measiuns - Mathbox for Thierry Arnoux

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Theorem measiuns 34197

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 34198 and meascnbl 34199. (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 32805	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	4	fveq2d 6910	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
6		measiuns.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
7		measbase 34177	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑆 ∈ ∪ ran sigAlgebra)
10		measiuns.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆)
11		simpll 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
12		fzossnn 13747	. . . . . . . . . . 11 ⊢ (1..^𝑛) ⊆ ℕ
13		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → 𝑁 = ℕ)
14	12, 13	sseqtrrid 4048	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → (1..^𝑛) ⊆ 𝑁)
15		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ 𝑁)
16		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑁 = (1..^𝐼))
17	15, 16	eleqtrd 2840	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ (1..^𝐼))
18		elfzouz2 13710	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1..^𝐼) → 𝐼 ∈ (ℤ_≥‘𝑛))
19		fzoss2 13723	. . . . . . . . . . . 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	sseqtrrd 4036	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → (1..^𝑛) ⊆ 𝑁)
22	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)))
23	14, 21, 22	mpjaodan 960	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (1..^𝑛) ⊆ 𝑁)
24	23	sselda 3994	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ 𝑁)
25	10	sbimi 2071	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) → [𝑘 / 𝑛]𝐴 ∈ 𝑆)
26		sban 2077	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁))
27		sbv 2085	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
28		clelsb1 2865	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝑛 ∈ 𝑁 ↔ 𝑘 ∈ 𝑁)
29	27, 28	anbi12i 628	. . . . . . . . . 10 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
30	26, 29	bitri 275	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
31		sbsbc 3794	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ [𝑘 / 𝑛]𝐴 ∈ 𝑆)
32		sbcel1g 4421	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆))
33	32	elv 3482	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆)
34		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐴
35	34, 1, 2	cbvcsbw 3917	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐵
36		csbid 3920	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
37	35, 36	eqtri 2762	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = 𝐵
38	37	eleq1i 2829	. . . . . . . . . 10 ⊢ (⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
39	31, 33, 38	3bitri 297	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
40	25, 30, 39	3imtr3i 291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐵 ∈ 𝑆)
41	11, 24, 40	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝐵 ∈ 𝑆)
42	41	ralrimiva 3143	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
43		sigaclfu2 34101	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
44	9, 42, 43	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
45		difelsiga 34113	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
46	9, 10, 44, 45	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
47	46	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
48		eqimss 4053	. . . . . 6 ⊢ (𝑁 = ℕ → 𝑁 ⊆ ℕ)
49		fzossnn 13747	. . . . . . 7 ⊢ (1..^𝐼) ⊆ ℕ
50		sseq1 4020	. . . . . . 7 ⊢ (𝑁 = (1..^𝐼) → (𝑁 ⊆ ℕ ↔ (1..^𝐼) ⊆ ℕ))
51	49, 50	mpbiri 258	. . . . . 6 ⊢ (𝑁 = (1..^𝐼) → 𝑁 ⊆ ℕ)
52	48, 51	jaoi 857	. . . . 5 ⊢ ((𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)) → 𝑁 ⊆ ℕ)
53	3, 52	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ)
54		nnct 14018	. . . 4 ⊢ ℕ ≼ ω
55		ssct 9089	. . . 4 ⊢ ((𝑁 ⊆ ℕ ∧ ℕ ≼ ω) → 𝑁 ≼ ω)
56	53, 54, 55	sylancl 586	. . 3 ⊢ (𝜑 → 𝑁 ≼ ω)
57	1, 2, 3	iundisj2cnt 32806	. . 3 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
58		measvuni 34194	. . 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 1373	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ^*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
60	5, 59	eqtrd 2774	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 206 ∧ wa 395 ∨ wo 847 = wceq 1536 [wsb 2061 ∈ wcel 2105 Ⅎwnfc 2887 ∀wral 3058 Vcvv 3477 [wsbc 3790 ⦋csb 3907 ∖ cdif 3959 ⊆ wss 3962 ∪ cuni 4911 ∪ ciun 4995 Disj wdisj 5114 class class class wbr 5147 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ωcom 7886 ≼ cdom 8981 1c1 11153 ℕcn 12263 ℤ_≥cuz 12875 ..^cfzo 13690 Σ^*cesum 34007 sigAlgebracsiga 34088 measurescmeas 34175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-ac2 10500 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-disj 5115 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-pm 8867 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-dju 9938 df-card 9976 df-acn 9979 df-ac 10153 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-ordt 17547 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-ps 18623 df-tsr 18624 df-plusf 18664 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-submnd 18809 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-subrng 20562 df-subrg 20586 df-abv 20826 df-lmod 20876 df-scaf 20877 df-sra 21189 df-rgmod 21190 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-perf 23160 df-cn 23250 df-cnp 23251 df-haus 23338 df-tx 23585 df-hmeo 23778 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-tmd 24095 df-tgp 24096 df-tsms 24150 df-trg 24183 df-xms 24345 df-ms 24346 df-tms 24347 df-nm 24610 df-ngp 24611 df-nrg 24613 df-nlm 24614 df-ii 24916 df-cncf 24917 df-limc 25915 df-dv 25916 df-log 26612 df-esum 34008 df-siga 34089 df-meas 34176

This theorem is referenced by: measiun 34198 meascnbl 34199

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