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Theorem ovolfs2 24935

Description: Alternative expression for the interval length function. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypothesis**
Ref	Expression
ovolfs2.1	⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)

**Assertion**
Ref	Expression
ovolfs2	⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))

Proof of Theorem ovolfs2 Dummy variables 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolfcl 24830	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))))
2		ovolioo 24932	. . . . 5 ⊢ (((1^st ‘(𝐹‘𝑛)) ∈ ℝ ∧ (2^nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1^st ‘(𝐹‘𝑛)) ≤ (2^nd ‘(𝐹‘𝑛))) → (vol*‘((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))) = ((2^nd ‘(𝐹‘𝑛)) − (1^st ‘(𝐹‘𝑛))))
3	1, 2	syl 17	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (vol*‘((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))) = ((2^nd ‘(𝐹‘𝑛)) − (1^st ‘(𝐹‘𝑛))))
4		inss2 4189	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5		rexpssxrxp 11200	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
6	4, 5	sstri 3953	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
7		ffvelcdm 7032	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sselid 3942	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ^* × ℝ^*))
9		1st2nd2 7960	. . . . . . . 8 ⊢ ((𝐹‘𝑛) ∈ (ℝ^* × ℝ^*) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
11	10	fveq2d 6846	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
12		df-ov 7360	. . . . . 6 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
13	11, 12	eqtr4di 2794	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
14	13	fveq2d 6846	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (vol‘((,)‘(𝐹‘𝑛))) = (vol‘((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛)))))
15		ovolfs2.1	. . . . 5 ⊢ 𝐺 = ((abs ∘ − ) ∘ 𝐹)
16	15	ovolfsval 24834	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ((2^nd ‘(𝐹‘𝑛)) − (1^st ‘(𝐹‘𝑛))))
17	3, 14, 16	3eqtr4rd 2787	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘((,)‘(𝐹‘𝑛))))
18	17	mpteq2dva 5205	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
19	15	ovolfsf 24835	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
20	19	feqmptd 6910	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)))
21		id 22	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
22	21	feqmptd 6910	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
23		ioof 13364	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
24	23	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
25	24	ffvelcdmda 7035	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ (ℝ^* × ℝ^*)) → ((,)‘𝑥) ∈ 𝒫 ℝ)
26	24	feqmptd 6910	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,) = (𝑥 ∈ (ℝ^* × ℝ^*) ↦ ((,)‘𝑥)))
27		ovolf 24846	. . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
28	27	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol*:𝒫 ℝ⟶(0[,]+∞))
29	28	feqmptd 6910	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol* = (𝑦 ∈ 𝒫 ℝ ↦ (vol*‘𝑦)))
30		fveq2 6842	. . . 4 ⊢ (𝑦 = ((,)‘𝑥) → (vol‘𝑦) = (vol‘((,)‘𝑥)))
31	25, 26, 29, 30	fmptco 7075	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol* ∘ (,)) = (𝑥 ∈ (ℝ^* × ℝ^) ↦ (vol‘((,)‘𝑥))))
32		2fveq3 6847	. . 3 ⊢ (𝑥 = (𝐹‘𝑛) → (vol‘((,)‘𝑥)) = (vol‘((,)‘(𝐹‘𝑛))))
33	8, 22, 31, 32	fmptco 7075	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((vol* ∘ (,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
34	18, 20, 33	3eqtr4d 2786	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 𝒫 cpw 4560 ⟨cop 4592 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 1^st c1st 7919 2^nd c2nd 7920 ℝcr 11050 0cc0 11051 +∞cpnf 11186 ℝ^*cxr 11188 ≤ cle 11190 − cmin 11385 ℕcn 12153 (,)cioo 13264 [,)cico 13266 [,]cicc 13267 abscabs 15119 vol*covol 24826

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-rest 17304 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296 df-cmp 22738 df-ovol 24828 df-vol 24829

This theorem is referenced by: uniioombllem2 24947

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