Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > meaiuninc | Structured version Visualization version GIF version |
Description: Measures are continuous from below (bounded case): if 𝐸 is a sequence of non-decreasing measurable sets (with bounded measure) then the measure of the union is the limit of the measures. This is Proposition 112C (e) of [Fremlin1] p. 16. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
Ref | Expression |
---|---|
meaiuninc.m | ⊢ (𝜑 → 𝑀 ∈ Meas) |
meaiuninc.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
meaiuninc.z | ⊢ 𝑍 = (ℤ≥‘𝑁) |
meaiuninc.e | ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) |
meaiuninc.i | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) |
meaiuninc.x | ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) |
meaiuninc.s | ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
Ref | Expression |
---|---|
meaiuninc | ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meaiuninc.s | . . . 4 ⊢ 𝑆 = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) | |
2 | fveq2 6103 | . . . . . 6 ⊢ (𝑛 = 𝑚 → (𝐸‘𝑛) = (𝐸‘𝑚)) | |
3 | 2 | fveq2d 6107 | . . . . 5 ⊢ (𝑛 = 𝑚 → (𝑀‘(𝐸‘𝑛)) = (𝑀‘(𝐸‘𝑚))) |
4 | 3 | cbvmptv 4678 | . . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) |
5 | 1, 4 | eqtri 2632 | . . 3 ⊢ 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → 𝑆 = (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚)))) |
7 | meaiuninc.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ Meas) | |
8 | meaiuninc.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
9 | meaiuninc.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑁) | |
10 | meaiuninc.e | . . 3 ⊢ (𝜑 → 𝐸:𝑍⟶dom 𝑀) | |
11 | meaiuninc.i | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ (𝐸‘(𝑛 + 1))) | |
12 | meaiuninc.x | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 (𝑀‘(𝐸‘𝑛)) ≤ 𝑥) | |
13 | 5, 1 | eqtr3i 2634 | . . 3 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) = (𝑛 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑛))) |
14 | fveq2 6103 | . . . . . . 7 ⊢ (𝑘 = 𝑖 → (𝐸‘𝑘) = (𝐸‘𝑖)) | |
15 | 14 | cbviunv 4495 | . . . . . 6 ⊢ ∪ 𝑘 ∈ (𝑁..^𝑚)(𝐸‘𝑘) = ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖) |
16 | 15 | difeq2i 3687 | . . . . 5 ⊢ ((𝐸‘𝑚) ∖ ∪ 𝑘 ∈ (𝑁..^𝑚)(𝐸‘𝑘)) = ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖)) |
17 | 16 | mpteq2i 4669 | . . . 4 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝑚) ∖ ∪ 𝑘 ∈ (𝑁..^𝑚)(𝐸‘𝑘))) = (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖))) |
18 | fveq2 6103 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝐸‘𝑚) = (𝐸‘𝑛)) | |
19 | oveq2 6557 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑁..^𝑚) = (𝑁..^𝑛)) | |
20 | 19 | iuneq1d 4481 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖) = ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖)) |
21 | 18, 20 | difeq12d 3691 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖)) = ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) |
22 | 21 | cbvmptv 4678 | . . . 4 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝑚) ∖ ∪ 𝑖 ∈ (𝑁..^𝑚)(𝐸‘𝑖))) = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) |
23 | 17, 22 | eqtri 2632 | . . 3 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐸‘𝑚) ∖ ∪ 𝑘 ∈ (𝑁..^𝑚)(𝐸‘𝑘))) = (𝑛 ∈ 𝑍 ↦ ((𝐸‘𝑛) ∖ ∪ 𝑖 ∈ (𝑁..^𝑛)(𝐸‘𝑖))) |
24 | 7, 8, 9, 10, 11, 12, 13, 23 | meaiuninclem 39373 | . 2 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝑀‘(𝐸‘𝑚))) ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
25 | 6, 24 | eqbrtrd 4605 | 1 ⊢ (𝜑 → 𝑆 ⇝ (𝑀‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∖ cdif 3537 ⊆ wss 3540 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 1c1 9816 + caddc 9818 ≤ cle 9954 ℤcz 11254 ℤ≥cuz 11563 ..^cfzo 12334 ⇝ cli 14063 Meascmea 39342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-acn 8651 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-xadd 11823 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-salg 39205 df-sumge0 39256 df-mea 39343 |
This theorem is referenced by: meaiuninc2 39375 |
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