omeiunle - Mathbox for Glauco Siliprandi

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Theorem omeiunle 45233

Description: The outer measure of the indexed union of a countable set is the less than or equal to the extended sum of the outer measures. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
omeiunle.nph	⊢ Ⅎ𝑛𝜑
omeiunle.ne	⊢ Ⅎ𝑛𝐸
omeiunle.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
omeiunle.x	⊢ 𝑋 = ∪ dom 𝑂
omeiunle.z	⊢ 𝑍 = (ℤ_≥‘𝑁)
omeiunle.e	⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)

**Assertion**
Ref	Expression
omeiunle	⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))

Distinct variable groups: 𝑛,𝑂 𝑛,𝑋 𝑛,𝑍 Allowed substitution hints: 𝜑(𝑛) 𝐸(𝑛) 𝑁(𝑛)

Proof of Theorem omeiunle Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13407	. . 3 ⊢ (0[,]+∞) ⊆ ℝ^*
2		omeiunle.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		omeiunle.x	. . . 4 ⊢ 𝑋 = ∪ dom 𝑂
4		omeiunle.nph	. . . . . 6 ⊢ Ⅎ𝑛𝜑
5		omeiunle.e	. . . . . . . . 9 ⊢ (𝜑 → 𝐸:𝑍⟶𝒫 𝑋)
6	5	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ∈ 𝒫 𝑋)
7		elpwi 4610	. . . . . . . 8 ⊢ ((𝐸‘𝑛) ∈ 𝒫 𝑋 → (𝐸‘𝑛) ⊆ 𝑋)
8	6, 7	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) ⊆ 𝑋)
9	8	ex 414	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝐸‘𝑛) ⊆ 𝑋))
10	4, 9	ralrimi 3255	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
11		iunss 5049	. . . . 5 ⊢ (∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋 ↔ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
12	10, 11	sylibr 233	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) ⊆ 𝑋)
13	2, 3, 12	omecl 45219	. . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ (0[,]+∞))
14	1, 13	sselid 3981	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ∈ ℝ^*)
15	5	ffnd 6719	. . . . 5 ⊢ (𝜑 → 𝐸 Fn 𝑍)
16		omeiunle.z	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑁)
17	16	fvexi 6906	. . . . . 6 ⊢ 𝑍 ∈ V
18	17	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ V)
19		fnex 7219	. . . . 5 ⊢ ((𝐸 Fn 𝑍 ∧ 𝑍 ∈ V) → 𝐸 ∈ V)
20	15, 18, 19	syl2anc 585	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
21		rnexg 7895	. . . 4 ⊢ (𝐸 ∈ V → ran 𝐸 ∈ V)
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → ran 𝐸 ∈ V)
23	2, 3	omef 45212	. . . 4 ⊢ (𝜑 → 𝑂:𝒫 𝑋⟶(0[,]+∞))
24	5	frnd 6726	. . . 4 ⊢ (𝜑 → ran 𝐸 ⊆ 𝒫 𝑋)
25	23, 24	fssresd 6759	. . 3 ⊢ (𝜑 → (𝑂 ↾ ran 𝐸):ran 𝐸⟶(0[,]+∞))
26	22, 25	sge0xrcl 45101	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ ran 𝐸)) ∈ ℝ^*)
27	2	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑂 ∈ OutMeas)
28	27, 3, 8	omecl 45219	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑂‘(𝐸‘𝑛)) ∈ (0[,]+∞))
29		eqid 2733	. . . 4 ⊢ (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
30	4, 28, 29	fmptdf 7117	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))):𝑍⟶(0[,]+∞))
31	18, 30	sge0xrcl 45101	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))) ∈ ℝ^*)
32		fvex 6905	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
33	32	rgenw 3066	. . . . . . 7 ⊢ ∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ V
34		dfiun3g 5964	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑍 (𝐸‘𝑛) ∈ V → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
35	33, 34	ax-mp 5	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))
36	35	a1i 11	. . . . 5 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
37	5	feqmptd 6961	. . . . . . . 8 ⊢ (𝜑 → 𝐸 = (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚)))
38		omeiunle.ne	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝐸
39		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑛𝑚
40	38, 39	nffv 6902	. . . . . . . . . 10 ⊢ Ⅎ𝑛(𝐸‘𝑚)
41		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐸‘𝑛)
