measiuns - Mathbox for Thierry Arnoux

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Theorem measiuns 31154

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 31155 and meascnbl 31156. (Contributed by Thierry Arnoux, 22-Jan-2017.) (Proof shortened by Thierry Arnoux, 7-Feb-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
measiuns	⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = Σ^*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))

Distinct variable groups: 𝐴,𝑘 𝑘,𝑛,𝐼 𝑛,𝑀 𝑘,𝑁,𝑛 𝑆,𝑘,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑘,𝑛) 𝑀(𝑘)

**Proof of Theorem measiuns**
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 30295	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	4	fveq2d 6501	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
6		measiuns.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
7		measbase 31134	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
9	8	adantr 473	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑆 ∈ ∪ ran sigAlgebra)
10		measiuns.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆)
11		simpll 755	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
12		fzossnn 12900	. . . . . . . . . . 11 ⊢ (1..^𝑛) ⊆ ℕ
13		simpr 477	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → 𝑁 = ℕ)
14	12, 13	syl5sseqr 3905	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → (1..^𝑛) ⊆ 𝑁)
15		simplr 757	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ 𝑁)
16		simpr 477	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑁 = (1..^𝐼))
17	15, 16	eleqtrd 2863	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ (1..^𝐼))
18		elfzouz2 12867	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1..^𝐼) → 𝐼 ∈ (ℤ_≥‘𝑛))
19		fzoss2 12879	. . . . . . . . . . . 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 3893	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → (1..^𝑛) ⊆ 𝑁)
22	3	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)))
23	14, 21, 22	mpjaodan 942	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (1..^𝑛) ⊆ 𝑁)
24	23	sselda 3853	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ 𝑁)
25	10	sbimi 2026	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) → [𝑘 / 𝑛]𝐴 ∈ 𝑆)
26		sban 2032	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁))
27		sbv 2041	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
28		clelsb3 2888	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝑛 ∈ 𝑁 ↔ 𝑘 ∈ 𝑁)
29	27, 28	anbi12i 618	. . . . . . . . . 10 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
30	26, 29	bitri 267	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
31		sbsbc 3680	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ [𝑘 / 𝑛]𝐴 ∈ 𝑆)
32		sbcel1g 4246	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆))
33	32	elv 3415	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆)
34		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐴
35	34, 1, 2	cbvcsb 3786	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐵
36		csbid 3789	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
37	35, 36	eqtri 2797	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = 𝐵
38	37	eleq1i 2851	. . . . . . . . . 10 ⊢ (⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
39	31, 33, 38	3bitri 289	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
40	25, 30, 39	3imtr3i 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐵 ∈ 𝑆)
41	11, 24, 40	syl2anc 576	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝐵 ∈ 𝑆)
42	41	ralrimiva 3127	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
43		sigaclfu2 31058	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
44	9, 42, 43	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
45		difelsiga 31070	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
46	9, 10, 44, 45	syl3anc 1352	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
47	46	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
48		eqimss 3908	. . . . . 6 ⊢ (𝑁 = ℕ → 𝑁 ⊆ ℕ)
49		fzossnn 12900	. . . . . . 7 ⊢ (1..^𝐼) ⊆ ℕ
50		sseq1 3877	. . . . . . 7 ⊢ (𝑁 = (1..^𝐼) → (𝑁 ⊆ ℕ ↔ (1..^𝐼) ⊆ ℕ))
51	49, 50	mpbiri 250	. . . . . 6 ⊢ (𝑁 = (1..^𝐼) → 𝑁 ⊆ ℕ)
52	48, 51	jaoi 844	. . . . 5 ⊢ ((𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)) → 𝑁 ⊆ ℕ)
53	3, 52	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ)
54		nnct 13163	. . . 4 ⊢ ℕ ≼ ω
55		ssct 8393	. . . 4 ⊢ ((𝑁 ⊆ ℕ ∧ ℕ ≼ ω) → 𝑁 ≼ ω)
56	53, 54, 55	sylancl 578	. . 3 ⊢ (𝜑 → 𝑁 ≼ ω)
57	1, 2, 3	iundisj2cnt 30296	. . 3 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
58		measvuni 31151	. . 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)))
59	6, 47, 56, 57, 58	syl112anc 1355	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ^*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
60	5, 59	eqtrd 2809	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 198 ∧ wa 387 ∨ wo 834 = wceq 1508 [wsb 2016 ∈ wcel 2051 Ⅎwnfc 2911 ∀wral 3083 Vcvv 3410 [wsbc 3676 ⦋csb 3781 ∖ cdif 3821 ⊆ wss 3824 ∪ cuni 4709 ∪ ciun 4789 Disj wdisj 4894 class class class wbr 4926 ran crn 5405 ‘cfv 6186 (class class class)co 6975 ωcom 7395 ≼ cdom 8303 1c1 10335 ℕcn 11438 ℤ_≥cuz 12057 ..^cfzo 12848 Σ^*cesum 30963 sigAlgebracsiga 31044 measurescmeas 31132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-inf2 8897 ax-ac2 9682 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 ax-addf 10413 ax-mulf 10414

