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Theorem ovolicc2 25491

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 25444	. . . . 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 4876	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 = ∪ ran ((,) ∘ 𝑓))
5	4	sseq2d 3968	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓)))
6		pweq 4570	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘ 𝑓))
7	6	ineq1d 4173	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
8	7	rexeqdv 3299	. . . . . . . . . . . . 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 24782	. . . . . . . . . . . . . . 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 24717	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) ∈ Top
17		iccssre 13357	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
18	10, 11, 17	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
19		uniretop 24718	. . . . . . . . . . . . . . . 16 ⊢ ℝ = ∪ (topGen‘ran (,))
20	19	cmpsub 23356	. . . . . . . . . . . . . . 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 13375	. . . . . . . . . . . . . . . . . . 19 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
25		ffn 6670	. . . . . . . . . . . . . . . . . . 19 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
26	24, 25	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (,) Fn (ℝ^* × ℝ^*)
27		dffn3 6682	. . . . . . . . . . . . . . . . . 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 25443	. . . . . . . . . . . . . . . . . . 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 4192	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
33		rexpssxrxp 11189	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
34	32, 33	sstri 3945	. . . . . . . . . . . . . . . . . 18 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
35		fss 6686	. . . . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ ran (,))
41		retopbas 24716	. . . . . . . . . . . . . . 15 ⊢ ran (,) ∈ TopBases
42		bastg 22922	. . . . . . . . . . . . . . 15 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
43	41, 42	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ran (,) ⊆ (topGen‘ran (,))
44	40, 43	sstrdi 3948	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
45		fvex 6855	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ V
46	45	elpw2 5281	. . . . . . . . . . . . 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 3579	. . . . . . . . . . 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 3919	. . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓))
56	55	sseld 3934	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓)))
57	38	ffnd 6671	. . . . . . . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
64	63	ac6sfi 9196	. . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . 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 25490	. . . . . . . . . . . . . . 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 5104	. . . . . . . . 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 4036	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
97	11, 10	resubcld 11577	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
98	97	rexrd 11194	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
99		infxrgelb 13263	. . . 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 25442	. . 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 5128	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 3401 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 𝒫 cpw 4556 ∪ cuni 4865 class class class wbr 5100 × cxp 5630 ran crn 5633 ∘ ccom 5636 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 Fincfn 8895 supcsup 9355 infcinf 9356 ℝcr 11037 1c1 11039 + caddc 11041 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 − cmin 11376 ℕcn 12157 (,)cioo 13273 [,]cicc 13276 seqcseq 13936 abscabs 15169 ↾_t crest 17352 topGenctg 17369 Topctop 22849 TopBasesctb 22901 Compccmp 23342 vol*covol 25431

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-rest 17354 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-top 22850 df-topon 22867 df-bases 22902 df-cmp 23343 df-ovol 25433

This theorem is referenced by: ovolicc 25492

W3C validator