fiunelcarsg - Mathbox for Thierry Arnoux

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Theorem fiunelcarsg 34448

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 4851	. . 3 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2		eqidd 2736	. . 3 ⊢ (𝑎 = ∅ → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
3	1, 2	eleq12d 2829	. 2 ⊢ (𝑎 = ∅ → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ ∅ ∈ (toCaraSiga‘𝑀)))
4		unieq 4851	. . 3 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
5		eqidd 2736	. . 3 ⊢ (𝑎 = 𝑏 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
6	4, 5	eleq12d 2829	. 2 ⊢ (𝑎 = 𝑏 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)))
7		unieq 4851	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
8		eqidd 2736	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
9	7, 8	eleq12d 2829	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀)))
10		unieq 4851	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
11		eqidd 2736	. . 3 ⊢ (𝑎 = 𝐴 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
12	10, 11	eleq12d 2829	. 2 ⊢ (𝑎 = 𝐴 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝐴 ∈ (toCaraSiga‘𝑀)))
13		uni0 4868	. . . 4 ⊢ ∪ ∅ = ∅
14		difid 4306	. . . 4 ⊢ (𝑂 ∖ 𝑂) = ∅
15	13, 14	eqtr4i 2761	. . 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 34438	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
20	16, 17, 19	difelcarsg 34442	. . 3 ⊢ (𝜑 → (𝑂 ∖ 𝑂) ∈ (toCaraSiga‘𝑀))
21	15, 20	eqeltrid 2839	. 2 ⊢ (𝜑 → ∪ ∅ ∈ (toCaraSiga‘𝑀))
22		uniun 4863	. . . . 5 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
23		unisnv 4860	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
24	23	uneq2i 4097	. . . . 5 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
25	22, 24	eqtri 2758	. . . 4 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
26	16	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑂 ∈ 𝑉)
27	17	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑀:𝒫 𝑂⟶(0[,]+∞))
28		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → ∪ 𝑏 ∈ (toCaraSiga‘𝑀))
29		simpll 767	. . . . . 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 34447	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ∧ 𝑓 ∈ 𝒫 𝑂) → (𝑀‘(𝑒 ∪ 𝑓)) ≤ ((𝑀‘𝑒) +_𝑒 (𝑀‘𝑓)))
32	29, 31	syl3an1 1164	. . . . 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 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝐴 ⊆ (toCaraSiga‘𝑀))
35		simplrr 778	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ (𝐴 ∖ 𝑏))
36	35	eldifad 3897	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ 𝐴)
37	34, 36	sseldd 3918	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ (toCaraSiga‘𝑀))
38	26, 27, 28, 32, 37	unelcarsg 34444	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → (∪ 𝑏 ∪ 𝑥) ∈ (toCaraSiga‘𝑀))
39	25, 38	eqeltrid 2839	. . 3 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀))
40	39	ex 412	. 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 9090	1 ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∖ cdif 3882 ∪ cun 3883 ⊆ wss 3885 ∅c0 4263 𝒫 cpw 4531 {csn 4557 ∪ cuni 4840 class class class wbr 5074 ⟶wf 6483 ‘cfv 6487 (class class class)co 7356 ωcom 7806 ≼ cdom 8880 Fincfn 8882 0cc0 11027 +∞cpnf 11165 ≤ cle 11169 +_𝑒 cxad 13050 [,]cicc 13290 Σ^*cesum 34159 toCaraSigaccarsg 34433

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-inf2 9551 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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-ordt 17454 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-ps 18521 df-tsr 18522 df-plusf 18596 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18740 df-submnd 18741 df-grp 18901 df-minusg 18902 df-sbg 18903 df-mulg 19033 df-subg 19088 df-cntz 19281 df-cmn 19746 df-abl 19747 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-cring 20206 df-subrng 20512 df-subrg 20536 df-abv 20775 df-lmod 20846 df-scaf 20847 df-sra 21157 df-rgmod 21158 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-tmd 24025 df-tgp 24026 df-tsms 24080 df-trg 24113 df-xms 24273 df-ms 24274 df-tms 24275 df-nm 24535 df-ngp 24536 df-nrg 24538 df-nlm 24539 df-ii 24832 df-cncf 24833 df-limc 25821 df-dv 25822 df-log 26508 df-esum 34160 df-carsg 34434

This theorem is referenced by: carsgclctunlem1 34449 carsgclctunlem2 34451 carsgclctunlem3 34452

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