Theorem ovoliun 25540

Description: The Lebesgue outer measure function is countably sub-additive. (Many books allow +∞ as a value for one of the sets in the sum, but in our setup we can't do arithmetic on infinity, and in any case the volume of a union containing an infinitely large set is already infinitely large by monotonicity ovolss 25520, so we need not consider this case here, although we do allow the sum itself to be infinite.) (Contributed by Mario Carneiro, 12-Jun-2014.)

Hypotheses

Ref	Expression
ovoliun.t	⊢ 𝑇 = seq1( + , 𝐺)
ovoliun.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
ovoliun.a	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v	⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)

Assertion

Ref	Expression
ovoliun	⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))

Distinct variable group:   𝜑,𝑛

Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun

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

Step	Hyp	Ref	Expression
1		mnfxr 11318	. . . . . 6 ⊢ -∞ ∈ ℝ*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ*)
3		nnuz 12921	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 12648	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
5		ovoliun.v	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
6		ovoliun.g	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
7	5, 6	fmptd 7134	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelcdmda 7104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 14072	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6727	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 12277	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelcdm 7101	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11311	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ*)
17	12	frnd 6744	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11305	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
19	17, 18	sstrdi 3996	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
20		supxrcl 13357	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
22	15	mnfltd 13166	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 7100	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 586	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13366	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
27	19, 25, 26	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
28	2, 16, 21, 22, 27	xrltletrd 13203	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29		xrrebnd 13210	. . . . 5 ⊢ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
30	21, 29	syl 17	. . . 4 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
31	28, 30	mpbirand 707	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3923	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3913	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 5036	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6909	. . . . . . 7 ⊢ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) = (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6916	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6910	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5253	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2765	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 3976	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 4015	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3306	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
50	44, 49	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
51	50	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3251	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3146	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3306	. . . . . . . . . . 11 ⊢ (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
58	53, 57	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
59	58	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3251	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
62		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
63	10, 42, 52, 60, 61, 62	ovoliunlem3 25539	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
64	36, 63	eqbrtrid 5178	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
65	64	ralrimiva 3146	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66		iunss 5045	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 25513	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70		xralrple 13247	. . . . . 6 ⊢ (((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥)))
71	69, 70	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥)))
72	65, 71	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
73	72	ex 412	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
74	31, 73	sylbird 260	. 2 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
75		nltpnft 13206	. . . 4 ⊢ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
76	21, 75	syl 17	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
77		pnfge 13172	. . . . 5 ⊢ ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5147	. . . 4 ⊢ (sup(ran 𝑇, ℝ*, < ) = +∞ → ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
80	78, 79	syl5ibrcom 247	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
81	76, 80	sylbird 260	. 2 ⊢ (𝜑 → (¬ sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
82	74, 81	pm2.61d 179	1 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⦋csb 3899   ⊆ wss 3951  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  supcsup 9480  ℝcr 11154  1c1 11156   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296  ℕcn 12266  ℝ⁺crp 13034  seqcseq 14042  vol*covol 25497

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-cc 10475  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

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-op 4633  df-uni 4908  df-int 4947  df-iun 4993  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-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  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-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-ovol 25499

This theorem is referenced by:  ovoliun2  25541  voliunlem2  25586  voliunlem3  25587  ex-ovoliunnfl  37670