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Theorem ovolgelb 25515

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 1138	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) ∈ ℝ)
2		simp3 1139	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ⁺)
3	1, 2	ltaddrpd 13110	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) < ((vol*‘𝐴) + 𝐵))
4	2	rpred 13077	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
5	1, 4	readdcld 11290	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ)
6	1, 5	ltnled 11408	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) < ((vol‘𝐴) + 𝐵) ↔ ¬ ((vol‘𝐴) + 𝐵) ≤ (vol*‘𝐴)))
7	3, 6	mpbid 232	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ¬ ((vol‘𝐴) + 𝐵) ≤ (vol*‘𝐴))
8		eqid 2737	. . . . . . . 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 25508	. . . . . . 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 1134	. . . . . 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 4080	. . . . . . 7 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^
13	5	rexrd 11311	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ^*)
14		infxrgelb 13377	. . . . . . 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 587	. . . . . 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 2741	. . . . . . . . . . 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 5948	. . . . . . . . . . . . 13 ⊢ ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
19	18	supeq1i 9487	. . . . . . . . . . . 12 ⊢ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )
20	19	eqeq2i 2750	. . . . . . . . . . 11 ⊢ (𝑥 = sup(ran 𝑆, ℝ^, < ) ↔ 𝑥 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))
21	16, 20	bitr4di 289	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ) ↔ 𝑥 = sup(ran 𝑆, ℝ^, < )))
22	21	anbi2d 630	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < ))))
23	22	rexbidv 3179	. . . . . . . 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 3699	. . . . . . 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 3289	. . . . . . . 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 3183	. . . . . . . . 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 3093	. . . . . . . 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 464	. . . . . . . . . . . 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 450	. . . . . . . . . . . 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 275	. . . . . . . . . . 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 3093	. . . . . . . . . 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 25509	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
34	33, 17	ovolsf 25507	. . . . . . . . . . . . . . 15 ⊢ (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
35	32, 34	sylbi 217	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → 𝑆:ℕ⟶(0[,)+∞))
36	35	frnd 6744	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37		icossxr 13472	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ^*
38	36, 37	sstrdi 3996	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ ℝ^*)
39		supxrcl 13357	. . . . . . . . . . . 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 340	. . . . . . . . . . . 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 3522	. . . . . . . . . . 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 283	. . . . . . . . 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 3091	. . . . . . . 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 301	. . . . . . 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 275	. . . . . 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 288	. . . . 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 282	. . . 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 324	. . 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 3102	. . 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 234	. 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 11328	. . . . . 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 13191	. . . . . 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 260	. . . . 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 597	. . . 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 612	. . 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 3168	. 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ran crn 5686 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 supcsup 9480 infcinf 9481 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 +∞cpnf 11292 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℝ⁺crp 13034 (,)cioo 13387 [,)cico 13389 seqcseq 14042 abscabs 15273 vol*covol 25497

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-ovol 25499

This theorem is referenced by: ovolunlem2 25533 ovoliunlem3 25539 ovolscalem2 25549 ioombl1 25597 uniioombl 25624 mblfinlem3 37666 mblfinlem4 37667

W3C validator