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Theorem ovolgelb 25494

Description: The outer volume is the greatest lower bound on the sum of all interval coverings of 𝐴. (Contributed by Mario Carneiro, 15-Jun-2014.)

**Hypothesis**
Ref	Expression
ovolgelb.1	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))

**Assertion**
Ref	Expression
ovolgelb	⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ^, < ) ≤ ((vol*‘𝐴) + 𝐵)))

Distinct variable groups: 𝐴,𝑔 𝐵,𝑔 Allowed substitution hint: 𝑆(𝑔)

Proof of Theorem ovolgelb Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1134	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) ∈ ℝ)
2		simp3 1135	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ⁺)
3	1, 2	ltaddrpd 13094	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) < ((vol*‘𝐴) + 𝐵))
4	2	rpred 13061	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
5	1, 4	readdcld 11281	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ)
6	1, 5	ltnled 11399	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) < ((vol‘𝐴) + 𝐵) ↔ ¬ ((vol‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
7	3, 6	mpbid 231	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ¬ ((vol‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8		eqid 2726	. . . . . . . 8 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} = {𝑦 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^*, < ))}
9	8	ovolval 25487	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (vol‘𝐴) = inf({𝑦 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ))
10	9	3ad2ant1 1130	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) = inf({𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ))
11	10	breq2d 5155	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (((vol‘𝐴) + 𝐵) ≤ (vol‘𝐴) ↔ ((vol‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < )))
12		ssrab2 4073	. . . . . . 7 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^
13	5	rexrd 11302	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ^*)
14		infxrgelb 13359	. . . . . . 7 ⊢ (({𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^ ∧ ((vol‘𝐴) + 𝐵) ∈ ℝ^) → (((vol‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ^ ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ) ↔ ∀𝑥 ∈ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ((vol‘𝐴) + 𝐵) ≤ 𝑥))
15	12, 13, 14	sylancr 585	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (((vol‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < ) ↔ ∀𝑥 ∈ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ((vol‘𝐴) + 𝐵) ≤ 𝑥))
16		eqeq1 2730	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )))
17		ovolgelb.1	. . . . . . . . . . . . . 14 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝑔))
18	17	rneqi 5933	. . . . . . . . . . . . 13 ⊢ ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
19	18	supeq1i 9480	. . . . . . . . . . . 12 ⊢ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )
20	19	eqeq2i 2739	. . . . . . . . . . 11 ⊢ (𝑥 = sup(ran 𝑆, ℝ^, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))
21	16, 20	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ^, < )))
22	21	anbi2d 628	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < ))))
23	22	rexbidv 3169	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) ↔ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < ))))
24	23	ralrab 3686	. . . . . . 7 ⊢ (∀𝑥 ∈ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ((vol‘𝐴) + 𝐵) ≤ 𝑥 ↔ ∀𝑥 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥))
25		ralcom 3277	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ^* ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)∀𝑥 ∈ ℝ^* ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥))
26		r19.23v 3173	. . . . . . . . 9 ⊢ (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥))
27	26	ralbii 3083	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℝ^* ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑥 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥))
28		ancomst 463	. . . . . . . . . . . 12 ⊢ (((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ((𝑥 = sup(ran 𝑆, ℝ^, < ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → ((vol‘𝐴) + 𝐵) ≤ 𝑥))
29		impexp 449	. . . . . . . . . . . 12 ⊢ (((𝑥 = sup(ran 𝑆, ℝ^, < ) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑔)) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝑥 = sup(ran 𝑆, ℝ^, < ) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥)))
30	28, 29	bitri 274	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝑥 = sup(ran 𝑆, ℝ^, < ) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥)))
31	30	ralbii 3083	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ ℝ^* ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑥 ∈ ℝ^* (𝑥 = sup(ran 𝑆, ℝ^, < ) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥)))
32		elovolmlem 25488	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		eqid 2726	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
