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Theorem ovolgelb 24997

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 13049	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → (vol‘𝐴) < ((vol*‘𝐴) + 𝐵))
4	2	rpred 13016	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → 𝐵 ∈ ℝ)
5	1, 4	readdcld 11243	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ)
6	1, 5	ltnled 11361	. . . . 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 2733	. . . . . . . 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 24990	. . . . . . 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 5161	. . . . 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 4078	. . . . . . 7 ⊢ {𝑦 ∈ ℝ^* ∣ ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑦 = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < ))} ⊆ ℝ^
13	5	rexrd 11264	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol‘𝐴) ∈ ℝ ∧ 𝐵 ∈ ℝ⁺) → ((vol‘𝐴) + 𝐵) ∈ ℝ^*)
14		infxrgelb 13314	. . . . . . 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 588	. . . . . 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 2737	. . . . . . . . . . 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 5937	. . . . . . . . . . . . 13 ⊢ ran 𝑆 = ran seq1( + , ((abs ∘ − ) ∘ 𝑔))
19	18	supeq1i 9442	. . . . . . . . . . . 12 ⊢ sup(ran 𝑆, ℝ^, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ^, < )
20	19	eqeq2i 2746	. . . . . . . . . . 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 3690	. . . . . . 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 3287	. . . . . . . 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 3094	. . . . . . . 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 466	. . . . . . . . . . . 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 452	. . . . . . . . . . . 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 3094	. . . . . . . . . 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 24991	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) ↔ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
33		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
34	33, 17	ovolsf 24989	. . . . . . . . . . . . . . 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 6726	. . . . . . . . . . . . 13 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ (0[,)+∞))
37		icossxr 13409	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ ℝ^*
38	36, 37	sstrdi 3995	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑_m ℕ) → ran 𝑆 ⊆ ℝ^*)
39		supxrcl 13294	. . . . . . . . . . . 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 5153	. . . . . . . . . . . . 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 3514	. . . . . . . . . . 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 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 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 3103	. . 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 11281	. . . . . 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 13128	. . . . . 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 598	. . . 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 613	. . 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 3169	. 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 ∩ cin 3948 ⊆ wss 3949 ∪ cuni 4909 class class class wbr 5149 × cxp 5675 ran crn 5678 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 supcsup 9435 infcinf 9436 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 +∞cpnf 11245 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℝ⁺crp 12974 (,)cioo 13324 [,)cico 13326 seqcseq 13966 abscabs 15181 vol*covol 24979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-ovol 24981

This theorem is referenced by: ovolunlem2 25015 ovoliunlem3 25021 ovolscalem2 25031 ioombl1 25079 uniioombl 25106 mblfinlem3 36527 mblfinlem4 36528

W3C validator