| Step |
Hyp |
Ref |
Expression |
| 1 |
|
raleq |
⊢ ( 𝑥 = ∅ → ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ∀ 𝑘 ∈ ∅ ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) ) |
| 2 |
|
iuneq1 |
⊢ ( 𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵 ) |
| 3 |
2
|
fveq2d |
⊢ ( 𝑥 = ∅ → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) = ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) ) |
| 4 |
|
sumeq1 |
⊢ ( 𝑥 = ∅ → Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) = Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) ) |
| 5 |
3 4
|
breq12d |
⊢ ( 𝑥 = ∅ → ( ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ↔ ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) ≤ Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) ) ) |
| 6 |
1 5
|
imbi12d |
⊢ ( 𝑥 = ∅ → ( ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ) ↔ ( ∀ 𝑘 ∈ ∅ ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) ≤ Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) ) ) ) |
| 7 |
|
raleq |
⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) ) |
| 8 |
|
iuneq1 |
⊢ ( 𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵 ) |
| 9 |
8
|
fveq2d |
⊢ ( 𝑥 = 𝑦 → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) = ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ) |
| 10 |
|
sumeq1 |
⊢ ( 𝑥 = 𝑦 → Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) = Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) |
| 11 |
9 10
|
breq12d |
⊢ ( 𝑥 = 𝑦 → ( ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ↔ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) |
| 12 |
7 11
|
imbi12d |
⊢ ( 𝑥 = 𝑦 → ( ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ) ↔ ( ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ) |
| 13 |
|
raleq |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) ) |
| 14 |
|
iuneq1 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) |
| 15 |
14
|
fveq2d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) = ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ) |
| 16 |
|
sumeq1 |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) = Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) |
| 17 |
15 16
|
breq12d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ↔ ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) ) |
| 18 |
13 17
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 ∪ { 𝑧 } ) → ( ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ) ↔ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) ) ) |
| 19 |
|
raleq |
⊢ ( 𝑥 = 𝐴 → ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) ) |
| 20 |
|
iuneq1 |
⊢ ( 𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵 ) |
| 21 |
20
|
fveq2d |
⊢ ( 𝑥 = 𝐴 → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) = ( vol* ‘ ∪ 𝑘 ∈ 𝐴 𝐵 ) ) |
| 22 |
|
sumeq1 |
⊢ ( 𝑥 = 𝐴 → Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( vol* ‘ 𝐵 ) ) |
| 23 |
21 22
|
breq12d |
⊢ ( 𝑥 = 𝐴 → ( ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ↔ ( vol* ‘ ∪ 𝑘 ∈ 𝐴 𝐵 ) ≤ Σ 𝑘 ∈ 𝐴 ( vol* ‘ 𝐵 ) ) ) |
| 24 |
19 23
|
imbi12d |
⊢ ( 𝑥 = 𝐴 → ( ( ∀ 𝑘 ∈ 𝑥 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑥 𝐵 ) ≤ Σ 𝑘 ∈ 𝑥 ( vol* ‘ 𝐵 ) ) ↔ ( ∀ 𝑘 ∈ 𝐴 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝐴 𝐵 ) ≤ Σ 𝑘 ∈ 𝐴 ( vol* ‘ 𝐵 ) ) ) ) |
| 25 |
|
0le0 |
⊢ 0 ≤ 0 |
| 26 |
|
0iun |
⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ |
| 27 |
26
|
fveq2i |
⊢ ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) = ( vol* ‘ ∅ ) |
| 28 |
|
ovol0 |
⊢ ( vol* ‘ ∅ ) = 0 |
| 29 |
27 28
|
eqtri |
⊢ ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) = 0 |
| 30 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) = 0 |
| 31 |
25 29 30
|
3brtr4i |
⊢ ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) ≤ Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) |
| 32 |
31
|
a1i |
⊢ ( ∀ 𝑘 ∈ ∅ ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ ∅ 𝐵 ) ≤ Σ 𝑘 ∈ ∅ ( vol* ‘ 𝐵 ) ) |
| 33 |
|
ssun1 |
⊢ 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) |
| 34 |
|
ssralv |
⊢ ( 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) ) |
| 35 |
33 34
|
ax-mp |
⊢ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) |
| 36 |
35
|
imim1i |
⊢ ( ( ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) |
| 37 |
|
simprl |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) |
| 38 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 |
| 39 |
|
nfcv |
⊢ Ⅎ 𝑘 ℝ |
| 40 |
38 39
|
nfss |
⊢ Ⅎ 𝑘 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ |
| 41 |
|
nfcv |
⊢ Ⅎ 𝑘 vol* |
| 42 |
41 38
|
nffv |
⊢ Ⅎ 𝑘 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 43 |
42
|
nfel1 |
⊢ Ⅎ 𝑘 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ |
| 44 |
40 43
|
nfan |
⊢ Ⅎ 𝑘 ( ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 45 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑚 → 𝐵 = ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 46 |
45
|
sseq1d |
⊢ ( 𝑘 = 𝑚 → ( 𝐵 ⊆ ℝ ↔ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) ) |
| 47 |
45
|
fveq2d |
