ovolicc2 - Metamath Proof Explorer

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Theorem ovolicc2 23097

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 23050	. . . . 5 ⊢ (𝑧 ∈ 𝑀 ↔ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
3		ioof 12142	. . . . . . . . . . . . . . . . . 18 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
4		ffn 5958	. . . . . . . . . . . . . . . . . 18 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
5	3, 4	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (,) Fn (ℝ^* × ℝ^*)
6		dffn3 5967	. . . . . . . . . . . . . . . . 17 ⊢ ((,) Fn (ℝ^* × ℝ^) ↔ (,):(ℝ^ × ℝ^*)⟶ran (,))
7	5, 6	mpbi 219	. . . . . . . . . . . . . . . 16 ⊢ (,):(ℝ^* × ℝ^*)⟶ran (,)
8		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ))
9		reex 9906	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℝ ∈ V
10	9, 9	xpex 6860	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℝ × ℝ) ∈ V
11	10	inex2 4728	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ≤ ∩ (ℝ × ℝ)) ∈ V
12		nnex 10903	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ ∈ V
13	11, 12	elmap 7772	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ↔ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
14	8, 13	sylib 207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
15		inss2 3796	. . . . . . . . . . . . . . . . . 18 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
16		rexpssxrxp 9963	. . . . . . . . . . . . . . . . . 18 ⊢ (ℝ × ℝ) ⊆ (ℝ^* × ℝ^*)
17	15, 16	sstri 3577	. . . . . . . . . . . . . . . . 17 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^*)
18		fss 5969	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ^* × ℝ^)) → 𝑓:ℕ⟶(ℝ^ × ℝ^*))
19	14, 17, 18	sylancl 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → 𝑓:ℕ⟶(ℝ^* × ℝ^*))
20		fco 5971	. . . . . . . . . . . . . . . 16 ⊢ (((,):(ℝ^* × ℝ^)⟶ran (,) ∧ 𝑓:ℕ⟶(ℝ^ × ℝ^*)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
21	7, 19, 20	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → ((,) ∘ 𝑓):ℕ⟶ran (,))
22	21	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ((,) ∘ 𝑓):ℕ⟶ran (,))
23		frn 5966	. . . . . . . . . . . . . 14 ⊢ (((,) ∘ 𝑓):ℕ⟶ran (,) → ran ((,) ∘ 𝑓) ⊆ ran (,))
24	22, 23	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ ran (,))
25		retopbas 22374	. . . . . . . . . . . . . 14 ⊢ ran (,) ∈ TopBases
26		bastg 20581	. . . . . . . . . . . . . 14 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
27	25, 26	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ran (,) ⊆ (topGen‘ran (,))
28	24, 27	syl6ss 3580	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
29		fvex 6113	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) ∈ V
30	29	elpw2 4755	. . . . . . . . . . . 12 ⊢ (ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)) ↔ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,)))
31	28, 30	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)))
32		ovolicc.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
33		ovolicc.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ)
34		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
35		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵))
36	34, 35	icccmp 22436	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
37	32, 33, 36	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp)
38		retop 22375	. . . . . . . . . . . . . 14 ⊢ (topGen‘ran (,)) ∈ Top
39		iccssre 12126	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
40	32, 33, 39	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
41		uniretop 22376	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ (topGen‘ran (,))
42	41	cmpsub 21013	. . . . . . . . . . . . . 14 ⊢ (((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)))
43	38, 40, 42	sylancr 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ∈ Comp ↔ ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)))
44	37, 43	mpbid 221	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
45	44	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
46		simprr 792	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))
47		unieq 4380	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ∪ 𝑢 = ∪ ran ((,) ∘ 𝑓))
48	47	sseq2d 3596	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → ((𝐴[,]𝐵) ⊆ ∪ 𝑢 ↔ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓)))
49		pweq 4111	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → 𝒫 𝑢 = 𝒫 ran ((,) ∘ 𝑓))
50	49	ineq1d 3775	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (𝒫 𝑢 ∩ Fin) = (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
51	50	rexeqdv 3122	. . . . . . . . . . . . 13 ⊢ (𝑢 = ran ((,) ∘ 𝑓) → (∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣))
52	48, 51	imbi12d 333	. . . . . . . . . . . 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)))
53	52	rspcv 3278	. . . . . . . . . . 11 ⊢ (ran ((,) ∘ 𝑓) ∈ 𝒫 (topGen‘ran (,)) → (∀𝑢 ∈ 𝒫 (topGen‘ran (,))((𝐴[,]𝐵) ⊆ ∪ 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣) → ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)))
54	31, 45, 46, 53	syl3c 64	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → ∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣)
55		simprl 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin))
56		elin 3758	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ↔ (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
57	55, 56	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓) ∧ 𝑣 ∈ Fin))
58	57	simprd 478	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ Fin)
59	57	simpld 474	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ∈ 𝒫 ran ((,) ∘ 𝑓))
60	59	elpwid 4118	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → 𝑣 ⊆ ran ((,) ∘ 𝑓))
61	60	sseld 3567	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → 𝑡 ∈ ran ((,) ∘ 𝑓)))
62		ffn 5958	. . . . . . . . . . . . . . . . . . 19 ⊢ (((,) ∘ 𝑓):ℕ⟶ran (,) → ((,) ∘ 𝑓) Fn ℕ)
63	21, 62	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → ((,) ∘ 𝑓) Fn ℕ)
64	63	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ((,) ∘ 𝑓) Fn ℕ)
65		fvelrnb 6153	. . . . . . . . . . . . . . . . 17 ⊢ (((,) ∘ 𝑓) Fn ℕ → (𝑡 ∈ ran ((,) ∘ 𝑓) ↔ ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡))
