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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 ((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 11318 . . . . . 6 -∞ ∈ ℝ*
21a1i 11 . . . . 5 (𝜑 → -∞ ∈ ℝ*)
3 nnuz 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
4 1zzd 12648 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
5 ovoliun.v . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
6 ovoliun.g . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
75, 6fmptd 7134 . . . . . . . . . 10 (𝜑𝐺:ℕ⟶ℝ)
87ffvelcdmda 7104 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
93, 4, 8serfre 14072 . . . . . . . 8 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
10 ovoliun.t . . . . . . . . 9 𝑇 = seq1( + , 𝐺)
1110feq1i 6727 . . . . . . . 8 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
129, 11sylibr 234 . . . . . . 7 (𝜑𝑇:ℕ⟶ℝ)
13 1nn 12277 . . . . . . 7 1 ∈ ℕ
14 ffvelcdm 7101 . . . . . . 7 ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
1512, 13, 14sylancl 586 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ℝ)
1615rexrd 11311 . . . . 5 (𝜑 → (𝑇‘1) ∈ ℝ*)
1712frnd 6744 . . . . . . 7 (𝜑 → ran 𝑇 ⊆ ℝ)
18 ressxr 11305 . . . . . . 7 ℝ ⊆ ℝ*
1917, 18sstrdi 3996 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ*)
20 supxrcl 13357 . . . . . 6 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2119, 20syl 17 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2215mnfltd 13166 . . . . 5 (𝜑 → -∞ < (𝑇‘1))
2312ffnd 6737 . . . . . . 7 (𝜑𝑇 Fn ℕ)
24 fnfvelrn 7100 . . . . . . 7 ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
2523, 13, 24sylancl 586 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26 supxrub 13366 . . . . . 6 ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
2719, 25, 26syl2anc 584 . . . . 5 (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
282, 16, 21, 22, 27xrltletrd 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*, < ) < +∞)))
3021, 29syl 17 . . . 4 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3128, 30mpbirand 707 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32 nfcv 2905 . . . . . . . . 9 𝑚𝐴
33 nfcsb1v 3923 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
34 csbeq1a 3913 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
3532, 33, 34cbviun 5036 . . . . . . . 8 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
3635fveq2i 6909 . . . . . . 7 (vol*‘ 𝑛 ∈ ℕ 𝐴) = (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴)
37 nfcv 2905 . . . . . . . . . 10 𝑚(vol*‘𝐴)
38 nfcv 2905 . . . . . . . . . . 11 𝑛vol*
3938, 33nffv 6916 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
4034fveq2d 6910 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4137, 39, 40cbvmpt 5253 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
426, 41eqtri 2765 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
43 ovoliun.a . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4443ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45 nfv 1914 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
46 nfcv 2905 . . . . . . . . . . . . 13 𝑛
4733, 46nfss 3976 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
4834sseq1d 4015 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
4945, 47, 48cbvralw 3306 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5044, 49sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5150ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5251r19.21bi 3251 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
535ralrimiva 3146 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5437nfel1 2922 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5539nfel1 2922 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5640eleq1d 2826 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
5754, 55, 56cbvralw 3306 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
5853, 57sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
5958ad2antrr 726 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6059r19.21bi 3251 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ 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+)
6310, 42, 52, 60, 61, 62ovoliunlem3 25539 . . . . . . 7 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6436, 63eqbrtrid 5178 . . . . . 6 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6564ralrimiva 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)
6744, 66sylibr 234 . . . . . . 7 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68 ovolcl 25513 . . . . . . 7 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
6967, 68syl 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)))
7169, 70sylan 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)))
7265, 71mpbird 257 . . . 4 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
7372ex 412 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
7431, 73sylbird 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*, < ) < +∞))
7621, 75syl 17 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
77 pnfge 13172 . . . . 5 ((vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
7869, 77syl 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) ≤ +∞))
8078, 79syl5ibrcom 247 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8176, 80sylbird 260 . 2 (𝜑 → (¬ sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8274, 81pm2.61d 179 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
Colors of variables: wff setvar class
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  cr 11154  1c1 11156   + caddc 11158  +∞cpnf 11292  -∞cmnf 11293  *cxr 11294   < clt 11295  cle 11296  cn 12266  +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
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