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Theorem ovoliun2 25247
Description: The Lebesgue outer measure function is countably sub-additive. (This version is a little easier to read, but does not allow infinite values like ovoliun 25246.) (Contributed by Mario Carneiro, 12-Jun-2014.)
Hypotheses
Ref Expression
ovoliun.t 𝑇 = seq1( + , 𝐺)
ovoliun.g 𝐺 = (𝑛 ∈ β„• ↦ (vol*β€˜π΄))
ovoliun.a ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐴 βŠ† ℝ)
ovoliun.v ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜π΄) ∈ ℝ)
ovoliun2.t (πœ‘ β†’ 𝑇 ∈ dom ⇝ )
Assertion
Ref Expression
ovoliun2 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ Σ𝑛 ∈ β„• (vol*β€˜π΄))
Distinct variable group:   πœ‘,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun2
Dummy variables π‘˜ π‘š π‘₯ 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovoliun.t . . 3 𝑇 = seq1( + , 𝐺)
2 ovoliun.g . . 3 𝐺 = (𝑛 ∈ β„• ↦ (vol*β€˜π΄))
3 ovoliun.a . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐴 βŠ† ℝ)
4 ovoliun.v . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜π΄) ∈ ℝ)
51, 2, 3, 4ovoliun 25246 . 2 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ))
6 nnuz 12869 . . . . . . . 8 β„• = (β„€β‰₯β€˜1)
7 1zzd 12597 . . . . . . . 8 (πœ‘ β†’ 1 ∈ β„€)
8 fvex 6904 . . . . . . . . . . 11 (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ V
9 nfcv 2903 . . . . . . . . . . . . . 14 β„²π‘š(vol*β€˜π΄)
10 nfcv 2903 . . . . . . . . . . . . . . 15 Ⅎ𝑛vol*
11 nfcsb1v 3918 . . . . . . . . . . . . . . 15 β„²π‘›β¦‹π‘š / π‘›β¦Œπ΄
1210, 11nffv 6901 . . . . . . . . . . . . . 14 Ⅎ𝑛(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄)
13 csbeq1a 3907 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ 𝐴 = β¦‹π‘š / π‘›β¦Œπ΄)
1413fveq2d 6895 . . . . . . . . . . . . . 14 (𝑛 = π‘š β†’ (vol*β€˜π΄) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
159, 12, 14cbvmpt 5259 . . . . . . . . . . . . 13 (𝑛 ∈ β„• ↦ (vol*β€˜π΄)) = (π‘š ∈ β„• ↦ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
162, 15eqtri 2760 . . . . . . . . . . . 12 𝐺 = (π‘š ∈ β„• ↦ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
1716fvmpt2 7009 . . . . . . . . . . 11 ((π‘š ∈ β„• ∧ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ V) β†’ (πΊβ€˜π‘š) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
188, 17mpan2 689 . . . . . . . . . 10 (π‘š ∈ β„• β†’ (πΊβ€˜π‘š) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
1918adantl 482 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
204ralrimiva 3146 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘› ∈ β„• (vol*β€˜π΄) ∈ ℝ)
219nfel1 2919 . . . . . . . . . . . 12 β„²π‘š(vol*β€˜π΄) ∈ ℝ
2212nfel1 2919 . . . . . . . . . . . 12 Ⅎ𝑛(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ
2314eleq1d 2818 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ ((vol*β€˜π΄) ∈ ℝ ↔ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ))
2421, 22, 23cbvralw 3303 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• (vol*β€˜π΄) ∈ ℝ ↔ βˆ€π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
2520, 24sylib 217 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
2625r19.21bi 3248 . . . . . . . . 9 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
2719, 26eqeltrd 2833 . . . . . . . 8 ((πœ‘ ∧ π‘š ∈ β„•) β†’ (πΊβ€˜π‘š) ∈ ℝ)