66	64, 65	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))
67	61, 66	sylibd 228	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → (𝑡 ∈ 𝑣 → ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡))
68	67	ralrimiv 2948	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) ∧ (𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin) ∧ (𝐴[,]𝐵) ⊆ ∪ 𝑣)) → ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡)
69		fveq2 6103	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = (𝑔‘𝑡) → (((,) ∘ 𝑓)‘𝑘) = (((,) ∘ 𝑓)‘(𝑔‘𝑡)))
70	69	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑔‘𝑡) → ((((,) ∘ 𝑓)‘𝑘) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
71	70	ac6sfi 8089	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Fin ∧ ∀𝑡 ∈ 𝑣 ∃𝑘 ∈ ℕ (((,) ∘ 𝑓)‘𝑘) = 𝑡) → ∃𝑔(𝑔:𝑣⟶ℕ ∧ ∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡))
72	58, 68, 71	syl2anc 691	. . . . . . . . . . . . 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))
73	32	ad2antrr 758	. . . . . . . . . . . . . . . 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)
74	33	ad2antrr 758	. . . . . . . . . . . . . . . 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)
75		ovolicc.3	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
76	75	ad2antrr 758	. . . . . . . . . . . . . . . 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)
77		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
78	14	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)))
79		simprll 798	. . . . . . . . . . . . . . . 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))
80		simprlr 799	. . . . . . . . . . . . . . . 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)
81		simprrl 800	. . . . . . . . . . . . . . . 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)
82		simprrr 801	. . . . . . . . . . . . . . . . 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)
83		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (𝑔‘𝑡) = (𝑔‘𝑥))
84	83	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = (((,) ∘ 𝑓)‘(𝑔‘𝑥)))
85		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → 𝑡 = 𝑥)
86	84, 85	eqeq12d 2625	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → ((((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ↔ (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥))
87	86	rspccva 3281	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑡 ∈ 𝑣 (((,) ∘ 𝑓)‘(𝑔‘𝑡)) = 𝑡 ∧ 𝑥 ∈ 𝑣) → (((,) ∘ 𝑓)‘(𝑔‘𝑥)) = 𝑥)
88	82, 87	sylan 487	. . . . . . . . . . . . . . . 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)
89		eqid 2610	. . . . . . . . . . . . . . . 16 ⊢ {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} = {𝑢 ∈ 𝑣 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
90	73, 74, 76, 77, 78, 79, 80, 81, 88, 89	ovolicc2lem5 23096	. . . . . . . . . . . . . . 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^*, < ))
91	90	expr 641	. . . . . . . . . . . . . 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^*, < )))
92	91	exlimdv 1848	. . . . . . . . . . . . 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^*, < )))
93	72, 92	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^*, < ))
94	93	rexlimdvaa 3014	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → (∃𝑣 ∈ (𝒫 ran ((,) ∘ 𝑓) ∩ Fin)(𝐴[,]𝐵) ⊆ ∪ 𝑣 → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )))
95	94	adantrr 749	. . . . . . . . . 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^*, < )))
96	54, 95	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ))
97		breq2 4587	. . . . . . . . 9 ⊢ (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ) → ((𝐵 − 𝐴) ≤ 𝑧 ↔ (𝐵 − 𝐴) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < )))
98	96, 97	syl5ibrcom 236	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ) ∧ (𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓))) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧))
99	98	expr 641	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → ((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) → (𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < ) → (𝐵 − 𝐴) ≤ 𝑧)))
100	99	impd 446	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)) → (((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → (𝐵 − 𝐴) ≤ 𝑧))
101	100	rexlimdva 3013	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑧 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^*, < )) → (𝐵 − 𝐴) ≤ 𝑧))
102	2, 101	syl5bi 231	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑀 → (𝐵 − 𝐴) ≤ 𝑧))
103	102	ralrimiv 2948	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧)
104		ssrab2 3650	. . . . 5 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_𝑚 ℕ)((𝐴[,]𝐵) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ^, < ))} ⊆ ℝ^
105	1, 104	eqsstri 3598	. . . 4 ⊢ 𝑀 ⊆ ℝ^*
106	33, 32	resubcld 10337	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
107	106	rexrd 9968	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ^*)
108		infxrgelb 12037	. . . 4 ⊢ ((𝑀 ⊆ ℝ^* ∧ (𝐵 − 𝐴) ∈ ℝ^) → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
109	105, 107, 108	sylancr 694	. . 3 ⊢ (𝜑 → ((𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ) ↔ ∀𝑧 ∈ 𝑀 (𝐵 − 𝐴) ≤ 𝑧))
110	103, 109	mpbird 246	. 2 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ inf(𝑀, ℝ^*, < ))
111	1	ovolval 23049	. . 3 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → (vol‘(𝐴[,]𝐵)) = inf(𝑀, ℝ^, < ))
112	40, 111	syl 17	. 2 ⊢ (𝜑 → (vol‘(𝐴[,]𝐵)) = inf(𝑀, ℝ^, < ))
113	110, 112	breqtrrd 4611	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (vol*‘(𝐴[,]𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 × cxp 5036 ran crn 5039 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑_𝑚 cmap 7744 Fincfn 7841 supcsup 8229 infcinf 8230 ℝcr 9814 1c1 9816 + caddc 9818 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 (,)cioo 12046 [,]cicc 12049 seqcseq 12663 abscabs 13822 ↾_t crest 15904 topGenctg 15921 Topctop 20517 TopBasesctb 20520 Compccmp 20999 vol*covol 23038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-rest 15906 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cmp 21000 df-ovol 23040

This theorem is referenced by: ovolicc 23098

W3C validator