measiuns - Mathbox for Thierry Arnoux

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Theorem measiuns 34516

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 34517 and meascnbl 34518. (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 33002	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑁 𝐴 = ∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
5	4	fveq2d 6873	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 𝐴) = (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
6		measiuns.3	. . 3 ⊢ (𝜑 → 𝑀 ∈ (measures‘𝑆))
7		measbase 34496	. . . . . . 7 ⊢ (𝑀 ∈ (measures‘𝑆) → 𝑆 ∈ ∪ ran sigAlgebra)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
9	8	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝑆 ∈ ∪ ran sigAlgebra)
10		measiuns.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → 𝐴 ∈ 𝑆)
11		simpll 776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝜑)
12		fzossnn 13719	. . . . . . . . . . 11 ⊢ (1..^𝑛) ⊆ ℕ
13		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → 𝑁 = ℕ)
14	12, 13	sseqtrrid 3981	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = ℕ) → (1..^𝑛) ⊆ 𝑁)
15		simplr 778	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ 𝑁)
16		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑁 = (1..^𝐼))
17	15, 16	eleqtrd 2866	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → 𝑛 ∈ (1..^𝐼))
18		elfzouz2 13682	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1..^𝐼) → 𝐼 ∈ (ℤ_≥‘𝑛))
19		fzoss2 13695	. . . . . . . . . . . 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 3975	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑁 = (1..^𝐼)) → (1..^𝑛) ⊆ 𝑁)
22	3	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)))
23	14, 21, 22	mpjaodan 971	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (1..^𝑛) ⊆ 𝑁)
24	23	sselda 3938	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝑘 ∈ 𝑁)
25	10	sbimi 2109	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) → [𝑘 / 𝑛]𝐴 ∈ 𝑆)
26		sban 2115	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ ([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁))
27		sbv 2123	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝜑 ↔ 𝜑)
28		clelsb1 2891	. . . . . . . . . . 11 ⊢ ([𝑘 / 𝑛]𝑛 ∈ 𝑁 ↔ 𝑘 ∈ 𝑁)
29	27, 28	anbi12i 637	. . . . . . . . . 10 ⊢ (([𝑘 / 𝑛]𝜑 ∧ [𝑘 / 𝑛]𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
30	26, 29	bitri 277	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛](𝜑 ∧ 𝑛 ∈ 𝑁) ↔ (𝜑 ∧ 𝑘 ∈ 𝑁))
31		sbsbc 3750	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ [𝑘 / 𝑛]𝐴 ∈ 𝑆)
32		sbcel1g 4372	. . . . . . . . . . 11 ⊢ (𝑘 ∈ V → ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆))
33	32	elv 3461	. . . . . . . . . 10 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ ⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆)
34		nfcv 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐴
35	34, 1, 2	cbvcsbw 3864	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = ⦋𝑘 / 𝑘⦌𝐵
36		csbid 3867	. . . . . . . . . . . 12 ⊢ ⦋𝑘 / 𝑘⦌𝐵 = 𝐵
37	35, 36	eqtri 2787	. . . . . . . . . . 11 ⊢ ⦋𝑘 / 𝑛⦌𝐴 = 𝐵
38	37	eleq1i 2855	. . . . . . . . . 10 ⊢ (⦋𝑘 / 𝑛⦌𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
39	31, 33, 38	3bitri 299	. . . . . . . . 9 ⊢ ([𝑘 / 𝑛]𝐴 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆)
40	25, 30, 39	3imtr3i 293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐵 ∈ 𝑆)
41	11, 24, 40	syl2anc 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑁) ∧ 𝑘 ∈ (1..^𝑛)) → 𝐵 ∈ 𝑆)
42	41	ralrimiva 3156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
43		sigaclfu2 34420	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ ∀𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
44	9, 42, 43	syl2anc 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆)
45		difelsiga 34432	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ ∪ 𝑘 ∈ (1..^𝑛)𝐵 ∈ 𝑆) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
46	9, 10, 44, 45	syl3anc 1392	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑁) → (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
47	46	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵) ∈ 𝑆)
48		eqimss 3996	. . . . . 6 ⊢ (𝑁 = ℕ → 𝑁 ⊆ ℕ)
49		fzossnn 13719	. . . . . . 7 ⊢ (1..^𝐼) ⊆ ℕ
50		sseq1 3963	. . . . . . 7 ⊢ (𝑁 = (1..^𝐼) → (𝑁 ⊆ ℕ ↔ (1..^𝐼) ⊆ ℕ))
51	49, 50	mpbiri 260	. . . . . 6 ⊢ (𝑁 = (1..^𝐼) → 𝑁 ⊆ ℕ)
52	48, 51	jaoi 868	. . . . 5 ⊢ ((𝑁 = ℕ ∨ 𝑁 = (1..^𝐼)) → 𝑁 ⊆ ℕ)
53	3, 52	syl 17	. . . 4 ⊢ (𝜑 → 𝑁 ⊆ ℕ)
54		nnct 13996	. . . 4 ⊢ ℕ ≼ ω
55		ssct 9032	. . . 4 ⊢ ((𝑁 ⊆ ℕ ∧ ℕ ≼ ω) → 𝑁 ≼ ω)
56	53, 54, 55	sylancl 595	. . 3 ⊢ (𝜑 → 𝑁 ≼ ω)
57	1, 2, 3	iundisj2cnt 33003	. . 3 ⊢ (𝜑 → Disj 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵))
58		measvuni 34513	. . 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 1395	. 2 ⊢ (𝜑 → (𝑀‘∪ 𝑛 ∈ 𝑁 (𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)) = Σ^*𝑛 ∈ 𝑁(𝑀‘(𝐴 ∖ ∪ 𝑘 ∈ (1..^𝑛)𝐵)))
60	5, 59	eqtrd 2799	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 208 ∧ wa 399 ∨ wo 858 = wceq 1562 [wsb 2092 ∈ wcel 2144 Ⅎwnfc 2911 ∀wral 3078 Vcvv 3456 [wsbc 3746 ⦋csb 3854 ∖ cdif 3903 ⊆ wss 3906 ∪ cuni 4867 ∪ ciun 4951 Disj wdisj 5069 class class class wbr 5102 ran crn 5650 ‘cfv 6523 (class class class)co 7398 ωcom 7848 ≼ cdom 8927 1c1 11076 ℕcn 12212 ℤ_≥cuz 12841 ..^cfzo 13661 Σ^*cesum 34326 sigAlgebracsiga 34407 measurescmeas 34494

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-ac2 10422 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9861 df-card 9899 df-acn 9902 df-ac 10074 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-mod 13882 df-seq 14017 df-exp 14077 df-fac 14289 df-bc 14318 df-hash 14346 df-shft 15082 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-limsup 15500 df-clim 15517 df-rlim 15518 df-sum 15716 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-ordt 17533 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-ps 18600 df-tsr 18601 df-plusf 18675 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-mhm 18819 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-subg 19167 df-cntz 19359 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-ring 20287 df-cring 20288 df-subrng 20598 df-subrg 20622 df-abv 20860 df-lmod 20931 df-scaf 20932 df-sra 21242 df-rgmod 21243 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-tmd 24134 df-tgp 24135 df-tsms 24189 df-trg 24222 df-xms 24382 df-ms 24383 df-tms 24384 df-nm 24644 df-ngp 24645 df-nrg 24647 df-nlm 24648 df-ii 24941 df-cncf 24942 df-limc 25930 df-dv 25931 df-log 26623 df-esum 34327 df-siga 34408 df-meas 34495

This theorem is referenced by: measiun 34517 meascnbl 34518

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