⊢ ( 𝑘 = 𝑚 → ( vol* ‘ 𝐵 ) = ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) |
| 48 |
47
|
eleq1d |
⊢ ( 𝑘 = 𝑚 → ( ( vol* ‘ 𝐵 ) ∈ ℝ ↔ ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) |
| 49 |
46 48
|
anbi12d |
⊢ ( 𝑘 = 𝑚 → ( ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ( ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) ) |
| 50 |
44 49
|
rspc |
⊢ ( 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) ) |
| 51 |
37 50
|
mpan9 |
⊢ ( ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ∧ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → ( ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) |
| 52 |
51
|
simpld |
⊢ ( ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ∧ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) |
| 53 |
52
|
ralrimiva |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ∀ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) |
| 54 |
|
iunss |
⊢ ( ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ↔ ∀ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) |
| 55 |
53 54
|
sylibr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) |
| 56 |
|
iunss1 |
⊢ ( 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) → ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 57 |
33 56
|
ax-mp |
⊢ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 |
| 58 |
57 55
|
sstrid |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) |
| 59 |
|
simpll |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → 𝑦 ∈ Fin ) |
| 60 |
|
elun1 |
⊢ ( 𝑚 ∈ 𝑦 → 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) |
| 61 |
51
|
simprd |
⊢ ( ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ∧ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 62 |
60 61
|
sylan2 |
⊢ ( ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ∧ 𝑚 ∈ 𝑦 ) → ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 63 |
59 62
|
fsumrecl |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 64 |
|
simprr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) |
| 65 |
|
nfcv |
⊢ Ⅎ 𝑚 𝐵 |
| 66 |
65 38 45
|
cbviun |
⊢ ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 |
| 67 |
66
|
fveq2i |
⊢ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) = ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 68 |
|
nfcv |
⊢ Ⅎ 𝑚 ( vol* ‘ 𝐵 ) |
| 69 |
47 68 42
|
cbvsum |
⊢ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) = Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 70 |
64 67 69
|
3brtr3g |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ≤ Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) |
| 71 |
|
ovollecl |
⊢ ( ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ∧ ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ≤ Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) → ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 72 |
58 63 70 71
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 73 |
|
ssun2 |
⊢ { 𝑧 } ⊆ ( 𝑦 ∪ { 𝑧 } ) |
| 74 |
|
vsnid |
⊢ 𝑧 ∈ { 𝑧 } |
| 75 |
73 74
|
sselii |
⊢ 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) |
| 76 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑧 / 𝑘 ⦌ 𝐵 |
| 77 |
76 39
|
nfss |
⊢ Ⅎ 𝑘 ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ |
| 78 |
41 76
|
nffv |
⊢ Ⅎ 𝑘 ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
| 79 |
78
|
nfel1 |
⊢ Ⅎ 𝑘 ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ |
| 80 |
77 79
|
nfan |
⊢ Ⅎ 𝑘 ( ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 81 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑧 → 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
| 82 |
81
|
sseq1d |
⊢ ( 𝑘 = 𝑧 → ( 𝐵 ⊆ ℝ ↔ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ) ) |
| 83 |
81
|
fveq2d |
⊢ ( 𝑘 = 𝑧 → ( vol* ‘ 𝐵 ) = ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) |
| 84 |
83
|
eleq1d |
⊢ ( 𝑘 = 𝑧 → ( ( vol* ‘ 𝐵 ) ∈ ℝ ↔ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) |
| 85 |
82 84
|
anbi12d |
⊢ ( 𝑘 = 𝑧 → ( ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ↔ ( ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) ) |
| 86 |
80 85
|
rspc |
⊢ ( 𝑧 ∈ ( 𝑦 ∪ { 𝑧 } ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) ) |
| 87 |
75 37 86
|
mpsyl |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) |
| 88 |
87
|
simprd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 89 |
72 88
|
readdcld |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ∈ ℝ ) |
| 90 |
|
iunxun |
⊢ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 = ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ∪ 𝑚 ∈ { 𝑧 } ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 91 |
|
vex |
⊢ 𝑧 ∈ V |
| 92 |
|
csbeq1 |
⊢ ( 𝑚 = 𝑧 → ⦋ 𝑚 / 𝑘 ⦌ 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
| 93 |
91 92
|
iunxsn |
⊢ ∪ 𝑚 ∈ { 𝑧 } ⦋ 𝑚 / 𝑘 ⦌ 𝐵 = ⦋ 𝑧 / 𝑘 ⦌ 𝐵 |
| 94 |
93
|
uneq2i |
⊢ ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ∪ 𝑚 ∈ { 𝑧 } ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
| 95 |
90 94
|
eqtri |
⊢ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 = ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) |
| 96 |
95
|
fveq2i |
