Theorem uniiccvol 24184

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 24158.) (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 24073	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4		ovolcl 24082	. . 3 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6		eqid 2798	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7		uniioombl.3	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8	6, 7	ovolsf 24076	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
10	9	frnd 6494	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11		icossxr 12810	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
12	10, 11	sstrdi 3927	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
13		supxrcl 12696	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15		ssid 3937	. . 3 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
16	7	ovollb2 24093	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
17	1, 15, 16	sylancl 589	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18		uniioombl.2	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
19	1, 18, 7	uniioovol 24183	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20		ioossicc 12811	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ⊆ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥)))
21		df-ov 7138	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
22		df-ov 7138	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
23	20, 21, 22	3sstr3i 3957	. . . . . . . . . . 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		ffvelrn 6826	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
26	25	elin2d 4126	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
27		1st2nd2 7710	. . . . . . . . . . . 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 6649	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
30	28	fveq2d 6649	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
31	24, 29, 30	3sstr4d 3962	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) ⊆ ([,]‘(𝐹‘𝑥)))
32		fvco3 6737	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33		fvco3 6737	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
34	31, 32, 33	3sstr4d 3962	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
35	1, 34	sylan 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
36	35	ralrimiva 3149	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37		ss2iun 4899	. . . . . 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 12825	. . . . . . . 8 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40		ffn 6487	. . . . . . . 8 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (,) Fn (ℝ* × ℝ*)
42		inss2 4156	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43		rexpssxrxp 10675	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
44	42, 43	sstri 3924	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45		fss 6501	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
46	1, 44, 45	sylancl 589	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
47		fnfco 6517	. . . . . . 7 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
48	41, 46, 47	sylancr 590	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49		fniunfv 6984	. . . . . 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 12826	. . . . . . . 8 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52		ffn 6487	. . . . . . . 8 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
53	51, 52	ax-mp 5	. . . . . . 7 ⊢ [,] Fn (ℝ* × ℝ*)
54		fnfco 6517	. . . . . . 7 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
55	53, 46, 54	sylancr 590	. . . . . 6 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56		fniunfv 6984	. . . . . 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 3961	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
59		ovolss 24089	. . . 4 ⊢ ((∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
60	58, 3, 59	syl2anc 587	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
61	19, 60	eqbrtrrd 5054	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
62	5, 14, 17, 61	xrletrid 12536	1 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ⟨cop 4531  ∪ cuni 4800  ∪ ciun 4881  Disj wdisj 4995   class class class wbr 5030   × cxp 5517  ran crn 5520   ∘ ccom 5523   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  1^st c1st 7669  2^nd c2nd 7670  supcsup 8888  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529  +∞cpnf 10661  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  ℕcn 11625  (,)cioo 12726  [,)cico 12728  [,]cicc 12729  seqcseq 13364  abscabs 14585  vol*covol 24066

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-disj 4996  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fi 8859  df-sup 8890  df-inf 8891  df-oi 8958  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12886  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13426  df-hash 13687  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-rest 16688  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cmp 21992  df-ovol 24068  df-vol 24069

This theorem is referenced by:  mblfinlem2  35095