Theorem uniiccvol 25739

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 25713.) (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 25628	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4		ovolcl 25637	. . 3 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ ℝ → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
5	3, 4	syl 18	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ*)
6		eqid 2763	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7		uniioombl.3	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8	6, 7	ovolsf 25631	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
9	1, 8	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
10	9	frnd 6714	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11		icossxr 13454	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ*
12	10, 11	sstrdi 3949	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
13		supxrcl 13336	. . 3 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
14	12, 13	syl 18	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
15		ssid 3959	. . 3 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
16	7	ovollb2 25648	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
17	1, 15, 16	sylancl 597	. 2 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) ≤ sup(ran 𝑆, ℝ*, < ))
18		uniioombl.2	. . . 4 ⊢ (𝜑 → Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)))
19	1, 18, 7	uniioovol 25738	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))
20		ioossicc 13455	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) ⊆ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥)))
21		df-ov 7413	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))(,)(2nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
22		df-ov 7413	. . . . . . . . . . . 12 ⊢ ((1st ‘(𝐹‘𝑥))[,](2nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
23	20, 21, 22	3sstr3i 3987	. . . . . . . . . . 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 7076	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
26	25	elin2d 4158	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
27		1st2nd2 8021	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
28	26, 27	syl 18	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩)
29	28	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
30	28	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1st ‘(𝐹‘𝑥)), (2nd ‘(𝐹‘𝑥))⟩))
31	24, 29, 30	3sstr4d 3992	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) ⊆ ([,]‘(𝐹‘𝑥)))
32		fvco3 6981	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33		fvco3 6981	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
34	31, 32, 33	3sstr4d 3992	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
35	1, 34	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
36	35	ralrimiva 3157	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37		ss2iun 4975	. . . . . 6 ⊢ (∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥) → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
38	36, 37	syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥))
39		ioof 13469	. . . . . . . 8 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
40		ffn 6705	. . . . . . . 8 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (,) Fn (ℝ* × ℝ*)
42		inss2 4190	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43		rexpssxrxp 11249	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
44	42, 43	sstri 3946	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)
45		fss 6722	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* × ℝ*))
46	1, 44, 45	sylancl 597	. . . . . . 7 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ* × ℝ*))
47		fnfco 6743	. . . . . . 7 ⊢ (((,) Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ((,) ∘ 𝐹) Fn ℕ)
48	41, 46, 47	sylancr 598	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49		fniunfv 7245	. . . . . 6 ⊢ (((,) ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
50	48, 49	syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) = ∪ ran ((,) ∘ 𝐹))
51		iccf 13470	. . . . . . . 8 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ*
52		ffn 6705	. . . . . . . 8 ⊢ ([,]:(ℝ* × ℝ*)⟶𝒫 ℝ* → [,] Fn (ℝ* × ℝ*))
53	51, 52	ax-mp 5	. . . . . . 7 ⊢ [,] Fn (ℝ* × ℝ*)
54		fnfco 6743	. . . . . . 7 ⊢ (([,] Fn (ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* × ℝ*)) → ([,] ∘ 𝐹) Fn ℕ)
55	53, 46, 54	sylancr 598	. . . . . 6 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56		fniunfv 7245	. . . . . 6 ⊢ (([,] ∘ 𝐹) Fn ℕ → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
57	55, 56	syl 18	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ ℕ (([,] ∘ 𝐹)‘𝑥) = ∪ ran ([,] ∘ 𝐹))
58	38, 50, 57	3sstr3d 3991	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
59		ovolss 25644	. . . 4 ⊢ ((∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹) ∧ ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
60	58, 3, 59	syl2anc 595	. . 3 ⊢ (𝜑 → (vol*‘∪ ran ((,) ∘ 𝐹)) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
61	19, 60	eqbrtrrd 5135	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤ (vol*‘∪ ran ([,] ∘ 𝐹)))
62	5, 14, 17, 61	xrletrid 13175	1 ⊢ (𝜑 → (vol*‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  ⟨cop 4595  ∪ cuni 4872  ∪ ciun 4956  Disj wdisj 5076   class class class wbr 5109   × cxp 5659  ran crn 5662   ∘ ccom 5665   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  supcsup 9396  ℝcr 11094  0cc0 11095  1c1 11096   + caddc 11098  +∞cpnf 11235  ℝ^*cxr 11237   < clt 11238   ≤ cle 11239   − cmin 11436  ℕcn 12228  (,)cioo 13367  [,)cico 13369  [,]cicc 13370  seqcseq 14033  abscabs 15281  vol*covol 25621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-disj 5077  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9367  df-sup 9398  df-inf 9399  df-oi 9468  df-dju 9883  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-q 12968  df-rp 13012  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13371  df-ico 13373  df-icc 13374  df-fz 13531  df-fzo 13679  df-fl 13821  df-seq 14034  df-exp 14094  df-hash 14363  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-rlim 15536  df-sum 15734  df-rest 17470  df-topgen 17491  df-psmet 21514  df-xmet 21515  df-met 21516  df-bl 21517  df-mopn 21518  df-top 23051  df-topon 23068  df-bases 23103  df-cmp 23544  df-ovol 25623  df-vol 25624

This theorem is referenced by:  mblfinlem2  38309