Theorem ovollb2 24853

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 24843). (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 485	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
2		ovolficcss 24833	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
3	2	adantr 481	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ)
4	1, 3	sstrd 3954	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5		ovolcl 24842	. . . . . 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 481	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ∈ ℝ*)
8		pnfge 13051	. . . 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 485	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → sup(ran 𝑆, ℝ*, < ) = +∞)
11	9, 10	breqtrrd 5133	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12		eqid 2736	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13		ovollb2.1	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
14	12, 13	ovolsf 24836	. . . . . . . 8 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
15	14	adantr 481	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝑆:ℕ⟶(0[,)+∞))
16	15	frnd 6676	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17		rge0ssre 13373	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
18	16, 17	sstrdi 3956	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19		1nn 12164	. . . . . . . 8 ⊢ 1 ∈ ℕ
20	15	fdmd 6679	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
21	19, 20	eleqtrrid 2845	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
22	21	ne0d 4295	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23		dm0rn0 5880	. . . . . . 7 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
24	23	necon3bii 2996	. . . . . 6 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
25	22, 24	sylib 217	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26		supxrre2 13250	. . . . 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 478	. . 3 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ≠ +∞) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ)
29		2fveq3 6847	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (1st ‘(𝐹‘𝑚)) = (1st ‘(𝐹‘𝑛)))
30		oveq2 7365	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
31	30	oveq2d 7373	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
32	29, 31	oveq12d 7375	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33		2fveq3 6847	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝐹‘𝑚)) = (2nd ‘(𝐹‘𝑛)))
34	33, 31	oveq12d 7375	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
35	32, 34	opeq12d 4838	. . . . . . 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 5218	. . . . . 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 2736	. . . . . 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 773	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
39		simpllr 774	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹))
40		simpr 485	. . . . . 6 ⊢ ((((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
41		simplr 767	. . . . . 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 24852	. . . . 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 3143	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
44		xralrple 13124	. . . . 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 256	. . 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 3032	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ⟨cop 4592  ∪ cuni 4865   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634   ∘ ccom 5637  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920  supcsup 9376  ℝcr 11050  0cc0 11051  1c1 11052   + caddc 11054  +∞cpnf 11186  ℝ^*cxr 11188   < clt 11189   ≤ cle 11190   − cmin 11385   / cdiv 11812  ℕcn 12153  2c2 12208  ℝ⁺crp 12915  [,)cico 13266  [,]cicc 13267  seqcseq 13906  ↑cexp 13967  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-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  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-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  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-sum 15571  df-ovol 24828

This theorem is referenced by:  ovolctb  24854  ovolicc1  24880  ioombl1lem4  24925  uniiccvol  24944