Proof of Theorem ovolsplit
| Step | Hyp | Ref
| Expression |
| 1 | | inundif 4479 |
. . . . 5
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
| 2 | 1 | eqcomi 2746 |
. . . 4
⊢ 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) |
| 3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) |
| 4 | 3 | fveq2d 6910 |
. 2
⊢ (𝜑 → (vol*‘𝐴) = (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)))) |
| 5 | | ovolsplit.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 6 | 5 | ssinss1d 4247 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℝ) |
| 7 | 5 | ssdifssd 4147 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ ℝ) |
| 8 | 6, 7 | unssd 4192 |
. . . . . . 7
⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ ℝ) |
| 9 | | ovolcl 25513 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ ℝ →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈
ℝ*) |
| 10 | 8, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈
ℝ*) |
| 11 | | pnfge 13172 |
. . . . . 6
⊢
((vol*‘((𝐴
∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈ ℝ* →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
| 12 | 10, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
| 13 | 12 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
| 14 | | oveq1 7438 |
. . . . . 6
⊢
((vol*‘(𝐴
∩ 𝐵)) = +∞ →
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵))) = (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 15 | 14 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵))) = (+∞ +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
| 16 | | ovolcl 25513 |
. . . . . . . 8
⊢ ((𝐴 ∖ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
| 17 | 7, 16 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
| 18 | 17 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
| 19 | | reex 11246 |
. . . . . . . . . . . . . 14
⊢ ℝ
∈ V |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℝ ∈
V) |
| 21 | 20, 5 | ssexd 5324 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ V) |
| 22 | 21 | difexd 5331 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) |
| 23 | | elpwg 4603 |
. . . . . . . . . . 11
⊢ ((𝐴 ∖ 𝐵) ∈ V → ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ ↔ (𝐴 ∖ 𝐵) ⊆ ℝ)) |
| 24 | 22, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ ↔ (𝐴 ∖ 𝐵) ⊆ ℝ)) |
| 25 | 7, 24 | mpbird 257 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 ℝ) |
| 26 | | ovolf 25517 |
. . . . . . . . . 10
⊢
vol*:𝒫 ℝ⟶(0[,]+∞) |
| 27 | 26 | ffvelcdmi 7103 |
. . . . . . . . 9
⊢ ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ →
(vol*‘(𝐴 ∖
𝐵)) ∈
(0[,]+∞)) |
| 28 | 25, 27 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞)) |
| 29 | 28 | xrge0nemnfd 45343 |
. . . . . . 7
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) |
| 30 | 29 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) |
| 31 | | xaddpnf2 13269 |
. . . . . 6
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈
ℝ* ∧ (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) → (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = +∞) |
| 32 | 18, 30, 31 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = +∞) |
| 33 | 15, 32 | eqtr2d 2778 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → +∞ =
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
| 34 | 13, 33 | breqtrd 5169 |
. . 3
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 35 | | simpl 482 |
. . . 4
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → 𝜑) |
| 36 | 20, 6 | sselpwd 5328 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 ℝ) |
| 37 | 26 | ffvelcdmi 7103 |
. . . . . . 7
⊢ ((𝐴 ∩ 𝐵) ∈ 𝒫 ℝ →
(vol*‘(𝐴 ∩ 𝐵)) ∈
(0[,]+∞)) |
| 38 | 36, 37 | syl 17 |
. . . . . 6
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞)) |
| 39 | 38 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞)) |
| 40 | | neqne 2948 |
. . . . . 6
⊢ (¬
(vol*‘(𝐴 ∩ 𝐵)) = +∞ →
(vol*‘(𝐴 ∩ 𝐵)) ≠
+∞) |
| 41 | 40 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ≠ +∞) |
| 42 | | ge0xrre 45544 |
. . . . 5
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈
(0[,]+∞) ∧ (vol*‘(𝐴 ∩ 𝐵)) ≠ +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
| 43 | 39, 41, 42 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
| 44 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
| 45 | | oveq2 7439 |
. . . . . . . . 9
⊢
((vol*‘(𝐴
∖ 𝐵)) = +∞
→ ((vol*‘(𝐴
∩ 𝐵))
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒
+∞)) |
| 46 | 45 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒
+∞)) |
| 47 | | ovolcl 25513 |
. . . . . . . . . . 11
⊢ ((𝐴 ∩ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ*) |
| 48 | 6, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ*) |
| 49 | 38 | xrge0nemnfd 45343 |
. . . . . . . . . 10
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ≠ -∞) |
| 50 | | xaddpnf1 13268 |
. . . . . . . . . 10
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈
ℝ* ∧ (vol*‘(𝐴 ∩ 𝐵)) ≠ -∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
| 51 | 48, 49, 50 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
| 52 | 51 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
| 53 | 46, 52 | eqtr2d 2778 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → +∞ =
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
| 54 | 44, 53 | breqtrd 5169 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 55 | 54 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 56 | | simpll 767 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
𝜑) |
| 57 | | simplr 769 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ) |
| 58 | 28 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞)) |
| 59 | | neqne 2948 |
. . . . . . . . 9
⊢ (¬
(vol*‘(𝐴 ∖
𝐵)) = +∞ →
(vol*‘(𝐴 ∖
𝐵)) ≠
+∞) |
| 60 | 59 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ≠ +∞) |
| 61 | | ge0xrre 45544 |
. . . . . . . 8
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈
(0[,]+∞) ∧ (vol*‘(𝐴 ∖ 𝐵)) ≠ +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
| 62 | 58, 60, 61 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
| 63 | 62 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ) |
| 64 | 6 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (𝐴 ∩ 𝐵) ⊆ ℝ) |
| 65 | | simp2 1138 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
| 66 | 7 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (𝐴 ∖ 𝐵) ⊆ ℝ) |
| 67 | | simp3 1139 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
| 68 | | ovolun 25534 |
. . . . . . . 8
⊢ ((((𝐴 ∩ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ((𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ)) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
| 69 | 64, 65, 66, 67, 68 | syl22anc 839 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
| 70 | | rexadd 13274 |
. . . . . . . . 9
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈ ℝ
∧ (vol*‘(𝐴
∖ 𝐵)) ∈ ℝ)
→ ((vol*‘(𝐴
∩ 𝐵))
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
| 71 | 70 | eqcomd 2743 |
. . . . . . . 8
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈ ℝ
∧ (vol*‘(𝐴
∖ 𝐵)) ∈ ℝ)
→ ((vol*‘(𝐴
∩ 𝐵)) +
(vol*‘(𝐴 ∖
𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 72 | 71 | 3adant1 1131 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 73 | 69, 72 | breqtrd 5169 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 74 | 56, 57, 63, 73 | syl3anc 1373 |
. . . . 5
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 75 | 55, 74 | pm2.61dan 813 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 76 | 35, 43, 75 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 77 | 34, 76 | pm2.61dan 813 |
. 2
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
| 78 | 4, 77 | eqbrtrd 5165 |
1
⊢ (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |