Theorem voliun 25875

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 488	. . . . . 6 ⊢ ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐴 ∈ dom vol)
2	1	ralimi 3099	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
3	2	adantr 486	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
4		eqid 2760	. . . . 5 ⊢ (𝑛 ∈ ℕ ↦ 𝐴) = (𝑛 ∈ ℕ ↦ 𝐴)
5	4	fmpt 7110	. . . 4 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ↔ (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
6	3, 5	sylib 221	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
7	4	fvmpt2 7005	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
8	7	adantrr 730	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
9	8	ralimiaa 3098	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
10		disjeq2 5073	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴 → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
11	9, 10	syl 18	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
12	11	biimpar 483	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
13		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑖((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)
14		nffvmpt1 6896	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)
15		fveq2 6885	. . . . 5 ⊢ (𝑛 = 𝑖 → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
16	13, 14, 15	cbvdisj 5079	. . . 4 ⊢ (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
17	12, 16	sylib 221	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
18		eqid 2760	. . 3 ⊢ (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))))
19		eqid 2760	. . 3 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
20		nfcv 2922	. . . 4 ⊢ Ⅎ𝑚(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
21		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑛vol
22		nffvmpt1 6896	. . . . 5 ⊢ Ⅎ𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)
23	21, 22	nffv 6895	. . . 4 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))
24		2fveq3 6890	. . . 4 ⊢ (𝑛 = 𝑚 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
25	20, 23, 24	cbvmpt 5206	. . 3 ⊢ (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
26	7	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
27	26	eleq1d 2845	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘𝐴) ∈ ℝ))
28	27	biimprd 251	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘𝐴) ∈ ℝ → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ))
29	28	impr 460	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
30	29	ralimiaa 3098	. . . . 5 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
31	30	adantr 486	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
32		nfv 1946	. . . . 5 ⊢ Ⅎ𝑖(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ
33	21, 14	nffv 6895	. . . . . 6 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
34	33	nfel1 2938	. . . . 5 ⊢ Ⅎ𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ
35		2fveq3 6890	. . . . . 6 ⊢ (𝑛 = 𝑖 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)))
36	35	eleq1d 2845	. . . . 5 ⊢ (𝑛 = 𝑖 → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ))
37	32, 34, 36	cbvralw 3304	. . . 4 ⊢ (∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
38	31, 37	sylib 221	. . 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 25873	. 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 4987	. . . . 5 ⊢ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
41	3, 40	syl 18	. . . 4 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
42	4	rnmpt 5938	. . . . 5 ⊢ ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
43	42	unieqi 4878	. . . 4 ⊢ ∪ ran (𝑛 ∈ ℕ ↦ 𝐴) = ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
44	41, 43	eqtr4di 2813	. . 3 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∪ 𝑛 ∈ ℕ 𝐴 = ∪ ran (𝑛 ∈ ℕ ↦ 𝐴))
45	44	fveq2d 6889	. 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 2760	. . . . . . . 8 ⊢ ℕ = ℕ
49	26	adantrr 730	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
50	49	ralimiaa 3098	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
51	50	adantr 486	. . . . . . . 8 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
52		mpteq12 5192	. . . . . . . 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 599	. . . . . . 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 2814	. . . . . 6 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
55	54	seqeq3d 14152	. . . . 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 2807	. . . 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 5919	. . 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 9438	. 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 2805	1 ⊢ ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘∪ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∩ cin 4089  ∪ cuni 4866  ∪ ciun 4950  Disj wdisj 5069   ↦ cmpt 5185  dom cdm 5650  ran crn 5651  ⟶wf 6534  ‘cfv 6538  supcsup 9432  ℝcr 11199  1c1 11201   + caddc 11203  ℝ^*cxr 11342   < clt 11343  ℕcn 12335  seqcseq 14144  vol*covol 25783  volcvol 25784

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9642  ax-cc 10513  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-oi 9504  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-xadd 13242  df-ioo 13480  df-ico 13482  df-icc 13483  df-fz 13640  df-fzo 13789  df-fl 13932  df-seq 14145  df-exp 14205  df-hash 14475  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-clim 15655  df-rlim 15656  df-sum 15854  df-xmet 21671  df-met 21672  df-ovol 25785  df-vol 25786

This theorem is used by:  volsup  25877  vitalilem4  25932  voliune  34862  voliunsge0lem  47482