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Theorem ovoliun 24107
 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 24087, 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 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
ovoliun.v ((𝜑𝑛 ∈ 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.
StepHypRef Expression
1 mnfxr 10687 . . . . . 6 -∞ ∈ ℝ*
21a1i 11 . . . . 5 (𝜑 → -∞ ∈ ℝ*)
3 nnuz 12269 . . . . . . . . 9 ℕ = (ℤ‘1)
4 1zzd 12001 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
5 ovoliun.v . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
6 ovoliun.g . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
75, 6fmptd 6860 . . . . . . . . . 10 (𝜑𝐺:ℕ⟶ℝ)
87ffvelrnda 6833 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
93, 4, 8serfre 13395 . . . . . . . 8 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10 ovoliun.t . . . . . . . . 9 𝑇 = seq1( + , 𝐺)
1110feq1i 6485 . . . . . . . 8 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
129, 11sylibr 237 . . . . . . 7 (𝜑𝑇:ℕ⟶ℝ)
13 1nn 11636 . . . . . . 7 1 ∈ ℕ
14 ffvelrn 6831 . . . . . . 7 ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
1512, 13, 14sylancl 589 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ℝ)
1615rexrd 10680 . . . . 5 (𝜑 → (𝑇‘1) ∈ ℝ*)
1712frnd 6501 . . . . . . 7 (𝜑 → ran 𝑇 ⊆ ℝ)
18 ressxr 10674 . . . . . . 7 ℝ ⊆ ℝ*
1917, 18sstrdi 3954 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ*)
20 supxrcl 12696 . . . . . 6 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2119, 20syl 17 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2215mnfltd 12507 . . . . 5 (𝜑 → -∞ < (𝑇‘1))
2312ffnd 6495 . . . . . . 7 (𝜑𝑇 Fn ℕ)
24 fnfvelrn 6830 . . . . . . 7 ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
2523, 13, 24sylancl 589 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26 supxrub 12705 . . . . . 6 ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
2719, 25, 26syl2anc 587 . . . . 5 (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
282, 16, 21, 22, 27xrltletrd 12542 . . . 4 (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29 xrrebnd 12549 . . . . 5 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3021, 29syl 17 . . . 4 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3128, 30mpbirand 706 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32 nfcv 2979 . . . . . . . . 9 𝑚𝐴
33 nfcsb1v 3879 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
34 csbeq1a 3869 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
3532, 33, 34cbviun 4936 . . . . . . . 8 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
3635fveq2i 6655 . . . . . . 7 (vol*‘ 𝑛 ∈ ℕ 𝐴) = (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴)
37 nfcv 2979 . . . . . . . . . 10 𝑚(vol*‘𝐴)
38 nfcv 2979 . . . . . . . . . . 11 𝑛vol*
3938, 33nffv 6662 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
4034fveq2d 6656 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4137, 39, 40cbvmpt 5143 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
426, 41eqtri 2845 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
43 ovoliun.a . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4443ralrimiva 3174 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45 nfv 1915 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
46 nfcv 2979 . . . . . . . . . . . . 13 𝑛
4733, 46nfss 3934 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
4834sseq1d 3973 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
4945, 47, 48cbvralw 3415 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5044, 49sylib 221 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5150ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5251r19.21bi 3198 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
535ralrimiva 3174 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5437nfel1 2995 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5539nfel1 2995 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5640eleq1d 2898 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
5754, 55, 56cbvralw 3415 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
5853, 57sylib 221 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
5958ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6059r19.21bi 3198 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
61 simplr 768 . . . . . . . 8 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
62 simpr 488 . . . . . . . 8 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
6310, 42, 52, 60, 61, 62ovoliunlem3 24106 . . . . . . 7 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6436, 63eqbrtrid 5077 . . . . . 6 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6564ralrimiva 3174 . . . . 5 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66 iunss 4944 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
6744, 66sylibr 237 . . . . . . 7 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68 ovolcl 24080 . . . . . . 7 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
6967, 68syl 17 . . . . . 6 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70 xralrple 12586 . . . . . 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)))
7169, 70sylan 583 . . . . 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)))
7265, 71mpbird 260 . . . 4 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
7372ex 416 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
7431, 73sylbird 263 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
75 nltpnft 12545 . . . 4 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
7621, 75syl 17 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
77 pnfge 12513 . . . . 5 ((vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
7869, 77syl 17 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79 breq2 5046 . . . 4 (sup(ran 𝑇, ℝ*, < ) = +∞ → ((vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
8078, 79syl5ibrcom 250 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8176, 80sylbird 263 . 2 (𝜑 → (¬ sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8274, 81pm2.61d 182 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ⦋csb 3855   ⊆ wss 3908  ∪ ciun 4894   class class class wbr 5042   ↦ cmpt 5122  ran crn 5533   Fn wfn 6329  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  supcsup 8892  ℝcr 10525  1c1 10527   + caddc 10529  +∞cpnf 10661  -∞cmnf 10662  ℝ*cxr 10663   < clt 10664   ≤ cle 10665  ℕcn 11625  ℝ+crp 12377  seqcseq 13364  vol*covol 24064 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-inf2 9092  ax-cc 9846  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-ioo 12730  df-ico 12732  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  df-cj 14449  df-re 14450  df-im 14451  df-sqrt 14585  df-abs 14586  df-clim 14836  df-rlim 14837  df-sum 15034  df-ovol 24066 This theorem is referenced by:  ovoliun2  24108  voliunlem2  24153  voliunlem3  24154  ex-ovoliunnfl  35059
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