Theorem uniiccvol 25535

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 25509.) (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 25424	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4		ovolcl 25433	. . 3 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6		eqid 2734	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7		uniioombl.3	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8	6, 7	ovolsf 25427	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
10	9	frnd 6668	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11		icossxr 13346	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
12	10, 11	sstrdi 3944	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
13		supxrcl 13228	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15		ssid 3954	. . 3 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
16	7	ovollb2 25444	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
17	1, 15, 16	sylancl 586	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18		uniioombl.2	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
19	1, 18, 7	uniioovol 25534	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20		ioossicc 13347	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ⊆ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥)))
21		df-ov 7359	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
22		df-ov 7359	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
23	20, 21, 22	3sstr3i 3982	. . . . . . . . . . 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 7024	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
26	25	elin2d 4155	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
27		1st2nd2 7970	. . . . . . . . . . . 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 6836	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
30	28	fveq2d 6836	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
31	24, 29, 30	3sstr4d 3987	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) ⊆ ([,]‘(𝐹‘𝑥)))
32		fvco3 6931	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33		fvco3 6931	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
34	31, 32, 33	3sstr4d 3987	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
35	1, 34	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
36	35	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37		ss2iun 4963	. . . . . 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 13361	. . . . . . . 8 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40		ffn 6660	. . . . . . . 8 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (,) Fn (ℝ* × ℝ*)
42		inss2 4188	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43		rexpssxrxp 11175	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
44	42, 43	sstri 3941	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45		fss 6676	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
46	1, 44, 45	sylancl 586	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
47		fnfco 6697	. . . . . . 7 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
48	41, 46, 47	sylancr 587	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49		fniunfv 7191	. . . . . 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 13362	. . . . . . . 8 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52		ffn 6660	. . . . . . . 8 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
53	51, 52	ax-mp 5	. . . . . . 7 ⊢ [,] Fn (ℝ* × ℝ*)
54		fnfco 6697	. . . . . . 7 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
55	53, 46, 54	sylancr 587	. . . . . 6 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56		fniunfv 7191	. . . . . 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 3986	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
59		ovolss 25440	. . . 4 ⊢ ((∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
60	58, 3, 59	syl2anc 584	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
61	19, 60	eqbrtrrd 5120	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
62	5, 14, 17, 61	xrletrid 13067	1 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4552  ⟨cop 4584  ∪ cuni 4861  ∪ ciun 4944  Disj wdisj 5063   class class class wbr 5096   × cxp 5620  ran crn 5623   ∘ ccom 5626   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  supcsup 9341  ℝcr 11023  0cc0 11024  1c1 11025   + caddc 11027  +∞cpnf 11161  ℝ^*cxr 11163   < clt 11164   ≤ cle 11165   − cmin 11362  ℕcn 12143  (,)cioo 13259  [,)cico 13261  [,]cicc 13262  seqcseq 13922  abscabs 15155  vol*covol 25417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-inf2 9548  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-disj 5064  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-pm 8764  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fi 9312  df-sup 9343  df-inf 9344  df-oi 9413  df-dju 9811  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-q 12860  df-rp 12904  df-xneg 13024  df-xadd 13025  df-xmul 13026  df-ioo 13263  df-ico 13265  df-icc 13266  df-fz 13422  df-fzo 13569  df-fl 13710  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-clim 15409  df-rlim 15410  df-sum 15608  df-rest 17340  df-topgen 17361  df-psmet 21299  df-xmet 21300  df-met 21301  df-bl 21302  df-mopn 21303  df-top 22836  df-topon 22853  df-bases 22888  df-cmp 23329  df-ovol 25419  df-vol 25420

This theorem is referenced by:  mblfinlem2  37798