fiunelcarsg - Mathbox for Thierry Arnoux

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Theorem fiunelcarsg 34318

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 4918	. . 3 ⊢ (𝑎 = ∅ → ∪ 𝑎 = ∪ ∅)
2		eqidd 2738	. . 3 ⊢ (𝑎 = ∅ → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
3	1, 2	eleq12d 2835	. 2 ⊢ (𝑎 = ∅ → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ ∅ ∈ (toCaraSiga‘𝑀)))
4		unieq 4918	. . 3 ⊢ (𝑎 = 𝑏 → ∪ 𝑎 = ∪ 𝑏)
5		eqidd 2738	. . 3 ⊢ (𝑎 = 𝑏 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
6	4, 5	eleq12d 2835	. 2 ⊢ (𝑎 = 𝑏 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)))
7		unieq 4918	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → ∪ 𝑎 = ∪ (𝑏 ∪ {𝑥}))
8		eqidd 2738	. . 3 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
9	7, 8	eleq12d 2835	. 2 ⊢ (𝑎 = (𝑏 ∪ {𝑥}) → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ (𝑏 ∪ {𝑥}) ∈ (toCaraSiga‘𝑀)))
10		unieq 4918	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
11		eqidd 2738	. . 3 ⊢ (𝑎 = 𝐴 → (toCaraSiga‘𝑀) = (toCaraSiga‘𝑀))
12	10, 11	eleq12d 2835	. 2 ⊢ (𝑎 = 𝐴 → (∪ 𝑎 ∈ (toCaraSiga‘𝑀) ↔ ∪ 𝐴 ∈ (toCaraSiga‘𝑀)))
13		uni0 4935	. . . 4 ⊢ ∪ ∅ = ∅
14		difid 4376	. . . 4 ⊢ (𝑂 ∖ 𝑂) = ∅
15	13, 14	eqtr4i 2768	. . 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 34308	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (toCaraSiga‘𝑀))
20	16, 17, 19	difelcarsg 34312	. . 3 ⊢ (𝜑 → (𝑂 ∖ 𝑂) ∈ (toCaraSiga‘𝑀))
21	15, 20	eqeltrid 2845	. 2 ⊢ (𝜑 → ∪ ∅ ∈ (toCaraSiga‘𝑀))
22		uniun 4930	. . . . 5 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ ∪ {𝑥})
23		unisnv 4927	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
24	23	uneq2i 4165	. . . . 5 ⊢ (∪ 𝑏 ∪ ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
25	22, 24	eqtri 2765	. . . 4 ⊢ ∪ (𝑏 ∪ {𝑥}) = (∪ 𝑏 ∪ 𝑥)
26	16	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑂 ∈ 𝑉)
27	17	ad2antrr 726	. . . . 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 34317	. . . . . 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 726	. . . . . 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 3963	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ 𝐴)
37	34, 36	sseldd 3984	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → 𝑥 ∈ (toCaraSiga‘𝑀))
38	26, 27, 28, 32, 37	unelcarsg 34314	. . . 4 ⊢ (((𝜑 ∧ (𝑏 ⊆ 𝐴 ∧ 𝑥 ∈ (𝐴 ∖ 𝑏))) ∧ ∪ 𝑏 ∈ (toCaraSiga‘𝑀)) → (∪ 𝑏 ∪ 𝑥) ∈ (toCaraSiga‘𝑀))
39	25, 38	eqeltrid 2845	. . 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 9206	1 ⊢ (𝜑 → ∪ 𝐴 ∈ (toCaraSiga‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ≼ cdom 8983 Fincfn 8985 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 +_𝑒 cxad 13152 [,]cicc 13390 Σ^*cesum 34028 toCaraSigaccarsg 34303

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-ordt 17546 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-ps 18611 df-tsr 18612 df-plusf 18652 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-abv 20810 df-lmod 20860 df-scaf 20861 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tmd 24080 df-tgp 24081 df-tsms 24135 df-trg 24168 df-xms 24330 df-ms 24331 df-tms 24332 df-nm 24595 df-ngp 24596 df-nrg 24598 df-nlm 24599 df-ii 24903 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-esum 34029 df-carsg 34304

This theorem is referenced by: carsgclctunlem1 34319 carsgclctunlem2 34321 carsgclctunlem3 34322

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