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Theorem ovoliun 24892
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 24872, 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 ∈ 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)
Assertion
Ref Expression
ovoliun (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ))
Distinct variable group:   πœ‘,𝑛
Allowed substitution hints:   𝐴(𝑛)   𝑇(𝑛)   𝐺(𝑛)

Proof of Theorem ovoliun
Dummy variables π‘˜ π‘š π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mnfxr 11220 . . . . . 6 -∞ ∈ ℝ*
21a1i 11 . . . . 5 (πœ‘ β†’ -∞ ∈ ℝ*)
3 nnuz 12814 . . . . . . . . 9 β„• = (β„€β‰₯β€˜1)
4 1zzd 12542 . . . . . . . . 9 (πœ‘ β†’ 1 ∈ β„€)
5 ovoliun.v . . . . . . . . . . 11 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (vol*β€˜π΄) ∈ ℝ)
6 ovoliun.g . . . . . . . . . . 11 𝐺 = (𝑛 ∈ β„• ↦ (vol*β€˜π΄))
75, 6fmptd 7066 . . . . . . . . . 10 (πœ‘ β†’ 𝐺:β„•βŸΆβ„)
87ffvelcdmda 7039 . . . . . . . . 9 ((πœ‘ ∧ π‘˜ ∈ β„•) β†’ (πΊβ€˜π‘˜) ∈ ℝ)
93, 4, 8serfre 13946 . . . . . . . 8 (πœ‘ β†’ seq1( + , 𝐺):β„•βŸΆβ„)
10 ovoliun.t . . . . . . . . 9 𝑇 = seq1( + , 𝐺)
1110feq1i 6663 . . . . . . . 8 (𝑇:β„•βŸΆβ„ ↔ seq1( + , 𝐺):β„•βŸΆβ„)
129, 11sylibr 233 . . . . . . 7 (πœ‘ β†’ 𝑇:β„•βŸΆβ„)
13 1nn 12172 . . . . . . 7 1 ∈ β„•
14 ffvelcdm 7036 . . . . . . 7 ((𝑇:β„•βŸΆβ„ ∧ 1 ∈ β„•) β†’ (π‘‡β€˜1) ∈ ℝ)
1512, 13, 14sylancl 587 . . . . . 6 (πœ‘ β†’ (π‘‡β€˜1) ∈ ℝ)
1615rexrd 11213 . . . . 5 (πœ‘ β†’ (π‘‡β€˜1) ∈ ℝ*)
1712frnd 6680 . . . . . . 7 (πœ‘ β†’ ran 𝑇 βŠ† ℝ)
18 ressxr 11207 . . . . . . 7 ℝ βŠ† ℝ*
1917, 18sstrdi 3960 . . . . . 6 (πœ‘ β†’ ran 𝑇 βŠ† ℝ*)
20 supxrcl 13243 . . . . . 6 (ran 𝑇 βŠ† ℝ* β†’ sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2119, 20syl 17 . . . . 5 (πœ‘ β†’ sup(ran 𝑇, ℝ*, < ) ∈ ℝ*)
2215mnfltd 13053 . . . . 5 (πœ‘ β†’ -∞ < (π‘‡β€˜1))
2312ffnd 6673 . . . . . . 7 (πœ‘ β†’ 𝑇 Fn β„•)
24 fnfvelrn 7035 . . . . . . 7 ((𝑇 Fn β„• ∧ 1 ∈ β„•) β†’ (π‘‡β€˜1) ∈ ran 𝑇)
2523, 13, 24sylancl 587 . . . . . 6 (πœ‘ β†’ (π‘‡β€˜1) ∈ ran 𝑇)
26 supxrub 13252 . . . . . 6 ((ran 𝑇 βŠ† ℝ* ∧ (π‘‡β€˜1) ∈ ran 𝑇) β†’ (π‘‡β€˜1) ≀ sup(ran 𝑇, ℝ*, < ))
2719, 25, 26syl2anc 585 . . . . 5 (πœ‘ β†’ (π‘‡β€˜1) ≀ sup(ran 𝑇, ℝ*, < ))
282, 16, 21, 22, 27xrltletrd 13089 . . . 4 (πœ‘ β†’ -∞ < sup(ran 𝑇, ℝ*, < ))
29 xrrebnd 13096 . . . . 5 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* β†’ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3021, 29syl 17 . . . 4 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ (-∞ < sup(ran 𝑇, ℝ*, < ) ∧ sup(ran 𝑇, ℝ*, < ) < +∞)))
3128, 30mpbirand 706 . . 3 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑇, ℝ*, < ) < +∞))
32 nfcv 2904 . . . . . . . . 9 β„²π‘šπ΄
33 nfcsb1v 3884 . . . . . . . . 9 β„²π‘›β¦‹π‘š / π‘›β¦Œπ΄
34 csbeq1a 3873 . . . . . . . . 9 (𝑛 = π‘š β†’ 𝐴 = β¦‹π‘š / π‘›β¦Œπ΄)
3532, 33, 34cbviun 5000 . . . . . . . 8 βˆͺ 𝑛 ∈ β„• 𝐴 = βˆͺ π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄
3635fveq2i 6849 . . . . . . 7 (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) = (vol*β€˜βˆͺ π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄)
37 nfcv 2904 . . . . . . . . . 10 β„²π‘š(vol*β€˜π΄)
38 nfcv 2904 . . . . . . . . . . 11 Ⅎ𝑛vol*
