Step |
Hyp |
Ref |
Expression |
1 |
|
oms.m |
⊢ 𝑀 = ( toOMeas ‘ 𝑅 ) |
2 |
|
oms.o |
⊢ ( 𝜑 → 𝑄 ∈ 𝑉 ) |
3 |
|
oms.r |
⊢ ( 𝜑 → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
4 |
|
oms.d |
⊢ ( 𝜑 → ∅ ∈ dom 𝑅 ) |
5 |
|
oms.0 |
⊢ ( 𝜑 → ( 𝑅 ‘ ∅ ) = 0 ) |
6 |
1
|
fveq1i |
⊢ ( 𝑀 ‘ ∅ ) = ( ( toOMeas ‘ 𝑅 ) ‘ ∅ ) |
7 |
|
0ss |
⊢ ∅ ⊆ ∪ dom 𝑅 |
8 |
3
|
fdmd |
⊢ ( 𝜑 → dom 𝑅 = 𝑄 ) |
9 |
8
|
unieqd |
⊢ ( 𝜑 → ∪ dom 𝑅 = ∪ 𝑄 ) |
10 |
7 9
|
sseqtrid |
⊢ ( 𝜑 → ∅ ⊆ ∪ 𝑄 ) |
11 |
|
omsfval |
⊢ ( ( 𝑄 ∈ 𝑉 ∧ 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ∧ ∅ ⊆ ∪ 𝑄 ) → ( ( toOMeas ‘ 𝑅 ) ‘ ∅ ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) |
12 |
2 3 10 11
|
syl3anc |
⊢ ( 𝜑 → ( ( toOMeas ‘ 𝑅 ) ‘ ∅ ) = inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) ) |
13 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
14 |
|
xrltso |
⊢ < Or ℝ* |
15 |
|
soss |
⊢ ( ( 0 [,] +∞ ) ⊆ ℝ* → ( < Or ℝ* → < Or ( 0 [,] +∞ ) ) ) |
16 |
13 14 15
|
mp2 |
⊢ < Or ( 0 [,] +∞ ) |
17 |
16
|
a1i |
⊢ ( 𝜑 → < Or ( 0 [,] +∞ ) ) |
18 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
19 |
18
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( 0 [,] +∞ ) ) |
20 |
4
|
snssd |
⊢ ( 𝜑 → { ∅ } ⊆ dom 𝑅 ) |
21 |
|
p0ex |
⊢ { ∅ } ∈ V |
22 |
21
|
elpw |
⊢ ( { ∅ } ∈ 𝒫 dom 𝑅 ↔ { ∅ } ⊆ dom 𝑅 ) |
23 |
20 22
|
sylibr |
⊢ ( 𝜑 → { ∅ } ∈ 𝒫 dom 𝑅 ) |
24 |
|
0ss |
⊢ ∅ ⊆ ∪ { ∅ } |
25 |
|
0ex |
⊢ ∅ ∈ V |
26 |
|
snct |
⊢ ( ∅ ∈ V → { ∅ } ≼ ω ) |
27 |
25 26
|
ax-mp |
⊢ { ∅ } ≼ ω |
28 |
24 27
|
pm3.2i |
⊢ ( ∅ ⊆ ∪ { ∅ } ∧ { ∅ } ≼ ω ) |
29 |
23 28
|
jctir |
⊢ ( 𝜑 → ( { ∅ } ∈ 𝒫 dom 𝑅 ∧ ( ∅ ⊆ ∪ { ∅ } ∧ { ∅ } ≼ ω ) ) ) |
30 |
|
unieq |
⊢ ( 𝑧 = { ∅ } → ∪ 𝑧 = ∪ { ∅ } ) |
31 |
30
|
sseq2d |
⊢ ( 𝑧 = { ∅ } → ( ∅ ⊆ ∪ 𝑧 ↔ ∅ ⊆ ∪ { ∅ } ) ) |
32 |
|
breq1 |
⊢ ( 𝑧 = { ∅ } → ( 𝑧 ≼ ω ↔ { ∅ } ≼ ω ) ) |
33 |
31 32
|
anbi12d |
⊢ ( 𝑧 = { ∅ } → ( ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) ↔ ( ∅ ⊆ ∪ { ∅ } ∧ { ∅ } ≼ ω ) ) ) |
34 |
33
|
elrab |
⊢ ( { ∅ } ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↔ ( { ∅ } ∈ 𝒫 dom 𝑅 ∧ ( ∅ ⊆ ∪ { ∅ } ∧ { ∅ } ≼ ω ) ) ) |
35 |
29 34
|
sylibr |
⊢ ( 𝜑 → { ∅ } ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
36 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → 𝑦 = ∅ ) |
37 |
36
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ( 𝑅 ‘ 𝑦 ) = ( 𝑅 ‘ ∅ ) ) |
38 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ( 𝑅 ‘ ∅ ) = 0 ) |
39 |
37 38
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑦 = ∅ ) → ( 𝑅 ‘ 𝑦 ) = 0 ) |
40 |
39 4 19
|
esumsn |
⊢ ( 𝜑 → Σ* 𝑦 ∈ { ∅ } ( 𝑅 ‘ 𝑦 ) = 0 ) |
41 |
40
|
eqcomd |
⊢ ( 𝜑 → 0 = Σ* 𝑦 ∈ { ∅ } ( 𝑅 ‘ 𝑦 ) ) |
42 |
|
esumeq1 |
⊢ ( 𝑥 = { ∅ } → Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) = Σ* 𝑦 ∈ { ∅ } ( 𝑅 ‘ 𝑦 ) ) |
43 |
42
|
rspceeqv |
⊢ ( ( { ∅ } ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ∧ 0 = Σ* 𝑦 ∈ { ∅ } ( 𝑅 ‘ 𝑦 ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 0 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
44 |
35 41 43
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 0 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
45 |
|
0xr |
⊢ 0 ∈ ℝ* |
46 |
|