34	33, 17	ovolsf 25486	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
35	32, 34	sylbi 216	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑆:ℕ⟶(0[,)+∞))
36	35	frnd 6725	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37		icossxr 13454	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ^*
38	36, 37	sstrdi 3991	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ ℝ^*)
39		supxrcl 13339	. . . . . . . . . . . 12 ⊢ (ran 𝑆 ⊆ ℝ^* → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
40	38, 39	syl 17	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → sup(ran 𝑆, ℝ^, < ) ∈ ℝ^)
41		breq2 5147	. . . . . . . . . . . . 13 ⊢ (𝑥 = sup(ran 𝑆, ℝ^, < ) → (((vol‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
42	41	imbi2d 339	. . . . . . . . . . . 12 ⊢ (𝑥 = sup(ran 𝑆, ℝ^, < ) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ))))
43	42	ceqsralv 3504	. . . . . . . . . . 11 ⊢ (sup(ran 𝑆, ℝ^, < ) ∈ ℝ^ → (∀𝑥 ∈ ℝ^* (𝑥 = sup(ran 𝑆, ℝ^, < ) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥)) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ))))
44	40, 43	syl 17	. . . . . . . . . 10 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (∀𝑥 ∈ ℝ^* (𝑥 = sup(ran 𝑆, ℝ^, < ) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ 𝑥)) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ))))
45	31, 44	bitrid 282	. . . . . . . . 9 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → (∀𝑥 ∈ ℝ^* ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ))))
46	45	ralbiia 3081	. . . . . . . 8 ⊢ (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)∀𝑥 ∈ ℝ^* ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
47	25, 27, 46	3bitr3i 300	. . . . . . 7 ⊢ (∀𝑥 ∈ ℝ^* (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < )) → ((vol‘𝐴) + 𝐵) ≤ 𝑥) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
48	24, 47	bitri 274	. . . . . 6 ⊢ (∀𝑥 ∈ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ((vol‘𝐴) + 𝐵) ≤ 𝑥 ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
49	15, 48	bitr2di 287	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )) ↔ ((vol‘𝐴) + 𝐵) ≤ inf({𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))}, ℝ^, < )))
50	11, 49	bitr4d 281	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (((vol‘𝐴) + 𝐵) ≤ (vol‘𝐴) ↔ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^*, < ))))
51	7, 50	mtbid 323	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ¬ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^*, < )))
52		rexanali 3092	. . 3 ⊢ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ ¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )) ↔ ¬ ∀𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
53	51, 52	sylibr 233	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ ¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^*, < )))
54		xrltnle 11319	. . . . . 6 ⊢ ((sup(ran 𝑆, ℝ^, < ) ∈ ℝ^ ∧ ((vol‘𝐴) + 𝐵) ∈ ℝ^) → (sup(ran 𝑆, ℝ^, < ) < ((vol‘𝐴) + 𝐵) ↔ ¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
55		xrltle 13173	. . . . . 6 ⊢ ((sup(ran 𝑆, ℝ^, < ) ∈ ℝ^ ∧ ((vol‘𝐴) + 𝐵) ∈ ℝ^) → (sup(ran 𝑆, ℝ^, < ) < ((vol‘𝐴) + 𝐵) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + 𝐵)))
56	54, 55	sylbird 259	. . . . 5 ⊢ ((sup(ran 𝑆, ℝ^, < ) ∈ ℝ^ ∧ ((vol‘𝐴) + 𝐵) ∈ ℝ^) → (¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol‘𝐴) + 𝐵)))
57	40, 13, 56	syl2anr 595	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → (¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < ) → sup(ran 𝑆, ℝ^, < ) ≤ ((vol*‘𝐴) + 𝐵)))
58	57	anim2d 610	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ ¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ^, < ) ≤ ((vol*‘𝐴) + 𝐵))))
59	58	reximdva 3158	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ ¬ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ^, < ) ≤ ((vol*‘𝐴) + 𝐵))))
60	53, 59	mpd 15	1 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran 𝑆, ℝ^, < ) ≤ ((vol*‘𝐴) + 𝐵)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 {crab 3419 ∩ cin 3945 ⊆ wss 3946 ∪ cuni 4905 class class class wbr 5143 × cxp 5670 ran crn 5673 ∘ ccom 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7413 ↑_m cmap 8844 supcsup 9473 infcinf 9474 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 +∞cpnf 11283 ℝ^*cxr 11285 < clt 11286 ≤ cle 11287 − cmin 11482 ℕcn 12255 ℝ⁺crp 13019 (,)cioo 13369 [,)cico 13371 seqcseq 14012 abscabs 15231 vol*covol 25476

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9475 df-inf 9476 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-n0 12516 df-z 12602 df-uz 12866 df-rp 13020 df-ico 13375 df-fz 13530 df-seq 14013 df-exp 14073 df-cj 15096 df-re 15097 df-im 15098 df-sqrt 15232 df-abs 15233 df-ovol 25478

This theorem is referenced by: ovolunlem2 25512 ovoliunlem3 25518 ovolscalem2 25528 ioombl1 25576 uniioombl 25603 mblfinlem3 37370 mblfinlem4 37371

W3C validator