3938, 33nffv 6856 . . . . . . . . . 10 Ⅎ𝑛(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄)
4034fveq2d 6850 . . . . . . . . . 10 (𝑛 = π‘š β†’ (vol*β€˜π΄) = (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
4137, 39, 40cbvmpt 5220 . . . . . . . . 9 (𝑛 ∈ β„• ↦ (vol*β€˜π΄)) = (π‘š ∈ β„• ↦ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
426, 41eqtri 2761 . . . . . . . 8 𝐺 = (π‘š ∈ β„• ↦ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄))
43 ovoliun.a . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝐴 βŠ† ℝ)
4443ralrimiva 3140 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘› ∈ β„• 𝐴 βŠ† ℝ)
45 nfv 1918 . . . . . . . . . . . 12 β„²π‘š 𝐴 βŠ† ℝ
46 nfcv 2904 . . . . . . . . . . . . 13 Ⅎ𝑛ℝ
4733, 46nfss 3940 . . . . . . . . . . . 12 β„²π‘›β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ
4834sseq1d 3979 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ (𝐴 βŠ† ℝ ↔ β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ))
4945, 47, 48cbvralw 3288 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• 𝐴 βŠ† ℝ ↔ βˆ€π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
5044, 49sylib 217 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
5150ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ βˆ€π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
5251r19.21bi 3233 . . . . . . . 8 ((((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ β¦‹π‘š / π‘›β¦Œπ΄ βŠ† ℝ)
535ralrimiva 3140 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘› ∈ β„• (vol*β€˜π΄) ∈ ℝ)
5437nfel1 2920 . . . . . . . . . . . 12 β„²π‘š(vol*β€˜π΄) ∈ ℝ
5539nfel1 2920 . . . . . . . . . . . 12 Ⅎ𝑛(vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ
5640eleq1d 2819 . . . . . . . . . . . 12 (𝑛 = π‘š β†’ ((vol*β€˜π΄) ∈ ℝ ↔ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ))
5754, 55, 56cbvralw 3288 . . . . . . . . . . 11 (βˆ€π‘› ∈ β„• (vol*β€˜π΄) ∈ ℝ ↔ βˆ€π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
5853, 57sylib 217 . . . . . . . . . 10 (πœ‘ β†’ βˆ€π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
5958ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ βˆ€π‘š ∈ β„• (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
6059r19.21bi 3233 . . . . . . . 8 ((((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) ∧ π‘š ∈ β„•) β†’ (vol*β€˜β¦‹π‘š / π‘›β¦Œπ΄) ∈ ℝ)
61 simplr 768 . . . . . . . 8 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ sup(ran 𝑇, ℝ*, < ) ∈ ℝ)
62 simpr 486 . . . . . . . 8 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ π‘₯ ∈ ℝ+)
6310, 42, 52, 60, 61, 62ovoliunlem3 24891 . . . . . . 7 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ (vol*β€˜βˆͺ π‘š ∈ β„• β¦‹π‘š / π‘›β¦Œπ΄) ≀ (sup(ran 𝑇, ℝ*, < ) + π‘₯))
6436, 63eqbrtrid 5144 . . . . . 6 (((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) ∧ π‘₯ ∈ ℝ+) β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ (sup(ran 𝑇, ℝ*, < ) + π‘₯))
6564ralrimiva 3140 . . . . 5 ((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) β†’ βˆ€π‘₯ ∈ ℝ+ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ (sup(ran 𝑇, ℝ*, < ) + π‘₯))
66 iunss 5009 . . . . . . . 8 (βˆͺ 𝑛 ∈ β„• 𝐴 βŠ† ℝ ↔ βˆ€π‘› ∈ β„• 𝐴 βŠ† ℝ)
