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Theorem ovoliun 23025
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 23005, 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 nnuz 11558 . . . . . . . . . 10 ℕ = (ℤ‘1)
2 1zzd 11244 . . . . . . . . . 10 (𝜑 → 1 ∈ ℤ)
3 ovoliun.v . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
4 ovoliun.g . . . . . . . . . . . 12 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
53, 4fmptd 6277 . . . . . . . . . . 11 (𝜑𝐺:ℕ⟶ℝ)
65ffvelrnda 6252 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
71, 2, 6serfre 12650 . . . . . . . . 9 (𝜑 → seq1( + , 𝐺):ℕ⟶ℝ)
8 ovoliun.t . . . . . . . . . 10 𝑇 = seq1( + , 𝐺)
98feq1i 5935 . . . . . . . . 9 (𝑇:ℕ⟶ℝ ↔ seq1( + , 𝐺):ℕ⟶ℝ)
107, 9sylibr 223 . . . . . . . 8 (𝜑𝑇:ℕ⟶ℝ)
11 frn 5952 . . . . . . . 8 (𝑇:ℕ⟶ℝ → ran 𝑇 ⊆ ℝ)
1210, 11syl 17 . . . . . . 7 (𝜑 → ran 𝑇 ⊆ ℝ)
13 ressxr 9940 . . . . . . 7 ℝ ⊆ ℝ*
1412, 13syl6ss 3580 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ*)
15 supxrcl 11976 . . . . . 6 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
1614, 15syl 17 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
17 xrrebnd 11835 . . . . 5 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
1816, 17syl 17 . . . 4 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
19 mnfxr 11786 . . . . . . 7 -∞ ∈ ℝ*
2019a1i 11 . . . . . 6 (𝜑 → -∞ ∈ ℝ*)
21 1nn 10881 . . . . . . . 8 1 ∈ ℕ
22 ffvelrn 6250 . . . . . . . 8 ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
2310, 21, 22sylancl 693 . . . . . . 7 (𝜑 → (𝑇‘1) ∈ ℝ)
2423rexrd 9946 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ℝ*)
25 mnflt 11797 . . . . . . 7 ((𝑇‘1) ∈ ℝ → -∞ < (𝑇‘1))
2623, 25syl 17 . . . . . 6 (𝜑 → -∞ < (𝑇‘1))
27 ffn 5944 . . . . . . . . 9 (𝑇:ℕ⟶ℝ → 𝑇 Fn ℕ)
2810, 27syl 17 . . . . . . . 8 (𝜑𝑇 Fn ℕ)
29 fnfvelrn 6249 . . . . . . . 8 ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
3028, 21, 29sylancl 693 . . . . . . 7 (𝜑 → (𝑇‘1) ∈ ran 𝑇)
31 supxrub 11985 . . . . . . 7 ((ran 𝑇 ⊆ ℝ* ∧ (𝑇‘1) ∈ ran 𝑇) → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
3214, 30, 31syl2anc 691 . . . . . 6 (𝜑 → (𝑇‘1) ≤ sup(ran 𝑇, ℝ*, < ))
3320, 24, 16, 26, 32xrltletrd 11830 . . . . 5 (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
3433biantrurd 528 . . . 4 (𝜑 → (sup(ran 𝑇, ℝ*, < ) < +∞ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3518, 34bitr4d 270 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
36 nfcv 2751 . . . . . . . . 9 𝑚𝐴
37 nfcsb1v 3515 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
38 csbeq1a 3508 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
3936, 37, 38cbviun 4488 . . . . . . . 8 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
4039fveq2i 6091 . . . . . . 7 (vol*‘ 𝑛 ∈ ℕ 𝐴) = (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴)
41 nfcv 2751 . . . . . . . . . 10 𝑚(vol*‘𝐴)
42 nfcv 2751 . . . . . . . . . . 11 𝑛vol*
4342, 37nffv 6095 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
4438fveq2d 6092 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4541, 43, 44cbvmpt 4672 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
464, 45eqtri 2632 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
47 ovoliun.a . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4847ralrimiva 2949 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
49 nfv 1830 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
50 nfcv 2751 . . . . . . . . . . . . 13 𝑛
5137, 50nfss 3561 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
5238sseq1d 3595 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
5349, 51, 52cbvral 3143 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5448, 53sylib 207 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5554ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ 𝑚 / 𝑛𝐴 ⊆ ℝ)
5655r19.21bi 2916 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
573ralrimiva 2949 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5841nfel1 2765 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5943nfel1 2765 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
6044eleq1d 2672 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
6158, 59, 60cbvral 3143 . . . . . . . . . . 11 (∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ ↔ ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6257, 61sylib 207 . . . . . . . . . 10 (𝜑 → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6362ad2antrr 758 . . . . . . . . 9 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → ∀𝑚 ∈ ℕ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
6463r19.21bi 2916 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ)
65 simplr 788 . . . . . . . 8 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
66 simpr 476 . . . . . . . 8 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
678, 46, 56, 64, 65, 66ovoliunlem3 23024 . . . . . . 7 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6840, 67syl5eqbr 4613 . . . . . 6 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6968ralrimiva 2949 . . . . 5 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
70 iunss 4492 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
7148, 70sylibr 223 . . . . . . 7 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
72 ovolcl 22998 . . . . . . 7 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
7371, 72syl 17 . . . . . 6 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
74 xralrple 11872 . . . . . 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)))
7573, 74sylan 487 . . . . 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)))
7669, 75mpbird 246 . . . 4 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
7776ex 449 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) ∈ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
7835, 77sylbird 249 . 2 (𝜑 → (sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
79 nltpnft 11833 . . . 4 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
8016, 79syl 17 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ ¬ sup(ran 𝑇, ℝ*, < ) < +∞))
81 pnfge 11804 . . . . 5 ((vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
8273, 81syl 17 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
83 breq2 4582 . . . 4 (sup(ran 𝑇, ℝ*, < ) = +∞ → ((vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ) ↔ (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞))
8482, 83syl5ibrcom 236 . . 3 (𝜑 → (sup(ran 𝑇, ℝ*, < ) = +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8580, 84sylbird 249 . 2 (𝜑 → (¬ sup(ran 𝑇, ℝ*, < ) < +∞ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < )))
8678, 85pm2.61d 169 1 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ sup(ran 𝑇, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  csb 3499  wss 3540   ciun 4450   class class class wbr 4578  cmpt 4638  ran crn 5029   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  supcsup 8207  cr 9792  1c1 9794   + caddc 9796  +∞cpnf 9928  -∞cmnf 9929  *cxr 9930   < clt 9931  cle 9932  cn 10870  +crp 11667  seqcseq 12621  vol*covol 22983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cc 9118  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4368  df-int 4406  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-tr 4676  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-n0 11143  df-z 11214  df-uz 11523  df-q 11624  df-rp 11668  df-ioo 12009  df-ico 12011  df-fz 12156  df-fzo 12293  df-fl 12413  df-seq 12622  df-exp 12681  df-hash 12938  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-clim 14016  df-rlim 14017  df-sum 14214  df-ovol 22985
This theorem is referenced by:  ovoliun2  23026  voliunlem2  23071  voliunlem3  23072  ex-ovoliunnfl  32416
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