Theorem voliun 25482

Description: The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
voliun.1	⊢ 𝑆 = seq1( + , 𝐺)
voliun.2	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))

Assertion

Ref	Expression
voliun	⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))

Proof of Theorem voliun

Dummy variables 𝑖 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐴 ∈ dom vol)
2	1	ralimi 3069	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
3	2	adantr 480	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
4		eqid 2731	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ 𝐴) = (𝑛 ∈ ℕ ↦ 𝐴)
5	4	fmpt 7043	. . . 4 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ↔ (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
6	3, 5	sylib 218	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
7	4	fvmpt2 6940	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
8	7	adantrr 717	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
9	8	ralimiaa 3068	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
10		disjeq2 5060	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴 → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
11	9, 10	syl 17	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
12	11	biimpar 477	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
13		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑖((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)
14		nffvmpt1 6833	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)
15		fveq2 6822	. . . . 5 ⊢ (𝑛 = 𝑖 → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
16	13, 14, 15	cbvdisj 5066	. . . 4 ⊢ (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
17	12, 16	sylib 218	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
18		eqid 2731	. . 3 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))))
19		eqid 2731	. . 3 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
20		nfcv 2894	. . . 4 ⊢ Ⅎ𝑚(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
21		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑛vol
22		nffvmpt1 6833	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)
23	21, 22	nffv 6832	. . . 4 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))
24		2fveq3 6827	. . . 4 ⊢ (𝑛 = 𝑚 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
25	20, 23, 24	cbvmpt 5191	. . 3 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
26	7	fveq2d 6826	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
27	26	eleq1d 2816	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘𝐴) ∈ ℝ))
28	27	biimprd 248	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘𝐴) ∈ ℝ → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ))
29	28	impr 454	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
30	29	ralimiaa 3068	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
31	30	adantr 480	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
32		nfv 1915	. . . . 5 ⊢ Ⅎ𝑖(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ
33	21, 14	nffv 6832	. . . . . 6 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
34	33	nfel1 2911	. . . . 5 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ
35		2fveq3 6827	. . . . . 6 ⊢ (𝑛 = 𝑖 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)))
36	35	eleq1d 2816	. . . . 5 ⊢ (𝑛 = 𝑖 → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ))
37	32, 34, 36	cbvralw 3274	. . . 4 ⊢ (∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
38	31, 37	sylib 218	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
39	6, 17, 18, 19, 25, 38	voliunlem3 25480	. 2 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ ran (𝑛 ∈ ℕ ↦ 𝐴)) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))), ℝ*, < ))
40		dfiun2g 4978	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
41	3, 40	syl 17	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
42	4	rnmpt 5896	. . . . 5 ⊢ ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
43	42	unieqi 4868	. . . 4 ⊢ ∪ ran (𝑛 ∈ ℕ ↦ 𝐴) = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
44	41, 43	eqtr4di 2784	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ ran (𝑛 ∈ ℕ ↦ 𝐴))
45	44	fveq2d 6826	. 2 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = (vol‘∪ ran (𝑛 ∈ ℕ ↦ 𝐴)))
46		voliun.1	. . . . 5 ⊢ 𝑆 = seq1( + , 𝐺)
47		voliun.2	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
48		eqid 2731	. . . . . . . 8 ⊢ ℕ = ℕ
49	26	adantrr 717	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
50	49	ralimiaa 3068	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
51	50	adantr 480	. . . . . . . 8 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
52		mpteq12 5177	. . . . . . . 8 ⊢ ((ℕ = ℕ ∧ ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴)) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
53	48, 51, 52	sylancr 587	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
54	47, 53	eqtr4id 2785	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
55	54	seqeq3d 13916	. . . . 5 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
56	46, 55	eqtrid 2778	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
57	56	rneqd 5877	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ran 𝑆 = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
58	57	supeq1d 9330	. 2 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))), ℝ*, < ))
59	39, 45, 58	3eqtr4d 2776	1 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ∩ cin 3896  ∪ cuni 4856  ∪ ciun 4939  Disj wdisj 5056   ↦ cmpt 5170  dom cdm 5614  ran crn 5615  ⟶wf 6477  ‘cfv 6481  supcsup 9324  ℝcr 11005  1c1 11007   + caddc 11009  ℝ^*cxr 11145   < clt 11146  ℕcn 12125  seqcseq 13908  vol*covol 25390  volcvol 25391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531  ax-cc 10326  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-disj 5057  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-sup 9326  df-inf 9327  df-oi 9396  df-dju 9794  df-card 9832  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-xadd 13012  df-ioo 13249  df-ico 13251  df-icc 13252  df-fz 13408  df-fzo 13555  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-xmet 21284  df-met 21285  df-ovol 25392  df-vol 25393

This theorem is referenced by:  volsup  25484  vitalilem4  25539  voliune  34242  voliunsge0lem  46569