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Theorem ovolicc2 25502

Description: The measure of a closed interval is upper bounded by its length. (Contributed by Mario Carneiro, 14-Jun-2014.)

**Hypotheses**
Ref	Expression
ovolicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ovolicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ovolicc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ovolicc2.m	⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}

**Assertion**
Ref	Expression
ovolicc2	⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵)))

Distinct variable groups: 𝑦,𝑓,𝐴 𝐵,𝑓,𝑦 𝑦,𝑀 𝜑,𝑓,𝑦 Allowed substitution hint: 𝑀(𝑓)

Proof of Theorem ovolicc2 Dummy variables 𝑔 𝑘 𝑡 𝑢 𝑣 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovolicc2.m	. . . . . 6 ⊢ 𝑀 = {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))}
2	1	elovolm 25455	. . . . 5 ⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		simprr 773	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))
4		unieq 4862	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 = ∪ ran ((,) ∘ 𝑓))
5	4	sseq2d 3955	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓)))
6		pweq 4556	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘ 𝑓))
7	6	ineq1d 4160	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
8	7	rexeqdv 3297	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
9	5, 8	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) ↔ ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)))
10		ovolicc.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
11		ovolicc.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℝ)
12		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
13		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
14	12, 13	icccmp 24804	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
15	10, 11, 14	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
16		retop 24739	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ Top
17		iccssre 13376	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
18	10, 11, 17	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
19		uniretop 24740	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
20	19	cmpsub 23378	. . . . . . . . . . . . . . 15 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) → (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)))
21	16, 18, 20	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)))
22	15, 21	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
24		ioof 13394	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
25		ffn 6663	. . . . . . . . . . . . . . . . . . 19 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (,) Fn (ℝ^* × ℝ^*)
27		dffn3 6675	. . . . . . . . . . . . . . . . . 18 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,):(ℝ^ × ℝ^*)⟶ran (,))
28	26, 27	mpbi 230	. . . . . . . . . . . . . . . . 17 ⊢ (,):(ℝ^* × ℝ^*)⟶ran (,)
29		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
30		elovolmlem 25454	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31	29, 30	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32		inss2 4179	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
33		rexpssxrxp 11184	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
34	32, 33	sstri 3932	. . . . . . . . . . . . . . . . . 18 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
35		fss 6679	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝑓:ℕ⟶(ℝ^ × ℝ^*))
36	31, 34, 35	sylancl 587	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
37		fco 6687	. . . . . . . . . . . . . . . . 17 ⊢ (((,):(ℝ^* × ℝ^)⟶ran (,) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
38	28, 36, 37	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
39	38	adantrr 718	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ((,) ∘ 𝑓):ℕ⟶ran (,))
40	39	frnd 6671	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ ran (,))
41		retopbas 24738	. . . . . . . . . . . . . . 15 ⊢ ran (,) ∈ TopBases
42		bastg 22944	. . . . . . . . . . . . . . 15 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran (,) ⊆ (topGen‘ran (,))
44	40, 43	sstrdi 3935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
45		fvex 6848	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ V
46	45	elpw2 5272	. . . . . . . . . . . . 13 ⊢ (ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
47	44, 46	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)))
48	9, 23, 47	rspcdva 3566	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
49	3, 48	mpd 15	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)
50		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
51		elin 3906	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ↔ (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
52	50, 51	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
53	52	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin)
54	52	simpld 494	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓))
55	54	elpwid 4551	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓))
56	55	sseld 3921	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓)))
57	38	ffnd 6664	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((,) ∘ 𝑓) Fn ℕ)
58	57	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ)
59		fvelrnb 6895	. . . . . . . . . . . . . . . . 17 ⊢ (((,) ∘ 𝑓) Fn ℕ → (𝑡 ∈ ran ((,) ∘ 𝑓) ↔ ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡))
60	58, 59	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ ran ((,) ∘ 𝑓) ↔ ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡))
61	56, 60	sylibd 239	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡))
62	61	ralrimiv 3129	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)
63		fveqeq2 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
64	63	ac6sfi 9188	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Fin ∧ ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
65	53, 62, 64	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
66	10	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐴 ∈ ℝ)
67	11	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐵 ∈ ℝ)
68		ovolicc.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
69	68	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝐴 ≤ 𝐵)
70		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
71	31	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
72		simprll 779	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
73		simprlr 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → (𝐴[,]𝐵) ⊆ ∪ 𝑣)
74		simprrl 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → 𝑔:𝑣⟶ℕ)
75		simprrr 782	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡)
76		2fveq3 6840	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥)))
77		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥)
78	76, 77	eqeq12d 2753	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥))
79	78	rspccva 3564	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)
80	75, 79	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)
81		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
82	66, 67, 69, 70, 71, 72, 73, 74, 80, 81	ovolicc2lem5 25501	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ ((𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣) ∧ (𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
83	82	expr 456	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
84	83	exlimdv 1935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
85	65, 84	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
86	85	rexlimdvaa 3140	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
87	86	adantrr 718	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
88	49, 87	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
89		breq2 5090	. . . . . . . . 9 ⊢ (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → ((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
90	88, 89	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧))
91	90	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧)))
92	91	impd 410	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → (𝐵 − 𝐴) ≤ 𝑧))
93	92	rexlimdva 3139	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → (𝐵 − 𝐴) ≤ 𝑧))
94	2, 93	biimtrid 242	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧))
95	94	ralrimiv 3129	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)
96	1	ssrab3 4023	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
97	11, 10	resubcld 11572	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
98	97	rexrd 11189	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
99		infxrgelb 13282	. . . 4 ⊢ ((𝑀 ⊆ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^) → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
100	96, 98, 99	sylancr 588	. . 3 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
101	95, 100	mpbird 257	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ))
102	1	ovolval 25453	. . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol‘(𝐴[,]𝐵)) = inf(𝑀, ℝ^, < ))
103	18, 102	syl 17	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) = inf(𝑀, ℝ^, < ))
104	101, 103	breqtrrd 5114	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3390 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 𝒫 cpw 4542 ∪ cuni 4851 class class class wbr 5086 × cxp 5623 ran crn 5626 ∘ ccom 5629 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8767 Fincfn 8887 supcsup 9347 infcinf 9348 ℝcr 11031 1c1 11033 + caddc 11035 ℝ^*cxr 11172 < clt 11173 ≤ cle 11174 − cmin 11371 ℕcn 12168 (,)cioo 13292 [,]cicc 13295 seqcseq 13957 abscabs 15190 ↾_t crest 17377 topGenctg 17394 Topctop 22871 TopBasesctb 22923 Compccmp 23364 vol*covol 25442

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-inf2 9556 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9857 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-n0 12432 df-z 12519 df-uz 12783 df-q 12893 df-rp 12937 df-xneg 13057 df-xadd 13058 df-xmul 13059 df-ioo 13296 df-ico 13298 df-icc 13299 df-fz 13456 df-fzo 13603 df-seq 13958 df-exp 14018 df-hash 14287 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-clim 15444 df-sum 15643 df-rest 17379 df-topgen 17400 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-top 22872 df-topon 22889 df-bases 22924 df-cmp 23365 df-ovol 25444

This theorem is referenced by: ovolicc 25503

W3C validator