ovolicc2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ovolicc2		Structured version Visualization version GIF version

Theorem ovolicc2 25046

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 24999	. . . . 5 ⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))
4		unieq 4919	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 = ∪ ran ((,) ∘ 𝑓))
5	4	sseq2d 4014	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓)))
6		pweq 4616	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘ 𝑓))
7	6	ineq1d 4211	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
8	7	rexeqdv 3326	. . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
13		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
14	12, 13	icccmp 24348	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
15	10, 11, 14	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
16		retop 24285	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ Top
17		iccssre 13408	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
18	10, 11, 17	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
19		uniretop 24286	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
20	19	cmpsub 22911	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
23	22	adantr 481	. . . . . . . . . . . 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 13426	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
25		ffn 6717	. . . . . . . . . . . . . . . . . . 19 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (,) Fn (ℝ^* × ℝ^*)
27		dffn3 6730	. . . . . . . . . . . . . . . . . 18 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,):(ℝ^ × ℝ^*)⟶ran (,))
28	26, 27	mpbi 229	. . . . . . . . . . . . . . . . 17 ⊢ (,):(ℝ^* × ℝ^*)⟶ran (,)
29		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ))
30		elovolmlem 24998	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
31	29, 30	sylib 217	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
32		inss2 4229	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
33		rexpssxrxp 11261	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
34	32, 33	sstri 3991	. . . . . . . . . . . . . . . . . 18 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
35		fss 6734	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝑓:ℕ⟶(ℝ^ × ℝ^*))
36	31, 34, 35	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
37		fco 6741	. . . . . . . . . . . . . . . . 17 ⊢ (((,):(ℝ^* × ℝ^)⟶ran (,) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
38	28, 36, 37	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
39	38	adantrr 715	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ((,) ∘ 𝑓):ℕ⟶ran (,))
40	39	frnd 6725	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ ran (,))
41		retopbas 24284	. . . . . . . . . . . . . . 15 ⊢ ran (,) ∈ TopBases
42		bastg 22476	. . . . . . . . . . . . . . 15 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran (,) ⊆ (topGen‘ran (,))
44	40, 43	sstrdi 3994	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
45		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ V
46	45	elpw2 5345	. . . . . . . . . . . . 13 ⊢ (ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
47	44, 46	sylibr 233	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)))
48	9, 23, 47	rspcdva 3613	. . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
51		elin 3964	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ↔ (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
52	50, 51	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
53	52	simprd 496	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin)
54	52	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓))
55	54	elpwid 4611	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓))
56	55	sseld 3981	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓)))
57	38	ffnd 6718	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((,) ∘ 𝑓) Fn ℕ)
58	57	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ)
59		fvelrnb 6952	. . . . . . . . . . . . . . . . 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 238	. . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)
63		fveqeq2 6900	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
64	63	ac6sfi 9289	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
71	31	adantr 481	. . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . 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 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))) → 𝑔:𝑣⟶ℕ)
75		simprrr 780	. . . . . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥)))
77		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥)
78	76, 77	eqeq12d 2748	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥))
79	78	rspccva 3611	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)
80	75, 79	sylan 580	. . . . . . . . . . . . . . . 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 2732	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
82	66, 67, 69, 70, 71, 72, 73, 74, 80, 81	ovolicc2lem5 25045	. . . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 1936	. . . . . . . . . . . . 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 3156	. . . . . . . . . . 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 715	. . . . . . . . . 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 5152	. . . . . . . . 9 ⊢ (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → ((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
90	88, 89	syl5ibrcom 246	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧))
91	90	expr 457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧)))
92	91	impd 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → (𝐵 − 𝐴) ≤ 𝑧))
93	92	rexlimdva 3155	. . . . 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 241	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧))
95	94	ralrimiv 3145	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)
96	1	ssrab3 4080	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
97	11, 10	resubcld 11644	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
98	97	rexrd 11266	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
99		infxrgelb 13316	. . . 4 ⊢ ((𝑀 ⊆ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^) → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
100	96, 98, 99	sylancr 587	. . 3 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
101	95, 100	mpbird 256	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ))
102	1	ovolval 24997	. . 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 5176	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 ran crn 5677 ∘ ccom 5680 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ↑_m cmap 8822 Fincfn 8941 supcsup 9437 infcinf 9438 ℝcr 11111 1c1 11113 + caddc 11115 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 − cmin 11446 ℕcn 12214 (,)cioo 13326 [,]cicc 13329 seqcseq 13968 abscabs 15183 ↾_t crest 17368 topGenctg 17385 Topctop 22402 TopBasesctb 22455 Compccmp 22897 vol*covol 24986

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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-n0 12475 df-z 12561 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-sum 15635 df-rest 17370 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-top 22403 df-topon 22420 df-bases 22456 df-cmp 22898 df-ovol 24988

This theorem is referenced by: ovolicc 25047

W3C validator