42		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (𝐸‘𝑚) = (𝐸‘𝑛))
43	40, 41, 42	cbvmpt 5260	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚)) = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛))
44	43	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ (𝐸‘𝑚)) = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
45	37, 44	eqtrd 2773	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
46	45	rneqd 5938	. . . . . 6 ⊢ (𝜑 → ran 𝐸 = ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
47	46	unieqd 4923	. . . . 5 ⊢ (𝜑 → ∪ ran 𝐸 = ∪ ran (𝑛 ∈ 𝑍 ↦ (𝐸‘𝑛)))
48	36, 47	eqtr4d 2776	. . . 4 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛) = ∪ ran 𝐸)
49	48	fveq2d 6896	. . 3 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) = (𝑂‘∪ ran 𝐸))
50		fnrndomg 10531	. . . . . 6 ⊢ (𝑍 ∈ V → (𝐸 Fn 𝑍 → ran 𝐸 ≼ 𝑍))
51	18, 15, 50	sylc 65	. . . . 5 ⊢ (𝜑 → ran 𝐸 ≼ 𝑍)
52	16	uzct 43750	. . . . . 6 ⊢ 𝑍 ≼ ω
53	52	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑍 ≼ ω)
54		domtr 9003	. . . . 5 ⊢ ((ran 𝐸 ≼ 𝑍 ∧ 𝑍 ≼ ω) → ran 𝐸 ≼ ω)
55	51, 53, 54	syl2anc 585	. . . 4 ⊢ (𝜑 → ran 𝐸 ≼ ω)
56	2, 3, 24, 55	omeunile 45221	. . 3 ⊢ (𝜑 → (𝑂‘∪ ran 𝐸) ≤ (Σ^{^}‘(𝑂 ↾ ran 𝐸)))
57	49, 56	eqbrtrd 5171	. 2 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑂 ↾ ran 𝐸)))
58		ltweuz 13926	. . . . . 6 ⊢ < We (ℤ_≥‘𝑁)
59		weeq2 5666	. . . . . . 7 ⊢ (𝑍 = (ℤ_≥‘𝑁) → ( < We 𝑍 ↔ < We (ℤ_≥‘𝑁)))
60	16, 59	ax-mp 5	. . . . . 6 ⊢ ( < We 𝑍 ↔ < We (ℤ_≥‘𝑁))
61	58, 60	mpbir 230	. . . . 5 ⊢ < We 𝑍
62	61	a1i 11	. . . 4 ⊢ (𝜑 → < We 𝑍)
63	18, 23, 5, 62	sge0resrn 45120	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ ran 𝐸)) ≤ (Σ^{^}‘(𝑂 ∘ 𝐸)))
64		fcompt 7131	. . . . . 6 ⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑂 ∘ 𝐸) = (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))))
65		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑂
66	65, 40	nffv 6902	. . . . . . . 8 ⊢ Ⅎ𝑛(𝑂‘(𝐸‘𝑚))
67		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑂‘(𝐸‘𝑛))
68		2fveq3 6897	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑂‘(𝐸‘𝑚)) = (𝑂‘(𝐸‘𝑛)))
69	66, 67, 68	cbvmpt 5260	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))
70	69	a1i 11	. . . . . 6 ⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑚 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑚))) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
71	64, 70	eqtrd 2773	. . . . 5 ⊢ ((𝑂:𝒫 𝑋⟶(0[,]+∞) ∧ 𝐸:𝑍⟶𝒫 𝑋) → (𝑂 ∘ 𝐸) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
72	23, 5, 71	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝑂 ∘ 𝐸) = (𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛))))
73	72	fveq2d 6896	. . 3 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ∘ 𝐸)) = (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
74	63, 73	breqtrd 5175	. 2 ⊢ (𝜑 → (Σ^{^}‘(𝑂 ↾ ran 𝐸)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))
75	14, 26, 31, 57, 74	xrletrd 13141	1 ⊢ (𝜑 → (𝑂‘∪ 𝑛 ∈ 𝑍 (𝐸‘𝑛)) ≤ (Σ^{^}‘(𝑛 ∈ 𝑍 ↦ (𝑂‘(𝐸‘𝑛)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ∀wral 3062 Vcvv 3475 ⊆ wss 3949 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 We wwe 5631 dom cdm 5677 ran crn 5678 ↾ cres 5679 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ωcom 7855 ≼ cdom 8937 0cc0 11110 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℤ_≥cuz 12822 [,]cicc 13327 Σ^{^}csumge0 45078 OutMeascome 45205

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-acn 9937 df-ac 10111 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-sumge0 45079 df-ome 45206

This theorem is referenced by: omeiunltfirp 45235 omeiunlempt 45236 caratheodorylem2 45243

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