Nearby theorems
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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > measun | Structured version Visualization version GIF version | ||
| Description: The measure the union of two disjoint sets is the sum of their measures. (Contributed by Thierry Arnoux, 10-Mar-2017.) |
| Ref | Expression |
|---|---|
| measun | ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1081 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑀 ∈ (measures‘𝑆)) | |
| 2 | measbase 30388 | . . . . 5 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra) | |
| 3 | 2 | 3ad2ant1 1102 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑆 ∈ ∪ ran sigAlgebra) |
| 4 | simp2l 1107 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑆) | |
| 5 | simp2r 1108 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑆) | |
| 6 | unelsiga 30325 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆) | |
| 7 | 3, 4, 5, 6 | syl3anc 1366 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ 𝑆) |
| 8 | ssun2 3810 | . . . 4 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵) | |
| 9 | 8 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵)) |
| 10 | measxun2 30401 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐵) +𝑒 (𝑀‘((𝐴 ∪ 𝐵) ∖ 𝐵)))) | |
| 11 | 1, 7, 5, 9, 10 | syl121anc 1371 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐵) +𝑒 (𝑀‘((𝐴 ∪ 𝐵) ∖ 𝐵)))) |
| 12 | difun2 4081 | . . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∖ 𝐵) = (𝐴 ∖ 𝐵) | |
| 13 | inundif 4079 | . . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 | |
| 14 | uneq1 3793 | . . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = (∅ ∪ (𝐴 ∖ 𝐵))) | |
| 15 | uncom 3790 | . . . . . . . . 9 ⊢ (∅ ∪ (𝐴 ∖ 𝐵)) = ((𝐴 ∖ 𝐵) ∪ ∅) | |
| 16 | un0 4000 | . . . . . . . . 9 ⊢ ((𝐴 ∖ 𝐵) ∪ ∅) = (𝐴 ∖ 𝐵) | |
| 17 | 15, 16 | eqtri 2673 | . . . . . . . 8 ⊢ (∅ ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵) |
| 18 | 14, 17 | syl6eq 2701 | . . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = (𝐴 ∖ 𝐵)) |
| 19 | 13, 18 | syl5reqr 2700 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
| 20 | 12, 19 | syl5eq 2697 | . . . . 5 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 ∪ 𝐵) ∖ 𝐵) = 𝐴) |
| 21 | 20 | fveq2d 6233 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝑀‘((𝐴 ∪ 𝐵) ∖ 𝐵)) = (𝑀‘𝐴)) |
| 22 | 21 | oveq2d 6706 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑀‘𝐵) +𝑒 (𝑀‘((𝐴 ∪ 𝐵) ∖ 𝐵))) = ((𝑀‘𝐵) +𝑒 (𝑀‘𝐴))) |
| 23 | 22 | 3ad2ant3 1104 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑀‘𝐵) +𝑒 (𝑀‘((𝐴 ∪ 𝐵) ∖ 𝐵))) = ((𝑀‘𝐵) +𝑒 (𝑀‘𝐴))) |
| 24 | iccssxr 12294 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 25 | measvxrge0 30396 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ (0[,]+∞)) | |
| 26 | 24, 25 | sseldi 3634 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝑀‘𝐵) ∈ ℝ*) |
| 27 | 1, 5, 26 | syl2anc 694 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘𝐵) ∈ ℝ*) |
| 28 | measvxrge0 30396 | . . . . 5 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ (0[,]+∞)) | |
| 29 | 24, 28 | sseldi 3634 | . . . 4 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ 𝐴 ∈ 𝑆) → (𝑀‘𝐴) ∈ ℝ*) |
| 30 | 1, 4, 29 | syl2anc 694 | . . 3 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘𝐴) ∈ ℝ*) |
| 31 | xaddcom 12109 | . . 3 ⊢ (((𝑀‘𝐵) ∈ ℝ* ∧ (𝑀‘𝐴) ∈ ℝ*) → ((𝑀‘𝐵) +𝑒 (𝑀‘𝐴)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) | |
| 32 | 27, 30, 31 | syl2anc 694 | . 2 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑀‘𝐵) +𝑒 (𝑀‘𝐴)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
| 33 | 11, 23, 32 | 3eqtrd 2689 | 1 ⊢ ((𝑀 ∈ (measures‘𝑆) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑀‘(𝐴 ∪ 𝐵)) = ((𝑀‘𝐴) +𝑒 (𝑀‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ∖ cdif 3604 ∪ cun 3605 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 ∪ cuni 4468 ran crn 5144 ‘cfv 5926 (class class class)co 6690 0cc0 9974 +∞cpnf 10109 ℝ*cxr 10111 +𝑒 cxad 11982 [,]cicc 12216 sigAlgebracsiga 30298 measurescmeas 30386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-ac2 9323 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 ax-addf 10053 ax-mulf 10054 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-supp 7341 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-pm 7902 df-ixp 7951 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-fsupp 8317 df-fi 8358 df-sup 8389 df-inf 8390 df-oi 8456 df-card 8803 df-acn 8806 df-ac 8977 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-q 11827 df-rp 11871 df-xneg 11984 df-xadd 11985 df-xmul 11986 df-ioo 12217 df-ioc 12218 df-ico 12219 df-icc 12220 df-fz 12365 df-fzo 12505 df-fl 12633 df-mod 12709 df-seq 12842 df-exp 12901 df-fac 13101 df-bc 13130 df-hash 13158 df-shft 13851 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-limsup 14246 df-clim 14263 df-rlim 14264 df-sum 14461 df-ef 14842 df-sin 14844 df-cos 14845 df-pi 14847 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-starv 16003 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-unif 16012 df-hom 16013 df-cco 16014 df-rest 16130 df-topn 16131 df-0g 16149 df-gsum 16150 df-topgen 16151 df-pt 16152 df-prds 16155 df-ordt 16208 df-xrs 16209 df-qtop 16214 df-imas 16215 df-xps 16217 df-mre 16293 df-mrc 16294 df-acs 16296 df-ps 17247 df-tsr 17248 df-plusf 17288 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-mhm 17382 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mulg 17588 df-subg 17638 df-cntz 17796 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-cring 18596 df-subrg 18826 df-abv 18865 df-lmod 18913 df-scaf 18914 df-sra 19220 df-rgmod 19221 df-psmet 19786 df-xmet 19787 df-met 19788 df-bl 19789 df-mopn 19790 df-fbas 19791 df-fg 19792 df-cnfld 19795 df-top 20747 df-topon 20764 df-topsp 20785 df-bases 20798 df-cld 20871 df-ntr 20872 df-cls 20873 df-nei 20950 df-lp 20988 df-perf 20989 df-cn 21079 df-cnp 21080 df-haus 21167 df-tx 21413 df-hmeo 21606 df-fil 21697 df-fm 21789 df-flim 21790 df-flf 21791 df-tmd 21923 df-tgp 21924 df-tsms 21977 df-trg 22010 df-xms 22172 df-ms 22173 df-tms 22174 df-nm 22434 df-ngp 22435 df-nrg 22437 df-nlm 22438 df-ii 22727 df-cncf 22728 df-limc 23675 df-dv 23676 df-log 24348 df-esum 30218 df-siga 30299 df-meas 30387 |
| This theorem is referenced by: measvuni 30405 measunl 30407 |
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