6744, 66sylibr 233 . . . . . . 7 (πœ‘ β†’ βˆͺ 𝑛 ∈ β„• 𝐴 βŠ† ℝ)
68 ovolcl 24865 . . . . . . 7 (βˆͺ 𝑛 ∈ β„• 𝐴 βŠ† ℝ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ∈ ℝ*)
6967, 68syl 17 . . . . . 6 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ∈ ℝ*)
70 xralrple 13133 . . . . . 6 (((vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ∈ ℝ* ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) β†’ ((vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ) ↔ βˆ€π‘₯ ∈ ℝ+ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ (sup(ran 𝑇, ℝ*, < ) + π‘₯)))
7169, 70sylan 581 . . . . 5 ((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) β†’ ((vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ) ↔ βˆ€π‘₯ ∈ ℝ+ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ (sup(ran 𝑇, ℝ*, < ) + π‘₯)))
7265, 71mpbird 257 . . . 4 ((πœ‘ ∧ sup(ran 𝑇, ℝ*, < ) ∈ ℝ) β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ))
7372ex 414 . . 3 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) ∈ ℝ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < )))
7431, 73sylbird 260 . 2 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) < +∞ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < )))
75 nltpnft 13092 . . . 4 (sup(ran 𝑇, ℝ*, < ) ∈ ℝ* β†’ (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ Β¬ sup(ran 𝑇, ℝ*, < ) < +∞))
7621, 75syl 17 . . 3 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) = +∞ ↔ Β¬ sup(ran 𝑇, ℝ*, < ) < +∞))
77 pnfge 13059 . . . . 5 ((vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ∈ ℝ* β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ +∞)
7869, 77syl 17 . . . 4 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ +∞)
79 breq2 5113 . . . 4 (sup(ran 𝑇, ℝ*, < ) = +∞ β†’ ((vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ) ↔ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ +∞))
8078, 79syl5ibrcom 247 . . 3 (πœ‘ β†’ (sup(ran 𝑇, ℝ*, < ) = +∞ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < )))
8176, 80sylbird 260 . 2 (πœ‘ β†’ (Β¬ sup(ran 𝑇, ℝ*, < ) < +∞ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < )))
8274, 81pm2.61d 179 1 (πœ‘ β†’ (vol*β€˜βˆͺ 𝑛 ∈ β„• 𝐴) ≀ sup(ran 𝑇, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  β¦‹csb 3859   βŠ† wss 3914  βˆͺ ciun 4958   class class class wbr 5109   ↦ cmpt 5192  ran crn 5638   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  supcsup 9384  β„cr 11058  1c1 11060   + caddc 11062  +∞cpnf 11194  -∞cmnf 11195  β„*cxr 11196   < clt 11197   ≀ cle 11198  β„•cn 12161  β„+crp 12923  seqcseq 13915  vol*covol 24849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cc 10379  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-pm 8774  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-oi 9454  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-ioo 13277  df-ico 13279  df-fz 13434  df-fzo 13577  df-fl 13706  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-rlim 15380  df-sum 15580  df-ovol 24851
This theorem is referenced by:  ovoliun2  24893  voliunlem2  24938  voliunlem3  24939  ex-ovoliunnfl  36171
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