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Mirrors > Home > MPE Home > Th. List > Mathboxes > measiuns | Structured version Visualization version GIF version |
Description: The measure of the union of a collection of sets, expressed as the sum of a disjoint set. This is used as a lemma for both measiun 30621 and meascnbl 30622. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.) |
Ref | Expression |
---|---|
measiuns.0 | ⊢ Ⅎ𝑛𝐵 |
measiuns.1 | ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) |
measiuns.2 | ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼))) |
measiuns.3 | ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) |
measiuns.4 | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆) |
Ref | Expression |
---|---|
measiuns | ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | measiuns.0 | . . . 4 ⊢ Ⅎ𝑛𝐵 | |
2 | measiuns.1 | . . . 4 ⊢ (𝑛 = 𝑘 → 𝐴 = 𝐵) | |
3 | measiuns.2 | . . . 4 ⊢ (𝜑 → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼))) | |
4 | 1, 2, 3 | iundisjcnt 29897 | . . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
5 | 4 | fveq2d 6336 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) |
6 | measiuns.3 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆)) | |
7 | measbase 30600 | . . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
8 | 6, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
9 | 8 | adantr 466 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑆 ∈ ∪ ran sigAlgebra) |
10 | measiuns.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆) | |
11 | simpll 742 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑) | |
12 | fzossnn 12725 | . . . . . . . . . . 11 ⊢ (1..^𝑛) ⊆ ℕ | |
13 | simpr 471 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → 𝑁 = ℕ) | |
14 | 12, 13 | syl5sseqr 3803 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → (1..^𝑛) ⊆ 𝑁) |
15 | simplr 744 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ 𝑁) | |
16 | simpr 471 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑁 = (1..^𝐼)) | |
17 | 15, 16 | eleqtrd 2852 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ (1..^𝐼)) |
18 | elfzouz2 12692 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1..^𝐼) → 𝐼 ∈ (ℤ≥‘𝑛)) | |
19 | fzoss2 12704 | . . . . . . . . . . . 12 ⊢ (𝐼 ∈ (ℤ≥‘𝑛) → (1..^𝑛) ⊆ (1..^𝐼)) | |
20 | 17, 18, 19 | 3syl 18 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → (1..^𝑛) ⊆ (1..^𝐼)) |
21 | 20, 16 | sseqtr4d 3791 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → (1..^𝑛) ⊆ 𝑁) |
22 | 3 | adantr 466 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼))) |
23 | 14, 21, 22 | mpjaodan 962 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (1..^𝑛) ⊆ 𝑁) |
24 | 23 | sselda 3752 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ 𝑁) |
25 | 10 | sbimi 2055 | . . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) → [𝑘 / 𝑛]𝐴 ∈ 𝑆) |
26 | sban 2546 | . . . . . . . . . 10 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁)) | |
27 | nfv 1995 | . . . . . . . . . . . 12 ⊢ Ⅎ𝑛𝜑 | |
28 | 27 | sbf 2527 | . . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑) |
29 | clelsb3 2878 | . . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝑛 ∈ 𝑁 ↔ 𝑘 ∈ 𝑁) | |
30 | 28, 29 | anbi12i 604 | . . . . . . . . . 10 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁)) |
31 | 26, 30 | bitri 264 | . . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁)) |
32 | sbsbc 3591 | . . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ [𝑘 / 𝑛]𝐴 ∈ 𝑆) | |
33 | vex 3354 | . . . . . . . . . . 11 ⊢ 𝑘 ∈ V | |
34 | sbcel1g 4131 | . . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆)) | |
35 | 33, 34 | ax-mp 5 | . . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆) |
36 | nfcv 2913 | . . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐴 | |
37 | 36, 1, 2 | cbvcsb 3687 | . . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐵 |
38 | csbid 3690 | . . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵 | |
39 | 37, 38 | eqtri 2793 | . . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = 𝐵 |
40 | 39 | eleq1i 2841 | . . . . . . . . . 10 ⊢ (⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆) |
41 | 32, 35, 40 | 3bitri 286 | . . . . . . . . 9 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆) |
42 | 25, 31, 41 | 3imtr3i 280 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐵 ∈ 𝑆) |
