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Theorem voliun 24941
Description: The Lebesgue measure function is countably additive. (Contributed by Mario Carneiro, 18-Mar-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
voliun.1 𝑆 = seq1( + , 𝐺)
voliun.2 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
Assertion
Ref Expression
voliun ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))

Proof of Theorem voliun
Dummy variables 𝑖 𝑚 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . . . 6 ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐴 ∈ dom vol)
21ralimi 3083 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
32adantr 482 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
4 eqid 2733 . . . . 5 (𝑛 ∈ ℕ ↦ 𝐴) = (𝑛 ∈ ℕ ↦ 𝐴)
54fmpt 7062 . . . 4 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ↔ (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
63, 5sylib 217 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
74fvmpt2 6963 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
87adantrr 716 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
98ralimiaa 3082 . . . . . 6 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
10 disjeq2 5078 . . . . . 6 (∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴 → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
119, 10syl 17 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑛 ∈ ℕ 𝐴))
1211biimpar 479 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
13 nfcv 2904 . . . . 5 𝑖((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)
14 nffvmpt1 6857 . . . . 5 𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)
15 fveq2 6846 . . . . 5 (𝑛 = 𝑖 → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
1613, 14, 15cbvdisj 5084 . . . 4 (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
1712, 16sylib 217 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
18 eqid 2733 . . 3 (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))))
19 eqid 2733 . . 3 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
20 nfcv 2904 . . . 4 𝑚(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
21 nfcv 2904 . . . . 5 𝑛vol
22 nffvmpt1 6857 . . . . 5 𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)
2321, 22nffv 6856 . . . 4 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))
24 2fveq3 6851 . . . 4 (𝑛 = 𝑚 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
2520, 23, 24cbvmpt 5220 . . 3 (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
267fveq2d 6850 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
2726eleq1d 2819 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘𝐴) ∈ ℝ))
2827biimprd 248 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘𝐴) ∈ ℝ → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ))
2928impr 456 . . . . . 6 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
3029ralimiaa 3082 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
3130adantr 482 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
32 nfv 1918 . . . . 5 𝑖(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ
3321, 14nffv 6856 . . . . . 6 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
3433nfel1 2920 . . . . 5 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ
35 2fveq3 6851 . . . . . 6 (𝑛 = 𝑖 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)))
3635eleq1d 2819 . . . . 5 (𝑛 = 𝑖 → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ))
3732, 34, 36cbvralw 3288 . . . 4 (∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
3831, 37sylib 217 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
396, 17, 18, 19, 25, 38voliunlem3 24939 . 2 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ ran (𝑛 ∈ ℕ ↦ 𝐴)) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))), ℝ*, < ))
40 dfiun2g 4994 . . . . 5 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑛 ∈ ℕ 𝐴 = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
413, 40syl 17 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑛 ∈ ℕ 𝐴 = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴})
424rnmpt 5914 . . . . 5 ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
4342unieqi 4882 . . . 4 ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
4441, 43eqtr4di 2791 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑛 ∈ ℕ 𝐴 = ran (𝑛 ∈ ℕ ↦ 𝐴))
4544fveq2d 6850 . 2 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = (vol‘ ran (𝑛 ∈ ℕ ↦ 𝐴)))
46 voliun.1 . . . . 5 𝑆 = seq1( + , 𝐺)
47 voliun.2 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
48 eqid 2733 . . . . . . . 8 ℕ = ℕ
4926adantrr 716 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
5049ralimiaa 3082 . . . . . . . . 9 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
5150adantr 482 . . . . . . . 8 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
52 mpteq12 5201 . . . . . . . 8 ((ℕ = ℕ ∧ ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴)) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
5348, 51, 52sylancr 588 . . . . . . 7 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
5447, 53eqtr4id 2792 . . . . . 6 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
5554seqeq3d 13923 . . . . 5 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
5646, 55eqtrid 2785 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
5756rneqd 5897 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ran 𝑆 = ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
5857supeq1d 9390 . 2 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → sup(ran 𝑆, ℝ*, < ) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))), ℝ*, < ))
5939, 45, 583eqtr4d 2783 1 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran 𝑆, ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  cin 3913   cuni 4869   ciun 4958  Disj wdisj 5074  cmpt 5192  dom cdm 5637  ran crn 5638  wf 6496  cfv 6500  supcsup 9384  cr 11058  1c1 11060   + caddc 11062  *cxr 11196   < clt 11197  cn 12161  seqcseq 13915  vol*covol 24849  volcvol 24850
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-disj 5075  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-of 7621  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-2o 8417  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-dju 9845  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-xadd 13042  df-ioo 13277  df-ico 13279  df-icc 13280  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-xmet 20812  df-met 20813  df-ovol 24851  df-vol 24852
This theorem is referenced by:  volsup  24943  vitalilem4  24998  voliune  32892  voliunsge0lem  44803
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