carsgsiga - Mathbox for Thierry Arnoux

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Theorem carsgsiga 31575

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 31557	. . 3 ⊢ (𝜑 → (toCaraSiga‘𝑀) ⊆ 𝒫 𝑂)
4		carsgsiga.1	. . . . 5 ⊢ (𝜑 → (𝑀‘∅) = 0)
5	1, 2, 4	baselcarsg 31559	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
6	1	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑂 ∈ 𝑉)
7	2	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
8		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → 𝑔 ∈ (toCaraSiga‘𝑀))
9	6, 7, 8	difelcarsg 31563	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (toCaraSiga‘𝑀)) → (𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀))
10	9	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ (toCaraSiga‘𝑀)(𝑂 ∖ 𝑔) ∈ (toCaraSiga‘𝑀))
11	1	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑂 ∈ 𝑉)
12	2	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
13	4	ad2antrr 724	. . . . . . 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 1173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
16	15	3adant1r 1173	. . . . . . 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 1173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
19	18	3adant1r 1173	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
20		simpr 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑔 ≼ ω)
21		elpwi 4551	. . . . . . . 8 ⊢ (𝑔 ∈ 𝒫 (toCaraSiga‘𝑀) → 𝑔 ⊆ (toCaraSiga‘𝑀))
22	21	ad2antlr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → 𝑔 ⊆ (toCaraSiga‘𝑀))
23	11, 12, 13, 16, 19, 20, 22	carsgclctun 31574	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) ∧ 𝑔 ≼ ω) → ∪ 𝑔 ∈ (toCaraSiga‘𝑀))
24	23	ex 415	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)) → (𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))
25	24	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑔 ∈ 𝒫 (toCaraSiga‘𝑀)(𝑔 ≼ ω → ∪ 𝑔 ∈ (toCaraSiga‘𝑀)))
26	5, 10, 25	3jca 1124	. . 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 514	. 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 6678	. . 3 ⊢ (toCaraSiga‘𝑀) ∈ V
29		issiga 31366	. . 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 236	1 ⊢ (𝜑 → (toCaraSiga‘𝑀) ∈ (sigAlgebra‘𝑂))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4832 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ωcom 7574 ≼ cdom 8501 0cc0 10531 +∞cpnf 10666 ≤ cle 10670 [,]cicc 12735 Σ^*cesum 31281 sigAlgebracsiga 31362 toCaraSigaccarsg 31554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-ordt 16768 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-ps 17804 df-tsr 17805 df-plusf 17845 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-abv 19582 df-lmod 19630 df-scaf 19631 df-sra 19938 df-rgmod 19939 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-tmd 22674 df-tgp 22675 df-tsms 22729 df-trg 22762 df-xms 22924 df-ms 22925 df-tms 22926 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 df-ii 23479 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-esum 31282 df-siga 31363 df-carsg 31555

This theorem is referenced by: omsmeas 31576

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