fiunelcarsg - Mathbox for Thierry Arnoux

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Theorem fiunelcarsg 34150

Description: The Caratheodory measurable sets are closed under finite union. (Contributed by Thierry Arnoux, 23-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))
fiunelcarsg.1	⊢ (𝜑 → 𝐴 ∈ Fin)
fiunelcarsg.2	⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))

**Assertion**
Ref	Expression
fiunelcarsg	⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))

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

Proof of Theorem fiunelcarsg Dummy variables 𝑎 𝑒 𝑏 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4924	. . 3 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2		eqidd 2727	. . 3 ⊢ (𝑎 = ∅ → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
3	1, 2	eleq12d 2820	. 2 ⊢ (𝑎 = ∅ → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ ∅ ∈ (toCaraSiga‘𝑀)))
4		unieq 4924	. . 3 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
5		eqidd 2727	. . 3 ⊢ (𝑎 = 𝑏 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
6	4, 5	eleq12d 2820	. 2 ⊢ (𝑎 = 𝑏 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)))
7		unieq 4924	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
8		eqidd 2727	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
9	7, 8	eleq12d 2820	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀)))
10		unieq 4924	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
11		eqidd 2727	. . 3 ⊢ (𝑎 = 𝐴 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
12	10, 11	eleq12d 2820	. 2 ⊢ (𝑎 = 𝐴 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝐴 ∈ (toCaraSiga‘𝑀)))
13		uni0 4943	. . . 4 ⊢ ∪ ∅ = ∅
14		difid 4375	. . . 4 ⊢ (𝑂 ∖ 𝑂) = ∅
15	13, 14	eqtr4i 2757	. . 3 ⊢ ∪ ∅ = (𝑂 ∖ 𝑂)
16		carsgval.1	. . . 4 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
17		carsgval.2	. . . 4 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
18		carsgsiga.1	. . . . 5 ⊢ (𝜑 → (𝑀‘∅) = 0)
19	16, 17, 18	baselcarsg 34140	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
20	16, 17, 19	difelcarsg 34144	. . 3 ⊢ (𝜑 → (𝑂 ∖ 𝑂) ∈ (toCaraSiga‘𝑀))
21	15, 20	eqeltrid 2830	. 2 ⊢ (𝜑 → ∪ ∅ ∈ (toCaraSiga‘𝑀))
22		uniun 4938	. . . . 5 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
23		unisnv 4935	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
24	23	uneq2i 4160	. . . . 5 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
25	22, 24	eqtri 2754	. . . 4 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
26	16	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑂 ∈ 𝑉)
27	17	ad2antrr 724	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
28		simpr 483	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
29		simpll 765	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝜑)
30		carsgsiga.2	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂) → (𝑀‘∪ 𝑥) ≤ Σ^*𝑦 ∈ 𝑥(𝑀‘𝑦))
31	16, 17, 18, 30	carsgsigalem 34149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +_𝑒 (𝑀‘𝑓)))
32	29, 31	syl3an1 1160	. . . . 5 ⊢ ((((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +_𝑒 (𝑀‘𝑓)))
33		fiunelcarsg.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (toCaraSiga‘𝑀))
34	33	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝐴 ⊆ (toCaraSiga‘𝑀))
35		simplrr 776	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ (𝐴 ∖ 𝑏))
36	35	eldifad 3959	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ 𝐴)
37	34, 36	sseldd 3980	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ (toCaraSiga‘𝑀))
38	26, 27, 28, 32, 37	unelcarsg 34146	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → (∪ 𝑏 ∪ 𝑥) ∈ (toCaraSiga‘𝑀))
39	25, 38	eqeltrid 2830	. . 3 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀))
40	39	ex 411	. 2 ⊢ ((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) → (∪ 𝑏 ∈ (toCaraSiga‘𝑀) → ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀)))
41		fiunelcarsg.1	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
42	3, 6, 9, 12, 21, 40, 41	findcard2d 9204	1 ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∖ cdif 3944 ∪ cun 3945 ⊆ wss 3947 ∅c0 4325 𝒫 cpw 4607 {csn 4633 ∪ cuni 4913 class class class wbr 5153 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ωcom 7876 ≼ cdom 8972 Fincfn 8974 0cc0 11158 +∞cpnf 11295 ≤ cle 11299 +_𝑒 cxad 13144 [,]cicc 13381 Σ^*cesum 33860 toCaraSigaccarsg 34135

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9684 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 ax-mulf 11238

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13539 df-fzo 13682 df-fl 13812 df-mod 13890 df-seq 14022 df-exp 14082 df-fac 14291 df-bc 14320 df-hash 14348 df-shft 15072 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-limsup 15473 df-clim 15490 df-rlim 15491 df-sum 15691 df-ef 16069 df-sin 16071 df-cos 16072 df-pi 16074 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-ordt 17516 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-ps 18591 df-tsr 18592 df-plusf 18632 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-mulg 19062 df-subg 19117 df-cntz 19311 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-cring 20219 df-subrng 20528 df-subrg 20553 df-abv 20788 df-lmod 20838 df-scaf 20839 df-sra 21151 df-rgmod 21152 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-tmd 24067 df-tgp 24068 df-tsms 24122 df-trg 24155 df-xms 24317 df-ms 24318 df-tms 24319 df-nm 24582 df-ngp 24583 df-nrg 24585 df-nlm 24586 df-ii 24888 df-cncf 24889 df-limc 25886 df-dv 25887 df-log 26583 df-esum 33861 df-carsg 34136

This theorem is referenced by: carsgclctunlem1 34151 carsgclctunlem2 34153 carsgclctunlem3 34154

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