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Theorem ovolgelb 25438

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 1137	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) ∈ ℝ)
2		simp3 1138	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ⁺)
3	1, 2	ltaddrpd 13089	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) < ((vol*‘𝐴) + 𝐵))
4	2	rpred 13056	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
5	1, 4	readdcld 11269	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ)
6	1, 5	ltnled 11387	. . . . 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 2736	. . . . . . . 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 25431	. . . . . . 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 1133	. . . . . 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 5136	. . . . 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 4060	. . . . . . 7 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^
13	5	rexrd 11290	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ^*)
14		infxrgelb 13357	. . . . . . 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 2740	. . . . . . . . . . 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 5922	. . . . . . . . . . . . 13 ⊢ ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
19	18	supeq1i 9464	. . . . . . . . . . . 12 ⊢ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )
20	19	eqeq2i 2749	. . . . . . . . . . 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 3165	. . . . . . . 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 3682	. . . . . . 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 3274	. . . . . . . 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 3169	. . . . . . . . 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 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 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 25432	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
34	33, 17	ovolsf 25430	. . . . . . . . . . . . . . 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 6719	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37		icossxr 13454	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ^*
38	36, 37	sstrdi 3976	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ ℝ^*)
39		supxrcl 13336	. . . . . . . . . . . 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 5128	. . . . . . . . . . . . 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 3506	. . . . . . . . . . 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 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 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 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 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 11307	. . . . . 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 13170	. . . . . 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 3154	. 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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ∃wrex 3061 {crab 3420 ∩ cin 3930 ⊆ wss 3931 ∪ cuni 4888 class class class wbr 5124 × cxp 5657 ran crn 5660 ∘ ccom 5663 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 supcsup 9457 infcinf 9458 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 +∞cpnf 11271 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 ℕcn 12245 ℝ⁺crp 13013 (,)cioo 13367 [,)cico 13369 seqcseq 14024 abscabs 15258 vol*covol 25420

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-ico 13373 df-fz 13530 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-ovol 25422

This theorem is referenced by: ovolunlem2 25456 ovoliunlem3 25462 ovolscalem2 25472 ioombl1 25520 uniioombl 25547 mblfinlem3 37688 mblfinlem4 37689

W3C validator