ovollb2 - Metamath Proof Explorer

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Theorem ovollb2 25411

Description: It is often more convenient to do calculations with *closed* coverings rather than open ones; here we show that it makes no difference (compare ovollb 25401). (Contributed by Mario Carneiro, 24-Mar-2015.)

**Hypothesis**
Ref	Expression
ovollb2.1	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))

**Assertion**
Ref	Expression
ovollb2	⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))

Proof of Theorem ovollb2 Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
2		ovolficcss 25391	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	2	adantr 480	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4	1, 3	sstrd 3988	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5		ovolcl 25400	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) ∈ ℝ^)
6	4, 5	syl 17	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol‘𝐴) ∈ ℝ^)
7	6	adantr 480	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) = +∞) → (vol‘𝐴) ∈ ℝ^*)
8		pnfge 13136	. . . 4 ⊢ ((vol‘𝐴) ∈ ℝ^ → (vol*‘𝐴) ≤ +∞)
9	7, 8	syl 17	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) = +∞) → (vol‘𝐴) ≤ +∞)
10		simpr 484	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) = +∞) → sup(ran 𝑆, ℝ^, < ) = +∞)
11	9, 10	breqtrrd 5170	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) = +∞) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^*, < ))
12		eqid 2728	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13		ovollb2.1	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
14	12, 13	ovolsf 25394	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝑆:ℕ⟶(0[,)+∞))
16	15	frnd 6724	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17		rge0ssre 13459	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
18	16, 17	sstrdi 3990	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19		1nn 12247	. . . . . . . 8 ⊢ 1 ∈ ℕ
20	15	fdmd 6727	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
21	19, 20	eleqtrrid 2836	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
22	21	ne0d 4331	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23		dm0rn0 5921	. . . . . . 7 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
24	23	necon3bii 2989	. . . . . 6 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
25	22, 24	sylib 217	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26		supxrre2 13336	. . . . 5 ⊢ ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅) → (sup(ran 𝑆, ℝ^, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ^, < ) ≠ +∞))
27	18, 25, 26	syl2anc 583	. . . 4 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (sup(ran 𝑆, ℝ^, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ^, < ) ≠ +∞))
28	27	biimpar 477	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ≠ +∞) → sup(ran 𝑆, ℝ^, < ) ∈ ℝ)
29		2fveq3 6896	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (1^st ‘(𝐹‘𝑚)) = (1^st ‘(𝐹‘𝑛)))
30		oveq2 7422	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
31	30	oveq2d 7430	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
32	29, 31	oveq12d 7432	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((1^st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1^st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33		2fveq3 6896	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (2^nd ‘(𝐹‘𝑚)) = (2^nd ‘(𝐹‘𝑛)))
34	33, 31	oveq12d 7432	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2^nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2^nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
35	32, 34	opeq12d 4877	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ⟨((1^st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2^nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩ = ⟨((1^st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2^nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
36	35	cbvmptv 5255	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2^nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1^st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2^nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
37		eqid 2728	. . . . . 6 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2^nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1^st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2^nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩)))
38		simplll 774	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
39		simpllr 775	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
40		simpr 484	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
41		simplr 768	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → sup(ran 𝑆, ℝ^, < ) ∈ ℝ)
42	13, 36, 37, 38, 39, 40, 41	ovollb2lem 25410	. . . . 5 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (vol‘𝐴) ≤ (sup(ran 𝑆, ℝ^*, < ) + 𝑥))
43	42	ralrimiva 3142	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ⁺ (vol‘𝐴) ≤ (sup(ran 𝑆, ℝ^*, < ) + 𝑥))
44		xralrple 13210	. . . . 5 ⊢ (((vol‘𝐴) ∈ ℝ^ ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) → ((vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ) ↔ ∀𝑥 ∈ ℝ⁺ (vol‘𝐴) ≤ (sup(ran 𝑆, ℝ^*, < ) + 𝑥)))
45	6, 44	sylan 579	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) → ((vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ) ↔ ∀𝑥 ∈ ℝ⁺ (vol‘𝐴) ≤ (sup(ran 𝑆, ℝ^*, < ) + 𝑥)))
46	43, 45	mpbird 257	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ∈ ℝ) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^*, < ))
47	28, 46	syldan 590	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ^, < ) ≠ +∞) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^*, < ))
48	11, 47	pm2.61dane 3025	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol‘𝐴) ≤ sup(ran 𝑆, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∩ cin 3944 ⊆ wss 3945 ∅c0 4318 ⟨cop 4630 ∪ cuni 4903 class class class wbr 5142 ↦ cmpt 5225 × cxp 5670 dom cdm 5672 ran crn 5673 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 supcsup 9457 ℝcr 11131 0cc0 11132 1c1 11133 + caddc 11135 +∞cpnf 11269 ℝ^*cxr 11271 < clt 11272 ≤ cle 11273 − cmin 11468 / cdiv 11895 ℕcn 12236 2c2 12291 ℝ⁺crp 13000 [,)cico 13352 [,]cicc 13353 seqcseq 13992 ↑cexp 14052 abscabs 15207 vol*covol 25384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15458 df-sum 15659 df-ovol 25386

This theorem is referenced by: ovolctb 25412 ovolicc1 25438 ioombl1lem4 25483 uniiccvol 25502

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