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Theorem voliun 24147
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 485 . . . . . 6 ((𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → 𝐴 ∈ dom vol)
21ralimi 3158 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
32adantr 483 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ 𝐴 ∈ dom vol)
4 eqid 2819 . . . . 5 (𝑛 ∈ ℕ ↦ 𝐴) = (𝑛 ∈ ℕ ↦ 𝐴)
54fmpt 6867 . . . 4 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ↔ (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
63, 5sylib 220 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ 𝐴):ℕ⟶dom vol)
74fvmpt2 6772 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
87adantrr 715 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
98ralimiaa 3157 . . . . . 6 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = 𝐴)
10 disjeq2 5026 . . . . . 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 480 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
13 nfcv 2975 . . . . 5 𝑖((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)
14 nffvmpt1 6674 . . . . 5 𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)
15 fveq2 6663 . . . . 5 (𝑛 = 𝑖 → ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) = ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
1613, 14, 15cbvdisj 5032 . . . 4 (Disj 𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛) ↔ Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
1712, 16sylib 220 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → Disj 𝑖 ∈ ℕ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
18 eqid 2819 . . 3 (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))) = (𝑚 ∈ ℕ ↦ (vol*‘(𝑥 ∩ ((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))))
19 eqid 2819 . . 3 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
20 nfcv 2975 . . . 4 𝑚(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))
21 nfcv 2975 . . . . 5 𝑛vol
22 nffvmpt1 6674 . . . . 5 𝑛((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)
2321, 22nffv 6673 . . . 4 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚))
24 2fveq3 6668 . . . 4 (𝑛 = 𝑚 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
2520, 23, 24cbvmpt 5158 . . 3 (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑚)))
267fveq2d 6667 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
2726eleq1d 2895 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘𝐴) ∈ ℝ))
2827biimprd 250 . . . . . . 7 ((𝑛 ∈ ℕ ∧ 𝐴 ∈ dom vol) → ((vol‘𝐴) ∈ ℝ → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ))
2928impr 457 . . . . . 6 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
3029ralimiaa 3157 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
3130adantr 483 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ)
32 nfv 1909 . . . . 5 𝑖(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ
3321, 14nffv 6673 . . . . . 6 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖))
3433nfel1 2992 . . . . 5 𝑛(vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ
35 2fveq3 6668 . . . . . 6 (𝑛 = 𝑖 → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)))
3635eleq1d 2895 . . . . 5 (𝑛 = 𝑖 → ((vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ))
3732, 34, 36cbvralw 3440 . . . 4 (∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) ∈ ℝ ↔ ∀𝑖 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑖)) ∈ ℝ)
3831, 37sylib 220 . . 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 24145 . 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 4946 . . . . 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 5820 . . . . 5 ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
4342unieqi 4839 . . . 4 ran (𝑛 ∈ ℕ ↦ 𝐴) = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = 𝐴}
4441, 43syl6eqr 2872 . . 3 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑛 ∈ ℕ 𝐴 = ran (𝑛 ∈ ℕ ↦ 𝐴))
4544fveq2d 6667 . 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 eqid 2819 . . . . . . . 8 ℕ = ℕ
4826adantrr 715 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ)) → (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
4948ralimiaa 3157 . . . . . . . . 9 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
5049adantr 483 . . . . . . . 8 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴))
51 mpteq12 5144 . . . . . . . 8 ((ℕ = ℕ ∧ ∀𝑛 ∈ ℕ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)) = (vol‘𝐴)) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
5247, 50, 51sylancr 589 . . . . . . 7 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))) = (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
53 voliun.2 . . . . . . 7 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
5452, 53syl6reqr 2873 . . . . . 6 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛))))
5554seqeq3d 13369 . . . . 5 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
5646, 55syl5eq 2866 . . . 4 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘((𝑛 ∈ ℕ ↦ 𝐴)‘𝑛)))))
5756rneqd 5801 . . 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 8902 . 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 2864 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 208  wa 398   = wceq 1531  wcel 2108  {cab 2797  wral 3136  wrex 3137  cin 3933   cuni 4830   ciun 4910  Disj wdisj 5022  cmpt 5137  dom cdm 5548  ran crn 5549  wf 6344  cfv 6348  supcsup 8896  cr 10528  1c1 10530   + caddc 10532  *cxr 10666   < clt 10667  cn 11630  seqcseq 13361  vol*covol 24055  volcvol 24056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cc 9849  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xadd 12500  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-rlim 14838  df-sum 15035  df-xmet 20530  df-met 20531  df-ovol 24057  df-vol 24058
This theorem is referenced by:  volsup  24149  vitalilem4  24204  voliune  31481  voliunsge0lem  42744
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