carsgsiga - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgsiga		Structured version Visualization version GIF version

Theorem carsgsiga 34335

Description: The Caratheodory measurable sets constructed from outer measures form a Sigma-algebra. Statement (iii) of Theorem 1.11.4 of [Bogachev] p. 42. (Contributed by Thierry Arnoux, 17-May-2020.)

**Hypotheses**
Ref	Expression
carsgval.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
carsgval.2	⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgsiga.1	⊢ (𝜑 → (𝑀‘∅) = 0)
carsgsiga.2	⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
carsgsiga.3	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))

**Assertion**
Ref	Expression
carsgsiga	⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))

Distinct variable groups: 𝑥,𝑦,𝑀 𝑥,𝑂,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝑉(𝑥,𝑦)

Proof of Theorem carsgsiga Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		carsgval.1	. . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
2		carsgval.2	. . . 4 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
3	1, 2	carsgcl 34317	. . 3 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
4		carsgsiga.1	. . . . 5 ⊢ (𝜑 → (𝑀‘∅) = 0)
5	1, 2, 4	baselcarsg 34319	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
6	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑂 ∈ 𝑉)
7	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
8		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑔 ∈ (toCaraSiga‘𝑀))
9	6, 7, 8	difelcarsg 34323	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → (𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀))
10	9	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀))
11	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑂 ∈ 𝑉)
12	2	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
13	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → (𝑀‘∅) = 0)
14		carsgsiga.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
15	14	3adant1r 1178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
16	15	3adant1r 1178	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
17		carsgsiga.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
18	17	3adant1r 1178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
19	18	3adant1r 1178	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
20		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑔 ≼ ω)
21		elpwi 4554	. . . . . . . 8 ⊢ (𝑔 ∈ 𝒫 (toCaraSiga‘𝑀) → 𝑔 ⊆ (toCaraSiga‘𝑀))
22	21	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑔 ⊆ (toCaraSiga‘𝑀))
23	11, 12, 13, 16, 19, 20, 22	carsgclctun 34334	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → ∪ 𝑔 ∈ (toCaraSiga‘𝑀))
24	23	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) → (𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))
25	24	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))
26	5, 10, 25	3jca 1128	. . 3 ⊢ (𝜑 → (𝑂 ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀))))
27	3, 26	jca 511	. 2 ⊢ (𝜑 → ((toCaraSiga‘𝑀) ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))))
28		fvex 6835	. . 3 ⊢ (toCaraSiga‘𝑀) ∈ V
29		issiga 34125	. . 3 ⊢ ((toCaraSiga‘𝑀) ∈ V → ((toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂) ↔ ((toCaraSiga‘𝑀) ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀))))))
30	28, 29	ax-mp 5	. 2 ⊢ ((toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂) ↔ ((toCaraSiga‘𝑀) ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀) ∧ ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))))
31	27, 30	sylibr 234	1 ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 Vcvv 3436 ∖ cdif 3894 ⊆ wss 3897 ∅c0 4280 𝒫 cpw 4547 ∪ cuni 4856 class class class wbr 5089 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ωcom 7796 ≼ cdom 8867 0cc0 11006 +∞cpnf 11143 ≤ cle 11147 [,]cicc 13248 Σ^*cesum 34040 sigAlgebracsiga 34121 toCaraSigaccarsg 34314

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-ac2 10354 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-disj 5057 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9794 df-card 9832 df-acn 9835 df-ac 10007 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-ordt 17405 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-ps 18472 df-tsr 18473 df-plusf 18547 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-subrng 20461 df-subrg 20485 df-abv 20724 df-lmod 20795 df-scaf 20796 df-sra 21107 df-rgmod 21108 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-tmd 23987 df-tgp 23988 df-tsms 24042 df-trg 24075 df-xms 24235 df-ms 24236 df-tms 24237 df-nm 24497 df-ngp 24498 df-nrg 24500 df-nlm 24501 df-ii 24797 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492 df-esum 34041 df-siga 34122 df-carsg 34315

This theorem is referenced by: omsmeas 34336

W3C validator