measiuns - Mathbox for Thierry Arnoux

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Theorem measiuns 34367

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 34368 and meascnbl 34369. (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 32861	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	4	fveq2d 6836	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
6		measiuns.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
7		measbase 34347	. . . . . . 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 13628	. . . . . . . . . . 11 ⊢ (1..^𝑛) ⊆ ℕ
13		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → 𝑁 = ℕ)
14	12, 13	sseqtrrid 3966	. . . . . . . . . 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 2839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ (1..^𝐼))
18		elfzouz2 13591	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1..^𝐼) → 𝐼 ∈ (ℤ_≥‘𝑛))
19		fzoss2 13604	. . . . . . . . . . . 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 3960	. . . . . . . . . 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 961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (1..^𝑛) ⊆ 𝑁)
24	23	sselda 3922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ 𝑁)
25	10	sbimi 2080	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) → [𝑘 / 𝑛]𝐴 ∈ 𝑆)
26		sban 2086	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁))
27		sbv 2094	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
28		clelsb1 2864	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝑛 ∈ 𝑁 ↔ 𝑘 ∈ 𝑁)
29	27, 28	anbi12i 629	. . . . . . . . . 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 3733	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ [𝑘 / 𝑛]𝐴 ∈ 𝑆)
32		sbcel1g 4357	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆))
33	32	elv 3435	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆)
34		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐴
35	34, 1, 2	cbvcsbw 3848	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐵
36		csbid 3851	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
37	35, 36	eqtri 2760	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = 𝐵
38	37	eleq1i 2828	. . . . . . . . . 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 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝐵 ∈ 𝑆)
42	41	ralrimiva 3130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
43		sigaclfu2 34271	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
44	9, 42, 43	syl2anc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
45		difelsiga 34283	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
46	9, 10, 44, 45	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
47	46	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
48		eqimss 3981	. . . . . 6 ⊢ (𝑁 = ℕ → 𝑁 ⊆ ℕ)
49		fzossnn 13628	. . . . . . 7 ⊢ (1..^𝐼) ⊆ ℕ
50		sseq1 3948	. . . . . . 7 ⊢ (𝑁 = (1..^𝐼) → (𝑁 ⊆ ℕ ↔ (1..^𝐼) ⊆ ℕ))
51	49, 50	mpbiri 258	. . . . . 6 ⊢ (𝑁 = (1..^𝐼) → 𝑁 ⊆ ℕ)
52	48, 51	jaoi 858	. . . . 5 ⊢ ((𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)) → 𝑁 ⊆ ℕ)
53	3, 52	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ)
54		nnct 13905	. . . 4 ⊢ ℕ ≼ ω
55		ssct 8987	. . . 4 ⊢ ((𝑁 ⊆ ℕ ∧ ℕ ≼ ω) → 𝑁 ≼ ω)
56	53, 54, 55	sylancl 587	. . 3 ⊢ (𝜑 → 𝑁 ≼ ω)
57	1, 2, 3	iundisj2cnt 32862	. . 3 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
58		measvuni 34364	. . 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 1377	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ^*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
60	5, 59	eqtrd 2772	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 848 = wceq 1542 [wsb 2068 ∈ wcel 2114 Ⅎwnfc 2884 ∀wral 3052 Vcvv 3430 [wsbc 3729 ⦋csb 3838 ∖ cdif 3887 ⊆ wss 3890 ∪ cuni 4851 ∪ ciun 4934 Disj wdisj 5053 class class class wbr 5086 ran crn 5623 ‘cfv 6490 (class class class)co 7358 ωcom 7808 ≼ cdom 8882 1c1 11028 ℕcn 12146 ℤ_≥cuz 12752 ..^cfzo 13571 Σ^*cesum 34177 sigAlgebracsiga 34258 measurescmeas 34345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-ac2 10374 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-acn 9855 df-ac 10027 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-q 12863 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-ioo 13266 df-ioc 13267 df-ico 13268 df-icc 13269 df-fz 13425 df-fzo 13572 df-fl 13713 df-mod 13791 df-seq 13926 df-exp 13986 df-fac 14198 df-bc 14227 df-hash 14255 df-shft 14991 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-limsup 15395 df-clim 15412 df-rlim 15413 df-sum 15611 df-ef 15991 df-sin 15993 df-cos 15994 df-pi 15996 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-rest 17343 df-topn 17344 df-0g 17362 df-gsum 17363 df-topgen 17364 df-pt 17365 df-prds 17368 df-ordt 17423 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-ps 18490 df-tsr 18491 df-plusf 18565 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-subrng 20481 df-subrg 20505 df-abv 20744 df-lmod 20815 df-scaf 20816 df-sra 21127 df-rgmod 21128 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-tmd 24015 df-tgp 24016 df-tsms 24070 df-trg 24103 df-xms 24263 df-ms 24264 df-tms 24265 df-nm 24525 df-ngp 24526 df-nrg 24528 df-nlm 24529 df-ii 24822 df-cncf 24823 df-limc 25811 df-dv 25812 df-log 26505 df-esum 34178 df-siga 34259 df-meas 34346

This theorem is referenced by: measiun 34368 meascnbl 34369

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