43 | 11, 24, 42 | syl2anc 565 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝐵 ∈ 𝑆) |
44 | 43 | ralrimiva 3115 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) |
45 | sigaclfu2 30524 | . . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) | |
46 | 9, 44, 45 | syl2anc 565 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) |
47 | difelsiga 30536 | . . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆) | |
48 | 9, 10, 46, 47 | syl3anc 1476 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆) |
49 | 48 | ralrimiva 3115 | . . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆) |
50 | eqimss 3806 | . . . . . 6 ⊢ (𝑁 = ℕ → 𝑁 ⊆ ℕ) | |
51 | fzossnn 12725 | . . . . . . 7 ⊢ (1..^𝐼) ⊆ ℕ | |
52 | sseq1 3775 | . . . . . . 7 ⊢ (𝑁 = (1..^𝐼) → (𝑁 ⊆ ℕ ↔ (1..^𝐼) ⊆ ℕ)) | |
53 | 51, 52 | mpbiri 248 | . . . . . 6 ⊢ (𝑁 = (1..^𝐼) → 𝑁 ⊆ ℕ) |
54 | 50, 53 | jaoi 836 | . . . . 5 ⊢ ((𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)) → 𝑁 ⊆ ℕ) |
55 | 3, 54 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ) |
56 | nnct 12988 | . . . 4 ⊢ ℕ ≼ ω | |
57 | ssct 8197 | . . . 4 ⊢ ((𝑁 ⊆ ℕ ∧ ℕ ≼ ω) → 𝑁 ≼ ω) | |
58 | 55, 56, 57 | sylancl 566 | . . 3 ⊢ (𝜑 → 𝑁 ≼ ω) |
59 | 1, 2, 3 | iundisj2cnt 29898 | . . 3 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) |
60 | measvuni 30617 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆 ∧ (𝑁 ≼ ω ∧ Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) | |
61 | 6, 49, 58, 59, 60 | syl112anc 1480 | . 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) |
62 | 5, 61 | eqtrd 2805 | 1 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = Σ*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∨ wo 826 = wceq 1631 [wsb 2049 ∈ wcel 2145 Ⅎwnfc 2900 ∀wral 3061 Vcvv 3351 [wsbc 3587 ⦋csb 3682 ∖ cdif 3720 ⊆ wss 3723 ∪ cuni 4574 ∪ ciun 4654 Disj wdisj 4754 class class class wbr 4786 ran crn 5250 ‘cfv 6031 (class class class)co 6793 ωcom 7212 ≼ cdom 8107 1c1 10139 ℕcn 11222 ℤ≥cuz 11888 ..^cfzo 12673 Σ*cesum 30429 sigAlgebracsiga 30510 measurescmeas 30598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-ac2 9487 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-addf 10217 ax-mulf 10218 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-disj 4755 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-supp 7447 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-ixp 8063 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-fsupp 8432 df-fi 8473 df-sup 8504 df-inf 8505 df-oi 8571 df-card 8965 df-acn 8968 df-ac 9139 df-cda 9192 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ioo 12384 df-ioc 12385 df-ico 12386 df-icc 12387 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-fac 13265 df-bc 13294 df-hash 13322 df-shft 14015 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-limsup 14410 df-clim 14427 df-rlim 14428 df-sum 14625 df-ef 15004 df-sin 15006 df-cos 15007 df-pi 15009 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-starv 16164 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-unif 16173 df-hom 16174 df-cco 16175 df-rest 16291 df-topn 16292 df-0g 16310 df-gsum 16311 df-topgen 16312 df-pt 16313 df-prds 16316 df-ordt 16369 df-xrs 16370 df-qtop 16375 df-imas 16376 df-xps 16378 df-mre 16454 df-mrc 16455 df-acs 16457 df-ps 17408 df-tsr 17409 df-plusf 17449 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-mhm 17543 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-mulg 17749 df-subg 17799 df-cntz 17957 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-cring 18758 df-subrg 18988 df-abv 19027 df-lmod 19075 df-scaf 19076 df-sra 19387 df-rgmod 19388 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-cnfld 19962 df-top 20919 df-topon 20936 df-topsp 20958 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-nei 21123 df-lp 21161 df-perf 21162 df-cn 21252 df-cnp 21253 df-haus 21340 df-tx 21586 df-hmeo 21779 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-tmd 22096 df-tgp 22097 df-tsms 22150 df-trg 22183 df-xms 22345 df-ms 22346 df-tms 22347 df-nm 22607 df-ngp 22608 df-nrg 22610 df-nlm 22611 df-ii 22900 df-cncf 22901 df-limc 23850 df-dv 23851 df-log 24524 df-esum 30430 df-siga 30511 df-meas 30599 |
This theorem is referenced by: measiun 30621 meascnbl 30622 |
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