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Theorem ovolgelb 24642

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 1136	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) ∈ ℝ)
2		simp3 1137	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ⁺)
3	1, 2	ltaddrpd 12804	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) < ((vol*‘𝐴) + 𝐵))
4	2	rpred 12771	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
5	1, 4	readdcld 11005	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ)
6	1, 5	ltnled 11122	. . . . 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 2740	. . . . . . . 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 24635	. . . . . . 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 1132	. . . . . 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 5091	. . . . 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 4018	. . . . . . 7 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^
13	5	rexrd 11026	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ^*)
14		infxrgelb 13068	. . . . . . 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 2744	. . . . . . . . . . 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 5845	. . . . . . . . . . . . 13 ⊢ ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
19	18	supeq1i 9184	. . . . . . . . . . . 12 ⊢ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )
20	19	eqeq2i 2753	. . . . . . . . . . 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 629	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )) ↔ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑥 = sup(ran 𝑆, ℝ^, < ))))
23	22	rexbidv 3228	. . . . . . . 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 3632	. . . . . . 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 3283	. . . . . . . 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 3210	. . . . . . . . 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 465	. . . . . . . . . . . 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 451	. . . . . . . . . . . 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 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 24636	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
34	33, 17	ovolsf 24634	. . . . . . . . . . . . . . 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 6606	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37		icossxr 13163	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ^*
38	36, 37	sstrdi 3938	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ ℝ^*)
39		supxrcl 13048	. . . . . . . . . . . 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 5083	. . . . . . . . . . . . 13 ⊢ (𝑥 = sup(ran 𝑆, ℝ^, < ) → (((vol‘𝐴) + 𝐵) ≤ 𝑥 ↔ ((vol‘𝐴) + 𝐵) ≤ sup(ran 𝑆, ℝ^, < )))
42	41	imbi2d 341	. . . . . . . . . . . 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 3468	. . . . . . . . . . 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	syl5bb 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 3092	. . . . . . . 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 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 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 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 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 3194	. . 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 11043	. . . . . 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 12882	. . . . . 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 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 3205	. 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 396 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ∃wrex 3067 {crab 3070 ∩ cin 3891 ⊆ wss 3892 ∪ cuni 4845 class class class wbr 5079 × cxp 5588 ran crn 5591 ∘ ccom 5594 ⟶wf 6428 ‘cfv 6432 (class class class)co 7271 ↑_m cmap 8598 supcsup 9177 infcinf 9178 ℝcr 10871 0cc0 10872 1c1 10873 + caddc 10875 +∞cpnf 11007 ℝ^*cxr 11009 < clt 11010 ≤ cle 11011 − cmin 11205 ℕcn 11973 ℝ⁺crp 12729 (,)cioo 13078 [,)cico 13080 seqcseq 13719 abscabs 14943 vol*covol 24624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-inf 9180 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12582 df-rp 12730 df-ico 13084 df-fz 13239 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-ovol 24626

This theorem is referenced by: ovolunlem2 24660 ovoliunlem3 24666 ovolscalem2 24676 ioombl1 24724 uniioombl 24751 mblfinlem3 35812 mblfinlem4 35813

W3C validator