Theorem voliun 25507

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 3073	. . . . 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 2735	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ 𝐴) = (𝑛 ∈ ℕ ↦ 𝐴)
5	4	fmpt 7100	. . . 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 6997	. . . . . . . 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 3072	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
10		disjeq2 5090	. . . . . 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 2898	. . . . 5 ⊢ Ⅎ𝑖((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)
14		nffvmpt1 6887	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)
15		fveq2 6876	. . . . 5 ⊢ (𝑛 = 𝑖 → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
16	13, 14, 15	cbvdisj 5096	. . . 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 2735	. . 3 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))))
19		eqid 2735	. . 3 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
20		nfcv 2898	. . . 4 ⊢ Ⅎ𝑚(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
21		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑛vol
22		nffvmpt1 6887	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)
23	21, 22	nffv 6886	. . . 4 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))
24		2fveq3 6881	. . . 4 ⊢ (𝑛 = 𝑚 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
25	20, 23, 24	cbvmpt 5223	. . 3 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
26	7	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
27	26	eleq1d 2819	. . . . . . . 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 3072	. . . . 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 1914	. . . . 5 ⊢ Ⅎ𝑖(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ
33	21, 14	nffv 6886	. . . . . 6 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
34	33	nfel1 2915	. . . . 5 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ
35		2fveq3 6881	. . . . . 6 ⊢ (𝑛 = 𝑖 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)))
36	35	eleq1d 2819	. . . . 5 ⊢ (𝑛 = 𝑖 → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ))
37	32, 34, 36	cbvralw 3286	. . . 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 25505	. 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 5006	. . . . 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 5937	. . . . 5 ⊢ ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
43	42	unieqi 4895	. . . 4 ⊢ ∪ ran (𝑛 ∈ ℕ ↦ 𝐴) = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
44	41, 43	eqtr4di 2788	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ ran (𝑛 ∈ ℕ ↦ 𝐴))
45	44	fveq2d 6880	. 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 2735	. . . . . . . 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 3072	. . . . . . . . 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 5208	. . . . . . . 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 2789	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
55	54	seqeq3d 14027	. . . . 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 2782	. . . 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 5918	. . 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 9458	. 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 2780	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 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060   ∩ cin 3925  ∪ cuni 4883  ∪ ciun 4967  Disj wdisj 5086   ↦ cmpt 5201  dom cdm 5654  ran crn 5655  ⟶wf 6527  ‘cfv 6531  supcsup 9452  ℝcr 11128  1c1 11130   + caddc 11132  ℝ^*cxr 11268   < clt 11269  ℕcn 12240  seqcseq 14019  vol*covol 25415  volcvol 25416

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cc 10449  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206  ax-pre-sup 11207

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-disj 5087  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-sup 9454  df-inf 9455  df-oi 9524  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-div 11895  df-nn 12241  df-2 12303  df-3 12304  df-n0 12502  df-z 12589  df-uz 12853  df-q 12965  df-rp 13009  df-xadd 13129  df-ioo 13366  df-ico 13368  df-icc 13369  df-fz 13525  df-fzo 13672  df-fl 13809  df-seq 14020  df-exp 14080  df-hash 14349  df-cj 15118  df-re 15119  df-im 15120  df-sqrt 15254  df-abs 15255  df-clim 15504  df-rlim 15505  df-sum 15703  df-xmet 21308  df-met 21309  df-ovol 25417  df-vol 25418

This theorem is referenced by:  volsup  25509  vitalilem4  25564  voliune  34260  voliunsge0lem  46501