⊢ ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( vol* ‘ ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) |
| 97 |
|
ovolun |
⊢ ( ( ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ∧ ( ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) ) → ( vol* ‘ ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ≤ ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 98 |
58 72 87 97
|
syl21anc |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ( ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ∪ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ≤ ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 99 |
96 98
|
eqbrtrid |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ≤ ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 100 |
|
ovollecl |
⊢ ( ( ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ⊆ ℝ ∧ ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ∈ ℝ ∧ ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ≤ ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 101 |
55 89 99 100
|
syl3anc |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 102 |
|
snfi |
⊢ { 𝑧 } ∈ Fin |
| 103 |
|
unfi |
⊢ ( ( 𝑦 ∈ Fin ∧ { 𝑧 } ∈ Fin ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 104 |
102 103
|
mpan2 |
⊢ ( 𝑦 ∈ Fin → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 105 |
104
|
ad2antrr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
| 106 |
105 61
|
fsumrecl |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℝ ) |
| 107 |
72 63 88 70
|
leadd1dd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ≤ ( Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 108 |
|
simplr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ¬ 𝑧 ∈ 𝑦 ) |
| 109 |
|
disjsn |
⊢ ( ( 𝑦 ∩ { 𝑧 } ) = ∅ ↔ ¬ 𝑧 ∈ 𝑦 ) |
| 110 |
108 109
|
sylibr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( 𝑦 ∩ { 𝑧 } ) = ∅ ) |
| 111 |
|
eqidd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( 𝑦 ∪ { 𝑧 } ) = ( 𝑦 ∪ { 𝑧 } ) ) |
| 112 |
61
|
recnd |
⊢ ( ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) ∧ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ∈ ℂ ) |
| 113 |
110 111 105 112
|
fsumsplit |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + Σ 𝑚 ∈ { 𝑧 } ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) ) |
| 114 |
88
|
recnd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℂ ) |
| 115 |
92
|
fveq2d |
⊢ ( 𝑚 = 𝑧 → ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) |
| 116 |
115
|
sumsn |
⊢ ( ( 𝑧 ∈ V ∧ ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ∈ ℂ ) → Σ 𝑚 ∈ { 𝑧 } ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) |
| 117 |
91 114 116
|
sylancr |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → Σ 𝑚 ∈ { 𝑧 } ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) |
| 118 |
117
|
oveq2d |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + Σ 𝑚 ∈ { 𝑧 } ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) = ( Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 119 |
113 118
|
eqtrd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) = ( Σ 𝑚 ∈ 𝑦 ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ) |
| 120 |
107 119
|
breqtrrd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( ( vol* ‘ ∪ 𝑚 ∈ 𝑦 ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) + ( vol* ‘ ⦋ 𝑧 / 𝑘 ⦌ 𝐵 ) ) ≤ Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) |
| 121 |
101 89 106 99 120
|
letrd |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ≤ Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) ) |
| 122 |
65 38 45
|
cbviun |
⊢ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 = ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 |
| 123 |
122
|
fveq2i |
⊢ ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) = ( vol* ‘ ∪ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 124 |
47 68 42
|
cbvsum |
⊢ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) = Σ 𝑚 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ ⦋ 𝑚 / 𝑘 ⦌ 𝐵 ) |
| 125 |
121 123 124
|
3brtr4g |
⊢ ( ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) ∧ ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ∧ ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) ) → ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) |
| 126 |
125
|
exp32 |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) → ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) ) ) |
| 127 |
126
|
a2d |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) ) ) |
| 128 |
36 127
|
syl5 |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( ∀ 𝑘 ∈ 𝑦 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝑦 𝐵 ) ≤ Σ 𝑘 ∈ 𝑦 ( vol* ‘ 𝐵 ) ) → ( ∀ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐵 ) ≤ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ( vol* ‘ 𝐵 ) ) ) ) |
| 129 |
6 12 18 24 32 128
|
findcard2s |
⊢ ( 𝐴 ∈ Fin → ( ∀ 𝑘 ∈ 𝐴 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) → ( vol* ‘ ∪ 𝑘 ∈ 𝐴 𝐵 ) ≤ Σ 𝑘 ∈ 𝐴 ( vol* ‘ 𝐵 ) ) ) |
| 130 |
129
|
imp |
⊢ ( ( 𝐴 ∈ Fin ∧ ∀ 𝑘 ∈ 𝐴 ( 𝐵 ⊆ ℝ ∧ ( vol* ‘ 𝐵 ) ∈ ℝ ) ) → ( vol* ‘ ∪ 𝑘 ∈ 𝐴 𝐵 ) ≤ Σ 𝑘 ∈ 𝐴 ( vol* ‘ 𝐵 ) ) |