Theorem ovollb2 25390

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 25380). (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 25370	. . . . . . . 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 3957	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5		ovolcl 25379	. . . . . 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 13090	. . . 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 5135	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12		eqid 2729	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13		ovollb2.1	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
14	12, 13	ovolsf 25373	. . . . . . . 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 6696	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17		rge0ssre 13417	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
18	16, 17	sstrdi 3959	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19		1nn 12197	. . . . . . . 8 ⊢ 1 ∈ ℕ
20	15	fdmd 6698	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
21	19, 20	eleqtrrid 2835	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
22	21	ne0d 4305	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23		dm0rn0 5888	. . . . . . 7 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
24	23	necon3bii 2977	. . . . . 6 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
25	22, 24	sylib 218	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26		supxrre2 13291	. . . . 5 ⊢ ((ran 𝑆 ⊆ ℝ ∧ ran 𝑆 ≠ ∅) → (sup(ran 𝑆, ℝ*, < ) ∈ ℝ ↔ sup(ran 𝑆, ℝ*, < ) ≠ +∞))
27	18, 25, 26	syl2anc 584	. . . 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 6863	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (1st ‘(𝐹‘𝑚)) = (1st ‘(𝐹‘𝑛)))
30		oveq2 7395	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
31	30	oveq2d 7403	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
32	29, 31	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33		2fveq3 6863	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝐹‘𝑚)) = (2nd ‘(𝐹‘𝑛)))
34	33, 31	oveq12d 7405	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
35	32, 34	opeq12d 4845	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ⟨((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩ = ⟨((1st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
36	35	cbvmptv 5211	. . . . . 6 ⊢ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩) = (𝑛 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛)))⟩)
37		eqid 2729	. . . . . 6 ⊢ seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚)))⟩))) = seq1( + , ((abs ∘ − ) ∘ (𝑚 ∈ ℕ ↦ ⟨((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))), ((2nd ‘(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 25389	. . . . 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 3125	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
44		xralrple 13165	. . . . 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 580	. . . 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 591	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
48	11, 47	pm2.61dane 3012	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  dom cdm 5638  ran crn 5639   ∘ ccom 5642  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  supcsup 9391  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071  +∞cpnf 11205  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209   − cmin 11405   / cdiv 11835  ℕcn 12186  2c2 12241  ℝ⁺crp 12951  [,)cico 13308  [,]cicc 13309  seqcseq 13966  ↑cexp 14026  abscabs 15200  vol*covol 25363

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

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 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-q 12908  df-rp 12952  df-ioo 13310  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-ovol 25365

This theorem is referenced by:  ovolctb  25391  ovolicc1  25417  ioombl1lem4  25462  uniiccvol  25481