Theorem uniiccvol 25554

Description: An almost-disjoint union of closed intervals (disjoint interiors) has volume equal to the sum of the volume of the intervals. (This proof does not use countable choice, unlike voliun 25528.) (Contributed by Mario Carneiro, 25-Mar-2015.)

Hypotheses

Ref	Expression
uniioombl.1	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
uniioombl.2	⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
uniioombl.3	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))

Assertion

Ref	Expression
uniiccvol	⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Distinct variable groups:   𝑥,𝐹   𝜑,𝑥

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem uniiccvol

Step	Hyp	Ref	Expression
1		uniioombl.1	. . . 4 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
2		ovolficcss 25443	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4		ovolcl 25452	. . 3 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6		eqid 2737	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7		uniioombl.3	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8	6, 7	ovolsf 25446	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
10	9	frnd 6680	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11		icossxr 13362	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
12	10, 11	sstrdi 3948	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
13		supxrcl 13244	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15		ssid 3958	. . 3 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
16	7	ovollb2 25463	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
17	1, 15, 16	sylancl 587	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18		uniioombl.2	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
19	1, 18, 7	uniioovol 25553	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20		ioossicc 13363	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ⊆ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥)))
21		df-ov 7373	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
22		df-ov 7373	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
23	20, 21, 22	3sstr3i 3986	. . . . . . . . . . 11 ⊢ ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩) ⊆ ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
25		ffvelcdm 7037	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
26	25	elin2d 4159	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
27		1st2nd2 7984	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
29	28	fveq2d 6848	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
30	28	fveq2d 6848	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
31	24, 29, 30	3sstr4d 3991	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) ⊆ ([,]‘(𝐹‘𝑥)))
32		fvco3 6943	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33		fvco3 6943	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
34	31, 32, 33	3sstr4d 3991	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
35	1, 34	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
36	35	ralrimiva 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37		ss2iun 4967	. . . . . 6 ⊢ (∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥) → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
38	36, 37	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
39		ioof 13377	. . . . . . . 8 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40		ffn 6672	. . . . . . . 8 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (,) Fn (ℝ* × ℝ*)
42		inss2 4192	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43		rexpssxrxp 11191	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
44	42, 43	sstri 3945	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45		fss 6688	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
46	1, 44, 45	sylancl 587	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
47		fnfco 6709	. . . . . . 7 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
48	41, 46, 47	sylancr 588	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49		fniunfv 7205	. . . . . 6 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
51		iccf 13378	. . . . . . . 8 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52		ffn 6672	. . . . . . . 8 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
53	51, 52	ax-mp 5	. . . . . . 7 ⊢ [,] Fn (ℝ* × ℝ*)
54		fnfco 6709	. . . . . . 7 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
55	53, 46, 54	sylancr 588	. . . . . 6 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56		fniunfv 7205	. . . . . 6 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
57	55, 56	syl 17	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
58	38, 50, 57	3sstr3d 3990	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
59		ovolss 25459	. . . 4 ⊢ ((∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
60	58, 3, 59	syl2anc 585	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
61	19, 60	eqbrtrrd 5124	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
62	5, 14, 17, 61	xrletrid 13083	1 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ⟨cop 4588  ∪ cuni 4865  ∪ ciun 4948  Disj wdisj 5067   class class class wbr 5100   × cxp 5632  ran crn 5635   ∘ ccom 5638   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370  1^st c1st 7943  2^nd c2nd 7944  supcsup 9357  ℝcr 11039  0cc0 11040  1c1 11041   + caddc 11043  +∞cpnf 11177  ℝ^*cxr 11179   < clt 11180   ≤ cle 11181   − cmin 11378  ℕcn 12159  (,)cioo 13275  [,)cico 13277  [,]cicc 13278  seqcseq 13938  abscabs 15171  vol*covol 25436

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-map 8779  df-pm 8780  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fi 9328  df-sup 9359  df-inf 9360  df-oi 9429  df-dju 9827  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-q 12876  df-rp 12920  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-ioo 13279  df-ico 13281  df-icc 13282  df-fz 13438  df-fzo 13585  df-fl 13726  df-seq 13939  df-exp 13999  df-hash 14268  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-clim 15425  df-rlim 15426  df-sum 15624  df-rest 17356  df-topgen 17377  df-psmet 21318  df-xmet 21319  df-met 21320  df-bl 21321  df-mopn 21322  df-top 22855  df-topon 22872  df-bases 22907  df-cmp 23348  df-ovol 25438  df-vol 25439

This theorem is referenced by:  mblfinlem2  37938