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Theorem uniiccvol 25097

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 25071.) (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 24986	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4		ovolcl 24995	. . 3 ⊢ (∪ ran ([,] ∘ 𝐹) ⊆ ℝ → (vol‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ^)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → (vol‘∪ ran ([,] ∘ 𝐹)) ∈ ℝ^)
6		eqid 2733	. . . . . . 7 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
7		uniioombl.3	. . . . . . 7 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
8	6, 7	ovolsf 24989	. . . . . 6 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
9	1, 8	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
10	9	frnd 6726	. . . 4 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
11		icossxr 13409	. . . 4 ⊢ (0[,)+∞) ⊆ ℝ^*
12	10, 11	sstrdi 3995	. . 3 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ^*)
13		supxrcl 13294	. . 3 ⊢ (ran 𝑆 ⊆ ℝ^* → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
14	12, 13	syl 17	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
15		ssid 4005	. . 3 ⊢ ∪ ran ([,] ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹)
16	7	ovollb2 25006	. . 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 25096	. . 3 ⊢ (𝜑 → (vol‘∪ ran ((,) ∘ 𝐹)) = sup(ran 𝑆, ℝ^, < ))
20		ioossicc 13410	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) ⊆ ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥)))
21		df-ov 7412	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐹‘𝑥))(,)(2^nd ‘(𝐹‘𝑥))) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
22		df-ov 7412	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝐹‘𝑥))[,](2^nd ‘(𝐹‘𝑥))) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
23	20, 21, 22	3sstr3i 4025	. . . . . . . . . . 11 ⊢ ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩) ⊆ ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
24	23	a1i 11	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩) ⊆ ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
25		ffvelcdm 7084	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ ( ≤ ∩ (ℝ × ℝ)))
26	25	elin2d 4200	. . . . . . . . . . . 12 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ (ℝ × ℝ))
27		1st2nd2 8014	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑥) ∈ (ℝ × ℝ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = ⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩)
29	28	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) = ((,)‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
30	28	fveq2d 6896	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ([,]‘(𝐹‘𝑥)) = ([,]‘⟨(1^st ‘(𝐹‘𝑥)), (2^nd ‘(𝐹‘𝑥))⟩))
31	24, 29, 30	3sstr4d 4030	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → ((,)‘(𝐹‘𝑥)) ⊆ ([,]‘(𝐹‘𝑥)))
32		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑥) = ((,)‘(𝐹‘𝑥)))
33		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ ℕ) → (([,] ∘ 𝐹)‘𝑥) = ([,]‘(𝐹‘𝑥)))
34	31, 32, 33	3sstr4d 4030	. . . . . . . 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 3147	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ℕ (((,) ∘ 𝐹)‘𝑥) ⊆ (([,] ∘ 𝐹)‘𝑥))
37		ss2iun 5016	. . . . . 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 13424	. . . . . . . 8 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
40		ffn 6718	. . . . . . . 8 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ (,) Fn (ℝ^* × ℝ^*)
42		inss2 4230	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
43		rexpssxrxp 11259	. . . . . . . . 9 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
44	42, 43	sstri 3992	. . . . . . . 8 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
45		fss 6735	. . . . . . . 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 6757	. . . . . . 7 ⊢ (((,) Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝐹) Fn ℕ)
48	41, 46, 47	sylancr 588	. . . . . 6 ⊢ (𝜑 → ((,) ∘ 𝐹) Fn ℕ)
49		fniunfv 7246	. . . . . 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 13425	. . . . . . . 8 ⊢ [,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^
52		ffn 6718	. . . . . . . 8 ⊢ ([,]:(ℝ^* × ℝ^)⟶𝒫 ℝ^ → [,] Fn (ℝ^* × ℝ^*))
53	51, 52	ax-mp 5	. . . . . . 7 ⊢ [,] Fn (ℝ^* × ℝ^*)
54		fnfco 6757	. . . . . . 7 ⊢ (([,] Fn (ℝ^* × ℝ^) ∧ 𝐹:ℕ⟶(ℝ^ × ℝ^*)) → ([,] ∘ 𝐹) Fn ℕ)
55	53, 46, 54	sylancr 588	. . . . . 6 ⊢ (𝜑 → ([,] ∘ 𝐹) Fn ℕ)
56		fniunfv 7246	. . . . . 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 4029	. . . 4 ⊢ (𝜑 → ∪ ran ((,) ∘ 𝐹) ⊆ ∪ ran ([,] ∘ 𝐹))
59		ovolss 25002	. . . 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 5173	. 2 ⊢ (𝜑 → sup(ran 𝑆, ℝ^, < ) ≤ (vol‘∪ ran ([,] ∘ 𝐹)))
62	5, 14, 17, 61	xrletrid 13134	1 ⊢ (𝜑 → (vol‘∪ ran ([,] ∘ 𝐹)) = sup(ran 𝑆, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∩ cin 3948 ⊆ wss 3949 𝒫 cpw 4603 ⟨cop 4635 ∪ cuni 4909 ∪ ciun 4998 Disj wdisj 5114 class class class wbr 5149 × cxp 5675 ran crn 5678 ∘ ccom 5681 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 supcsup 9435 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 (,)cioo 13324 [,)cico 13326 [,]cicc 13327 seqcseq 13966 abscabs 15181 vol*covol 24979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-rest 17368 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 df-cmp 22891 df-ovol 24981 df-vol 24982

This theorem is referenced by: mblfinlem2 36526

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