286, 7, 27serfre 14001 . . . . . . 7 (πœ‘ β†’ seq1( + , 𝐺):β„•βŸΆβ„)
291feq1i 6708 . . . . . . 7 (𝑇:β„•βŸΆβ„ ↔ seq1( + , 𝐺):β„•βŸΆβ„)
3028, 29sylibr 233 . . . . . 6 (πœ‘ β†’ 𝑇:β„•βŸΆβ„)
3130frnd 6725 . . . . 5 (πœ‘ β†’ ran 𝑇 βŠ† ℝ)
32 1nn 12227 . . . . . . . 8 1 ∈ β„•
3330fdmd 6728 . . . . . . . 8 (πœ‘ β†’ dom 𝑇 = β„•)
3432, 33eleqtrrid 2840 . . . . . . 7 (πœ‘ β†’ 1 ∈ dom 𝑇)
3534ne0d 4335 . . . . . 6 (πœ‘ β†’ dom 𝑇 β‰  βˆ…)
36 dm0rn0 5924 . . . . . . 7 (dom 𝑇 = βˆ… ↔ ran 𝑇 = βˆ…)
3736necon3bii 2993 . . . . . 6 (dom 𝑇 β‰  βˆ… ↔ ran 𝑇 β‰  βˆ…)
3835, 37sylib 217 . . . . 5 (πœ‘ β†’ ran 𝑇 β‰  βˆ…)
39 ovoliun2.t . . . . . . . . 9 (πœ‘ β†’ 𝑇 ∈ dom ⇝ )
401, 39eqeltrrid 2838 . . . . . . . 8 (πœ‘ β†’ seq1( + , 𝐺) ∈ dom ⇝ )
416, 7, 19, 26, 40isumrecl 15715 . . . . . . 7 (πœ‘ β†’ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
42 elfznn 13534 . . . . . . . . . . . . 13 (π‘š ∈ (1...π‘˜) β†’ π‘š ∈ β„•)
4342adantl 482 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ π‘š ∈ (1...π‘˜)) β†’ π‘š ∈ β„•)
4443, 18syl 17 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ π‘š ∈ (1...π‘˜)) β†’ (πΊβ€˜π‘š) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
45 simpr 485 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ β„•)
4645, 6eleqtrdi 2843 . . . . . . . . . . 11 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ π‘˜ ∈ (β„€β‰₯β€˜1))
47 simpl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ πœ‘)
4847, 42, 26syl2an 596 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ π‘š ∈ (1...π‘˜)) β†’ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
4948recnd 11246 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ β„•) ∧ π‘š ∈ (1...π‘˜)) β†’ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ β„‚)
5044, 46, 49fsumser 15680 . . . . . . . . . 10 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ Ξ£π‘š ∈ (1...π‘˜)(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) = (seq1( + , 𝐺)β€˜π‘˜))
511fveq1i 6892 . . . . . . . . . 10 (π‘‡β€˜π‘˜) = (seq1( + , 𝐺)β€˜π‘˜)
5250, 51eqtr4di 2790 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ Ξ£π‘š ∈ (1...π‘˜)(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) = (π‘‡β€˜π‘˜))
53 fzfid 13942 . . . . . . . . . . 11 (πœ‘ β†’ (1...π‘˜) ∈ Fin)
54 fz1ssnn 13536 . . . . . . . . . . . 12 (1...π‘˜) βŠ† β„•
5554a1i 11 . . . . . . . . . . 11 (πœ‘ β†’ (1...π‘˜) βŠ† β„•)
563ralrimiva 3146 . . . . . . . . . . . . . 14 (πœ‘ β†’ βˆ€π‘› ∈ β„• 𝐴 βŠ† ℝ)
57 nfv 1917 . . . . . . . . . . . . . . 15 β„²π‘š 𝐴 βŠ† ℝ
58 nfcv 2903 . . . . . . . . . . . . . . . 16 Ⅎ𝑛ℝ
5911, 58nfss 3974 . . . . . . . . . . . . . . 15 β„²π‘›β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ
6013sseq1d 4013 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ (𝐴 βŠ† ℝ ↔ β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ))
6157, 59, 60cbvralw 3303 . . . . . . . . . . . . . 14 (βˆ€π‘› ∈ β„• 𝐴 βŠ† ℝ ↔ βˆ€π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
6256, 61sylib 217 . . . . . . . . . . . . 13 (πœ‘ β†’ βˆ€π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
6362r19.21bi 3248 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘š ∈ β„•) β†’ β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
64 ovolge0 25222 . . . . . . . . . . . 12 (β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ β†’ 0 ≀ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
