Theorem ovollb2 25543

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 25533). (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 25523	. . . . . . . 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 4019	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 𝐴 ⊆ ℝ)
5		ovolcl 25532	. . . . . 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 13193	. . . 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 5194	. 2 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) = +∞) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
12		eqid 2740	. . . . . . . . 9 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
13		ovollb2.1	. . . . . . . . 9 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
14	12, 13	ovolsf 25526	. . . . . . . 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 6755	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ (0[,)+∞))
17		rge0ssre 13516	. . . . . 6 ⊢ (0[,)+∞) ⊆ ℝ
18	16, 17	sstrdi 4021	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ⊆ ℝ)
19		1nn 12304	. . . . . . . 8 ⊢ 1 ∈ ℕ
20	15	fdmd 6757	. . . . . . . 8 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 = ℕ)
21	19, 20	eleqtrrid 2851	. . . . . . 7 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → 1 ∈ dom 𝑆)
22	21	ne0d 4365	. . . . . 6 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → dom 𝑆 ≠ ∅)
23		dm0rn0 5949	. . . . . . 7 ⊢ (dom 𝑆 = ∅ ↔ ran 𝑆 = ∅)
24	23	necon3bii 2999	. . . . . 6 ⊢ (dom 𝑆 ≠ ∅ ↔ ran 𝑆 ≠ ∅)
25	22, 24	sylib 218	. . . . 5 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → ran 𝑆 ≠ ∅)
26		supxrre2 13393	. . . . 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 6925	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (1st ‘(𝐹‘𝑚)) = (1st ‘(𝐹‘𝑛)))
30		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (2↑𝑚) = (2↑𝑛))
31	30	oveq2d 7464	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → ((𝑥 / 2) / (2↑𝑚)) = ((𝑥 / 2) / (2↑𝑛)))
32	29, 31	oveq12d 7466	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((1st ‘(𝐹‘𝑚)) − ((𝑥 / 2) / (2↑𝑚))) = ((1st ‘(𝐹‘𝑛)) − ((𝑥 / 2) / (2↑𝑛))))
33		2fveq3 6925	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (2nd ‘(𝐹‘𝑚)) = (2nd ‘(𝐹‘𝑛)))
34	33, 31	oveq12d 7466	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((2nd ‘(𝐹‘𝑚)) + ((𝑥 / 2) / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑛)) + ((𝑥 / 2) / (2↑𝑛))))
35	32, 34	opeq12d 4905	. . . . . . 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 5279	. . . . . 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 2740	. . . . . 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 25542	. . . . 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 3152	. . . 4 ⊢ (((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) ∧ sup(ran 𝑆, ℝ*, < ) ∈ ℝ) → ∀𝑥 ∈ ℝ+ (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝑥))
44		xralrple 13267	. . . . 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 3035	1 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran ([,] ∘ 𝐹)) → (vol*‘𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  dom cdm 5700  ran crn 5701   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  supcsup 9509  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324   ≤ cle 11325   − cmin 11520   / cdiv 11947  ℕcn 12293  2c2 12348  ℝ⁺crp 13057  [,)cico 13409  [,]cicc 13410  seqcseq 14052  ↑cexp 14112  abscabs 15283  vol*covol 25516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-ovol 25518

This theorem is referenced by:  ovolctb  25544  ovolicc1  25570  ioombl1lem4  25615  uniiccvol  25634