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Theorem ovoliun 25553
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 25533, 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 11315 . . . . . 6 -∞ ∈ ℝ*
21a1i 11 . . . . 5 (𝜑 → -∞ ∈ ℝ*)
3 nnuz 12918 . . . . . . . . 9 ℕ = (ℤ‘1)
4 1zzd 12645 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
5 ovoliun.v . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → (vol*‘𝐴) ∈ ℝ)
6 ovoliun.g . . . . . . . . . . 11 𝐺 = (𝑛 ∈ ℕ ↦ (vol*‘𝐴))
75, 6fmptd 7133 . . . . . . . . . 10 (𝜑𝐺:ℕ⟶ℝ)
87ffvelcdmda 7103 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐺𝑘) ∈ ℝ)
93, 4, 8serfre 14068 . . . . . . . 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 12274 . . . . . . 7 1 ∈ ℕ
14 ffvelcdm 7100 . . . . . . 7 ((𝑇:ℕ⟶ℝ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ℝ)
1512, 13, 14sylancl 586 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ℝ)
1615rexrd 11308 . . . . 5 (𝜑 → (𝑇‘1) ∈ ℝ*)
1712frnd 6744 . . . . . . 7 (𝜑 → ran 𝑇 ⊆ ℝ)
18 ressxr 11302 . . . . . . 7 ℝ ⊆ ℝ*
1917, 18sstrdi 4007 . . . . . 6 (𝜑 → ran 𝑇 ⊆ ℝ*)
20 supxrcl 13353 . . . . . 6 (ran 𝑇 ⊆ ℝ* → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2119, 20syl 17 . . . . 5 (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2215mnfltd 13163 . . . . 5 (𝜑 → -∞ < (𝑇‘1))
2312ffnd 6737 . . . . . . 7 (𝜑𝑇 Fn ℕ)
24 fnfvelrn 7099 . . . . . . 7 ((𝑇 Fn ℕ ∧ 1 ∈ ℕ) → (𝑇‘1) ∈ ran 𝑇)
2523, 13, 24sylancl 586 . . . . . 6 (𝜑 → (𝑇‘1) ∈ ran 𝑇)
26 supxrub 13362 . . . . . 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 13199 . . . 4 (𝜑 → -∞ < sup(ran 𝑇, ℝ*, < ))
29 xrrebnd 13206 . . . . 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 2902 . . . . . . . . 9 𝑚𝐴
33 nfcsb1v 3932 . . . . . . . . 9 𝑛𝑚 / 𝑛𝐴
34 csbeq1a 3921 . . . . . . . . 9 (𝑛 = 𝑚𝐴 = 𝑚 / 𝑛𝐴)
3532, 33, 34cbviun 5040 . . . . . . . 8 𝑛 ∈ ℕ 𝐴 = 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴
3635fveq2i 6909 . . . . . . 7 (vol*‘ 𝑛 ∈ ℕ 𝐴) = (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴)
37 nfcv 2902 . . . . . . . . . 10 𝑚(vol*‘𝐴)
38 nfcv 2902 . . . . . . . . . . 11 𝑛vol*
3938, 33nffv 6916 . . . . . . . . . 10 𝑛(vol*‘𝑚 / 𝑛𝐴)
4034fveq2d 6910 . . . . . . . . . 10 (𝑛 = 𝑚 → (vol*‘𝐴) = (vol*‘𝑚 / 𝑛𝐴))
4137, 39, 40cbvmpt 5258 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ (vol*‘𝐴)) = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
426, 41eqtri 2762 . . . . . . . 8 𝐺 = (𝑚 ∈ ℕ ↦ (vol*‘𝑚 / 𝑛𝐴))
43 ovoliun.a . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → 𝐴 ⊆ ℝ)
4443ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
45 nfv 1911 . . . . . . . . . . . 12 𝑚 𝐴 ⊆ ℝ
46 nfcv 2902 . . . . . . . . . . . . 13 𝑛
4733, 46nfss 3987 . . . . . . . . . . . 12 𝑛𝑚 / 𝑛𝐴 ⊆ ℝ
4834sseq1d 4026 . . . . . . . . . . . 12 (𝑛 = 𝑚 → (𝐴 ⊆ ℝ ↔ 𝑚 / 𝑛𝐴 ⊆ ℝ))
4945, 47, 48cbvralw 3303 . . . . . . . . . . 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 3248 . . . . . . . 8 ((((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ ℕ) → 𝑚 / 𝑛𝐴 ⊆ ℝ)
535ralrimiva 3143 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ (vol*‘𝐴) ∈ ℝ)
5437nfel1 2919 . . . . . . . . . . . 12 𝑚(vol*‘𝐴) ∈ ℝ
5539nfel1 2919 . . . . . . . . . . . 12 𝑛(vol*‘𝑚 / 𝑛𝐴) ∈ ℝ
5640eleq1d 2823 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((vol*‘𝐴) ∈ ℝ ↔ (vol*‘𝑚 / 𝑛𝐴) ∈ ℝ))
5754, 55, 56cbvralw 3303 . . . . . . . . . . 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 3248 . . . . . . . 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 25552 . . . . . . 7 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑚 ∈ ℕ 𝑚 / 𝑛𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6436, 63eqbrtrid 5182 . . . . . 6 (((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
6564ralrimiva 3143 . . . . 5 ((𝜑 ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ (sup(ran 𝑇, ℝ*, < ) + 𝑥))
66 iunss 5049 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ ↔ ∀𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
6744, 66sylibr 234 . . . . . . 7 (𝜑 𝑛 ∈ ℕ 𝐴 ⊆ ℝ)
68 ovolcl 25526 . . . . . . 7 ( 𝑛 ∈ ℕ 𝐴 ⊆ ℝ → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
6967, 68syl 17 . . . . . 6 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
70 xralrple 13243 . . . . . 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 13202 . . . 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 13169 . . . . 5 ((vol*‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
7869, 77syl 17 . . . 4 (𝜑 → (vol*‘ 𝑛 ∈ ℕ 𝐴) ≤ +∞)
79 breq2 5151 . . . 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 1536  wcel 2105  wral 3058  csb 3907  wss 3962   ciun 4995   class class class wbr 5147  cmpt 5230  ran crn 5689   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  supcsup 9477  cr 11151  1c1 11153   + caddc 11155  +∞cpnf 11289  -∞cmnf 11290  *cxr 11291   < clt 11292  cle 11293  cn 12263  +crp 13031  seqcseq 14038  vol*covol 25510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-ioo 13387  df-ico 13389  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-ovol 25512
This theorem is referenced by:  ovoliun2  25554  voliunlem2  25599  voliunlem3  25600  ex-ovoliunnfl  37649
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