 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) |
10 | 5 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
11 | nfv 1912 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
12 | 2, 11 | nfan 1897 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) |
13 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆 | |
14 | 12, 13 | nfim 1894 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
15 | eleq1w 2822 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℝ ↔ 𝑏 ∈ ℝ)) | |
16 | 15 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝑏 ∈ ℝ))) |
17 | breq1 5151 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 < 𝐵 ↔ 𝑏 < 𝐵)) | |
18 | 17 | rabbidv 3441 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵}) |
19 | 18 | eleq1d 2824 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆)) |
20 | 16, 19 | imbi12d 344 | . . . . 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 2238 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
23 | 22 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) |
24 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
25 | 7, 8, 9, 10, 23, 24 | salpreimagtge 46681 | . 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 46663 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 {crab 3433 ∪ cuni 4912 class class class wbr 5148 ℝcr 11152 ℝ*cxr 11292 < clt 11293 SAlgcsalg 46264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-fl 13829 df-salg 46265 |
This theorem is referenced by: issmfgtlem 46711 |
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