ovolfs2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolfs2		Structured version Visualization version GIF version

Theorem ovolfs2 25488

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 25383	. . . . 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 25485	. . . . 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 4191	. . . . . . . . . 10 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
5		rexpssxrxp 11179	. . . . . . . . . 10 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
6	4, 5	sstri 3947	. . . . . . . . 9 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
7		ffvelcdm 7019	. . . . . . . . 9 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ( ≤ ∩ (ℝ × ℝ)))
8	6, 7	sselid 3935	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (ℝ^* × ℝ^*))
9		1st2nd2 7970	. . . . . . . 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 6830	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩))
12		df-ov 7356	. . . . . 6 ⊢ ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))) = ((,)‘⟨(1^st ‘(𝐹‘𝑛)), (2^nd ‘(𝐹‘𝑛))⟩)
13	11, 12	eqtr4di 2782	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((,)‘(𝐹‘𝑛)) = ((1^st ‘(𝐹‘𝑛))(,)(2^nd ‘(𝐹‘𝑛))))
14	13	fveq2d 6830	. . . 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 25387	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = ((2^nd ‘(𝐹‘𝑛)) − (1^st ‘(𝐹‘𝑛))))
17	3, 14, 16	3eqtr4rd 2775	. . 3 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = (vol*‘((,)‘(𝐹‘𝑛))))
18	17	mpteq2dva 5188	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (𝑛 ∈ ℕ ↦ (𝐺‘𝑛)) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
19	15	ovolfsf 25388	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺:ℕ⟶(0[,)+∞))
20	19	feqmptd 6895	. 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 6895	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐹 = (𝑛 ∈ ℕ ↦ (𝐹‘𝑛)))
23		ioof 13368	. . . . . 6 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
24	23	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ)
25	24	ffvelcdmda 7022	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝑥 ∈ (ℝ^* × ℝ^*)) → ((,)‘𝑥) ∈ 𝒫 ℝ)
26	24	feqmptd 6895	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (,) = (𝑥 ∈ (ℝ^* × ℝ^*) ↦ ((,)‘𝑥)))
27		ovolf 25399	. . . . . 6 ⊢ vol*:𝒫 ℝ⟶(0[,]+∞)
28	27	a1i 11	. . . . 5 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol*:𝒫 ℝ⟶(0[,]+∞))
29	28	feqmptd 6895	. . . 4 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → vol* = (𝑦 ∈ 𝒫 ℝ ↦ (vol*‘𝑦)))
30		fveq2 6826	. . . 4 ⊢ (𝑦 = ((,)‘𝑥) → (vol‘𝑦) = (vol‘((,)‘𝑥)))
31	25, 26, 29, 30	fmptco 7067	. . 3 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol* ∘ (,)) = (𝑥 ∈ (ℝ^* × ℝ^) ↦ (vol‘((,)‘𝑥))))
32		2fveq3 6831	. . 3 ⊢ (𝑥 = (𝐹‘𝑛) → (vol‘((,)‘𝑥)) = (vol‘((,)‘(𝐹‘𝑛))))
33	8, 22, 31, 32	fmptco 7067	. 2 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((vol* ∘ (,)) ∘ 𝐹) = (𝑛 ∈ ℕ ↦ (vol*‘((,)‘(𝐹‘𝑛)))))
34	18, 20, 33	3eqtr4d 2774	1 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝐺 = ((vol* ∘ (,)) ∘ 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∩ cin 3904 𝒫 cpw 4553 ⟨cop 4585 class class class wbr 5095 ↦ cmpt 5176 × cxp 5621 ∘ ccom 5627 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 ℝcr 11027 0cc0 11028 +∞cpnf 11165 ℝ^*cxr 11167 ≤ cle 11169 − cmin 11365 ℕcn 12146 (,)cioo 13266 [,)cico 13268 [,]cicc 13269 abscabs 15159 vol*covol 25379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 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 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 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 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612 df-rest 17344 df-topgen 17365 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-top 22797 df-topon 22814 df-bases 22849 df-cmp 23290 df-ovol 25381 df-vol 25382

This theorem is referenced by: uniioombllem2 25500

W3C validator