Theorem ovoliun 25559

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 25539, 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 11347	. . . . . 6 ⊢ -∞ ∈ ℝ*
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → -∞ ∈ ℝ*)
3		nnuz 12946	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
4		1zzd 12674	. . . . . . . . 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 7148	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺:ℕ⟶ℝ)
8	7	ffvelcdmda 7118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
9	3, 4, 8	serfre 14082	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10		ovoliun.t	. . . . . . . . 9 ⊢ 𝑇 = seq1( + , 𝐺)
11	10	feq1i 6738	. . . . . . . 8 ⊢ (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
12	9, 11	sylibr 234	. . . . . . 7 ⊢ (𝜑 → 𝑇:ℕ⟶ℝ)
13		1nn 12304	. . . . . . 7 ⊢ 1 ∈ ℕ
14		ffvelcdm 7115	. . . . . . 7 ⊢ ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
15	12, 13, 14	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ)
16	15	rexrd 11340	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ∈ ℝ*)
17	12	frnd 6755	. . . . . . 7 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ)
18		ressxr 11334	. . . . . . 7 ⊢ ℝ ⊆ ℝ*
19	17, 18	sstrdi 4021	. . . . . 6 ⊢ (𝜑 → ran 𝑇 ⊆ ℝ*)
20		supxrcl 13377	. . . . . 6 ⊢ (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
22	15	mnfltd 13187	. . . . 5 ⊢ (𝜑 → -∞ < (𝑇‘1))
23	12	ffnd 6748	. . . . . . 7 ⊢ (𝜑 → 𝑇 Fn ℕ)
24		fnfvelrn 7114	. . . . . . 7 ⊢ ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
25	23, 13, 24	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26		supxrub 13386	. . . . . 6 ⊢ ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
27	19, 25, 26	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
28	2, 16, 21, 22, 27	xrltletrd 13223	. . . 4 ⊢ (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29		xrrebnd 13230	. . . . 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 706	. . 3 ⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑚𝐴
33		nfcsb1v 3946	. . . . . . . . 9 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴
34		csbeq1a 3935	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴)
35	32, 33, 34	cbviun 5059	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ℕ 𝐴 = ∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴
36	35	fveq2i 6923	. . . . . . 7 ⊢ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) = (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴)
37		nfcv 2908	. . . . . . . . . 10 ⊢ Ⅎ𝑚(vol*‘𝐴)
38		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑛vol*
39	38, 33	nffv 6930	. . . . . . . . . 10 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴)
40	34	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘⦋𝑚 / 𝑛⦌𝐴))
41	37, 39, 40	cbvmpt 5277	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
42	6, 41	eqtri 2768	. . . . . . . 8 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘⦋𝑚 / 𝑛⦌𝐴))
43		ovoliun.a	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
44	43	ralrimiva 3152	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45		nfv 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝐴 ⊆ ℝ
46		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛ℝ
47	33, 46	nfss 4001	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ
48	34	sseq1d 4040	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ))
49	45, 47, 48	cbvralw 3312	. . . . . . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
52	51	r19.21bi 3257	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → ⦋𝑚 / 𝑛⦌𝐴 ⊆ ℝ)
53	5	ralrimiva 3152	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
54	37	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚(vol*‘𝐴) ∈ ℝ
55	39	nfel1 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ
56	40	eleq1d 2829	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ))
57	54, 55, 56	cbvralw 3312	. . . . . . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
60	59	r19.21bi 3257	. . . . . . . 8 ⊢ ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘⦋𝑚 / 𝑛⦌𝐴) ∈ ℝ)
61		simplr 768	. . . . . . . 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 25558	. . . . . . 7 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑚 ∈ ℕ ⦋𝑚 / 𝑛⦌𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
64	36, 63	eqbrtrid 5201	. . . . . 6 ⊢ (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
65	64	ralrimiva 3152	. . . . 5 ⊢ ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66		iunss 5068	. . . . . . . 8 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
67	44, 66	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68		ovolcl 25532	. . . . . . 7 ⊢ (∪ 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
69	67, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70		xralrple 13267	. . . . . 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 579	. . . . 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 13226	. . . 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 13193	. . . . 5 ⊢ ((vol*‘∪ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
78	69, 77	syl 17	. . . 4 ⊢ (𝜑 → (vol*‘∪ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79		breq2 5170	. . . 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 1537   ∈ wcel 2108  ∀wral 3067  ⦋csb 3921   ⊆ wss 3976  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  supcsup 9509  ℝcr 11183  1c1 11185   + caddc 11187  +∞cpnf 11321  -∞cmnf 11322  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325  ℕcn 12293  ℝ⁺crp 13057  seqcseq 14052  vol*covol 25516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-ioo 13411  df-ico 13413  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-ovol 25518

This theorem is referenced by:  ovoliun2  25560  voliunlem2  25605  voliunlem3  25606  ex-ovoliunnfl  37623