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-iin 4792 df-disj 4895 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-se 5364 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-of 7226 df-om 7396 df-1st 7500 df-2nd 7501 df-supp 7633 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-map 8207 df-pm 8208 df-ixp 8259 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-fsupp 8628 df-fi 8669 df-sup 8700 df-inf 8701 df-oi 8768 df-dju 9123 df-card 9161 df-acn 9164 df-ac 9335 df-cda 9387 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-uz 12058 df-q 12162 df-rp 12204 df-xneg 12323 df-xadd 12324 df-xmul 12325 df-ioo 12557 df-ioc 12558 df-ico 12559 df-icc 12560 df-fz 12708 df-fzo 12849 df-fl 12976 df-mod 13052 df-seq 13184 df-exp 13244 df-fac 13448 df-bc 13477 df-hash 13505 df-shft 14286 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-limsup 14688 df-clim 14705 df-rlim 14706 df-sum 14903 df-ef 15280 df-sin 15282 df-cos 15283 df-pi 15285 df-struct 16340 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-ress 16346 df-plusg 16433 df-mulr 16434 df-starv 16435 df-sca 16436 df-vsca 16437 df-ip 16438 df-tset 16439 df-ple 16440 df-ds 16442 df-unif 16443 df-hom 16444 df-cco 16445 df-rest 16551 df-topn 16552 df-0g 16570 df-gsum 16571 df-topgen 16572 df-pt 16573 df-prds 16576 df-ordt 16629 df-xrs 16630 df-qtop 16635 df-imas 16636 df-xps 16638 df-mre 16728 df-mrc 16729 df-acs 16731 df-ps 17681 df-tsr 17682 df-plusf 17722 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-mhm 17816 df-submnd 17817 df-grp 17907 df-minusg 17908 df-sbg 17909 df-mulg 18025 df-subg 18073 df-cntz 18231 df-cmn 18681 df-abl 18682 df-mgp 18976 df-ur 18988 df-ring 19035 df-cring 19036 df-subrg 19269 df-abv 19323 df-lmod 19371 df-scaf 19372 df-sra 19679 df-rgmod 19680 df-psmet 20255 df-xmet 20256 df-met 20257 df-bl 20258 df-mopn 20259 df-fbas 20260 df-fg 20261 df-cnfld 20264 df-top 21222 df-topon 21239 df-topsp 21261 df-bases 21274 df-cld 21347 df-ntr 21348 df-cls 21349 df-nei 21426 df-lp 21464 df-perf 21465 df-cn 21555 df-cnp 21556 df-haus 21643 df-tx 21890 df-hmeo 22083 df-fil 22174 df-fm 22266 df-flim 22267 df-flf 22268 df-tmd 22400 df-tgp 22401 df-tsms 22454 df-trg 22487 df-xms 22649 df-ms 22650 df-tms 22651 df-nm 22911 df-ngp 22912 df-nrg 22914 df-nlm 22915 df-ii 23204 df-cncf 23205 df-limc 24183 df-dv 24184 df-log 24857 df-esum 30964 df-siga 31045 df-meas 31133

This theorem is referenced by: measiun 31155 meascnbl 31156

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