6563, 64syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ π‘š ∈ β„•) β†’ 0 ≀ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
666, 7, 53, 55, 19, 26, 65, 40isumless 15795 . . . . . . . . . 10 (πœ‘ β†’ Ξ£π‘š ∈ (1...π‘˜)(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ≀ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
6766adantr 481 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ Ξ£π‘š ∈ (1...π‘˜)(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ≀ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
6852, 67eqbrtrrd 5172 . . . . . . . 8 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (π‘‡β€˜π‘˜) ≀ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
6968ralrimiva 3146 . . . . . . 7 (πœ‘ β†’ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
70 brralrspcev 5208 . . . . . . 7 ((Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ ∧ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄)) β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ π‘₯)
7141, 69, 70syl2anc 584 . . . . . 6 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ π‘₯)
7230ffnd 6718 . . . . . . . 8 (πœ‘ β†’ 𝑇 Fn β„•)
73 breq1 5151 . . . . . . . . 9 (𝑧 = (π‘‡β€˜π‘˜) β†’ (𝑧 ≀ π‘₯ ↔ (π‘‡β€˜π‘˜) ≀ π‘₯))
7473ralrn 7089 . . . . . . . 8 (𝑇 Fn β„• β†’ (βˆ€π‘§ ∈ ran 𝑇 𝑧 ≀ π‘₯ ↔ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ π‘₯))
7572, 74syl 17 . . . . . . 7 (πœ‘ β†’ (βˆ€π‘§ ∈ ran 𝑇 𝑧 ≀ π‘₯ ↔ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ π‘₯))
7675rexbidv 3178 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝑇 𝑧 ≀ π‘₯ ↔ βˆƒπ‘₯ ∈ ℝ βˆ€π‘˜ ∈ β„• (π‘‡β€˜π‘˜) ≀ π‘₯))
7771, 76mpbird 256 . . . . 5 (πœ‘ β†’ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝑇 𝑧 ≀ π‘₯)
78 supxrre 13310 . . . . 5 ((ran 𝑇 βŠ† ℝ ∧ ran 𝑇 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘§ ∈ ran 𝑇 𝑧 ≀ π‘₯) β†’ sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
7931, 38, 77, 78syl3anc 1371 . . . 4 (πœ‘ β†’ sup(ran 𝑇, ℝ*, < ) = sup(ran 𝑇, ℝ, < ))
806, 1, 7, 19, 26, 65, 71isumsup 15797 . . . 4 (πœ‘ β†’ Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) = sup(ran 𝑇, ℝ, < ))
8179, 80eqtr4d 2775 . . 3 (πœ‘ β†’ sup(ran 𝑇, ℝ*, < ) = Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
829, 12, 14cbvsumi 15647 . . 3 Σ𝑛 ∈ β„• (vol*β€˜π΄) = Ξ£π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄)
8381, 82eqtr4di 2790 . 2 (πœ‘ β†’ sup(ran 𝑇, ℝ*, < ) = Σ𝑛 ∈ β„• (vol*β€˜π΄))
845, 83breqtrd 5174 1 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ Σ𝑛 ∈ β„• (vol*β€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474  β¦‹csb 3893   βŠ† wss 3948  βˆ…c0 4322  βˆͺ ciun 4997   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  supcsup 9437  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115  β„*cxr 11251   < clt 11252   ≀ cle 11253  β„•cn 12216  β„€β‰₯cuz 12826  ...cfz 13488  seqcseq 13970   ⇝ cli 15432  Ξ£csu 15636  vol*covol 25203
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-ioo 13332  df-ico 13334  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-rlim 15437  df-sum 15637  df-ovol 25205
This theorem is referenced by:  ovoliunnul  25248  vitalilem5  25353
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