Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiininc | Structured version Visualization version GIF version |
Description: Measures are continuous from above: if 𝐸 is a nonincreasing sequence of measurable sets, and any of the sets has finite measure, then the measure of the intersection is the limit of the measures. This is Proposition 112C (f) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meaiininc.f | ⊢ Ⅎ𝑛𝜑 |
meaiininc.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaiininc.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
meaiininc.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
meaiininc.e | ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
meaiininc.i | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) |
meaiininc.k | ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) |
meaiininc.r | ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) |
meaiininc.s | ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
Ref | Expression |
---|---|
meaiininc | ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meaiininc.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
2 | meaiininc.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | meaiininc.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
4 | meaiininc.e | . . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) | |
5 | meaiininc.f | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
6 | nfv 1921 | . . . . . 6 ⊢ Ⅎ𝑛 𝑖 ∈ 𝑍 | |
7 | 5, 6 | nfan 1906 | . . . . 5 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑖 ∈ 𝑍) |
8 | nfv 1921 | . . . . 5 ⊢ Ⅎ𝑛(𝐸‘(𝑖 + 1)) ⊆ (𝐸‘𝑖) | |
9 | 7, 8 | nfim 1903 | . . . 4 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘(𝑖 + 1)) ⊆ (𝐸‘𝑖)) |
10 | eleq1w 2815 | . . . . . 6 ⊢ (𝑛 = 𝑖 → (𝑛 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍)) | |
11 | 10 | anbi2d 632 | . . . . 5 ⊢ (𝑛 = 𝑖 → ((𝜑 ∧ 𝑛 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍))) |
12 | fvoveq1 7193 | . . . . . 6 ⊢ (𝑛 = 𝑖 → (𝐸‘(𝑛 + 1)) = (𝐸‘(𝑖 + 1))) | |
13 | fveq2 6674 | . . . . . 6 ⊢ (𝑛 = 𝑖 → (𝐸‘𝑛) = (𝐸‘𝑖)) | |
14 | 12, 13 | sseq12d 3910 | . . . . 5 ⊢ (𝑛 = 𝑖 → ((𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛) ↔ (𝐸‘(𝑖 + 1)) ⊆ (𝐸‘𝑖))) |
15 | 11, 14 | imbi12d 348 | . . . 4 ⊢ (𝑛 = 𝑖 → (((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘(𝑖 + 1)) ⊆ (𝐸‘𝑖)))) |
16 | meaiininc.i | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘(𝑛 + 1)) ⊆ (𝐸‘𝑛)) | |
17 | 9, 15, 16 | chvarfv 2242 | . . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐸‘(𝑖 + 1)) ⊆ (𝐸‘𝑖)) |
18 | meaiininc.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑁)) | |
19 | meaiininc.r | . . 3 ⊢ (𝜑 → (𝑀‘(𝐸‘𝐾)) ∈ ℝ) | |
20 | 2fveq3 6679 | . . . 4 ⊢ (𝑚 = 𝑖 → (𝑀‘(𝐸‘𝑚)) = (𝑀‘(𝐸‘𝑖))) | |
21 | 20 | cbvmptv 5133 | . . 3 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) = (𝑖 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑖))) |
22 | 13 | difeq2d 4013 | . . . 4 ⊢ (𝑛 = 𝑖 → ((𝐸‘𝐾) ∖ (𝐸‘𝑛)) = ((𝐸‘𝐾) ∖ (𝐸‘𝑖))) |
23 | 22 | cbvmptv 5133 | . . 3 ⊢ (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛))) = (𝑖 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑖))) |
24 | fveq2 6674 | . . . 4 ⊢ (𝑚 = 𝑖 → ((𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))‘𝑚) = ((𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))‘𝑖)) | |
25 | 24 | cbviunv 4926 | . . 3 ⊢ ∪ 𝑚 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))‘𝑚) = ∪ 𝑖 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ ((𝐸‘𝐾) ∖ (𝐸‘𝑛)))‘𝑖) |
26 | 1, 2, 3, 4, 17, 18, 19, 21, 23, 25 | meaiininclem 43566 | . 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) ⇝ (𝑀‘∩ 𝑖 ∈ 𝑍 (𝐸‘𝑖))) |
27 | meaiininc.s | . . . . 5 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
28 | 2fveq3 6679 | . . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑚))) | |
29 | 28 | cbvmptv 5133 | . . . . 5 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) |
30 | 27, 29 | eqtri 2761 | . . . 4 ⊢ 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) |
31 | 30 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))) |
32 | 13 | cbviinv 4927 | . . . . 5 ⊢ ∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∩ 𝑖 ∈ 𝑍 (𝐸‘𝑖) |
33 | 32 | fveq2i 6677 | . . . 4 ⊢ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝑀‘∩ 𝑖 ∈ 𝑍 (𝐸‘𝑖)) |
34 | 33 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝑀‘∩ 𝑖 ∈ 𝑍 (𝐸‘𝑖))) |
35 | 31, 34 | breq12d 5043 | . 2 ⊢ (𝜑 → (𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ↔ (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) ⇝ (𝑀‘∩ 𝑖 ∈ 𝑍 (𝐸‘𝑖)))) |
36 | 26, 35 | mpbird 260 | 1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∩ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 ∖ cdif 3840 ⊆ wss 3843 ∪ ciun 4881 ∩ ciin 4882 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5525 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ℝcr 10614 1c1 10616 + caddc 10618 ℤcz 12062 ℤ≥cuz 12324 ⇝ cli 14931 Meascmea 43529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-omul 8136 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-oi 9047 df-card 9441 df-acn 9444 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-xadd 12591 df-ico 12827 df-icc 12828 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-clim 14935 df-sum 15136 df-salg 43392 df-sumge0 43443 df-mea 43530 |
This theorem is referenced by: meaiininc2 43568 vonicclem2 43764 |
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