ovolfs2 - Metamath Proof Explorer

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Theorem ovolfs2 23145

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 23042	. . . . 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 23143	. . . . 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 3796	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5		rexpssxrxp 9963	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
6	4, 5	sstri 3577	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
7		ffvelrn 6265	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sseldi 3566	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ^* × ℝ^*))
9		1st2nd2 7096	. . . . . . . 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 6107	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
12		df-ov 6552	. . . . . 6 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
13	11, 12	syl6eqr 2662	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
14	13	fveq2d 6107	. . . 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 23046	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ((2^nd ‘(𝐹‘𝑛)) − (1^st ‘(𝐹‘𝑛))))
17	3, 14, 16	3eqtr4rd 2655	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘((,)‘(𝐹‘𝑛))))
18	17	mpteq2dva 4672	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
19	15	ovolfsf 23047	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
20	19	feqmptd 6159	. 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 6159	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
23		ioof 12142	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
24	23	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
25	24	ffvelrnda 6267	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ (ℝ^* × ℝ^*)) → ((,)‘𝑥) ∈ 𝒫 ℝ)
26	24	feqmptd 6159	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,) = (𝑥 ∈ (ℝ^* × ℝ^*) ↦ ((,)‘𝑥)))
27		ovolf 23057	. . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
28	27	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol*:𝒫 ℝ⟶(0[,]+∞))
29	28	feqmptd 6159	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol* = (𝑦 ∈ 𝒫 ℝ ↦ (vol*‘𝑦)))
30		fveq2 6103	. . . 4 ⊢ (𝑦 = ((,)‘𝑥) → (vol‘𝑦) = (vol‘((,)‘𝑥)))
31	25, 26, 29, 30	fmptco 6303	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol* ∘ (,)) = (𝑥 ∈ (ℝ^* × ℝ^) ↦ (vol‘((,)‘𝑥))))
32		fveq2 6103	. . . 4 ⊢ (𝑥 = (𝐹‘𝑛) → ((,)‘𝑥) = ((,)‘(𝐹‘𝑛)))
33	32	fveq2d 6107	. . 3 ⊢ (𝑥 = (𝐹‘𝑛) → (vol‘((,)‘𝑥)) = (vol‘((,)‘(𝐹‘𝑛))))
34	8, 22, 31, 33	fmptco 6303	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((vol* ∘ (,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
35	18, 20, 34	3eqtr4d 2654	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 𝒫 cpw 4108 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ℝcr 9814 0cc0 9815 +∞cpnf 9950 ℝ^*cxr 9952 ≤ cle 9954 − cmin 10145 ℕcn 10897 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 abscabs 13822 vol*covol 23038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040 df-vol 23041

This theorem is referenced by: uniioombllem2 23157

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