eqid |
⊢ ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) = ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
47 |
46
|
elrnmpt |
⊢ ( 0 ∈ ℝ* → ( 0 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 0 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
48 |
45 47
|
ax-mp |
⊢ ( 0 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 0 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
49 |
44 48
|
sylibr |
⊢ ( 𝜑 → 0 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
50 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
51 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
52 |
51
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
53 |
52
|
nfcri |
⊢ Ⅎ 𝑥 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
54 |
50 53
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
55 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
56 |
|
vex |
⊢ 𝑥 ∈ V |
57 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
58 |
|
nfcv |
⊢ Ⅎ 𝑦 { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } |
59 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑥 |
60 |
59
|
nfesum1 |
⊢ Ⅎ 𝑦 Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) |
61 |
58 60
|
nfmpt |
⊢ Ⅎ 𝑦 ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
62 |
61
|
nfrn |
⊢ Ⅎ 𝑦 ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
63 |
62
|
nfcri |
⊢ Ⅎ 𝑦 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
64 |
57 63
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
65 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } |
66 |
64 65
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
67 |
60
|
nfeq2 |
⊢ Ⅎ 𝑦 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) |
68 |
66 67
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
69 |
3
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑅 : 𝑄 ⟶ ( 0 [,] +∞ ) ) |
70 |
|
ssrab2 |
⊢ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ⊆ 𝒫 dom 𝑅 |
71 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) |
72 |
70 71
|
sselid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ∈ 𝒫 dom 𝑅 ) |
73 |
8
|
pweqd |
⊢ ( 𝜑 → 𝒫 dom 𝑅 = 𝒫 𝑄 ) |
74 |
73
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝒫 dom 𝑅 = 𝒫 𝑄 ) |
75 |
72 74
|
eleqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ∈ 𝒫 𝑄 ) |
76 |
75
|
elpwid |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑥 ⊆ 𝑄 ) |
77 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑥 ) |
78 |
76 77
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ 𝑄 ) |
79 |
69 78
|
ffvelrnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ∧ 𝑦 ∈ 𝑥 ) → ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
80 |
79
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → ( 𝑦 ∈ 𝑥 → ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) ) |
81 |
68 80
|
ralrimi |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → ∀ 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
82 |
59
|
esumcl |
⊢ ( ( 𝑥 ∈ V ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
83 |
56 81 82
|
sylancr |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ∈ ( 0 [,] +∞ ) ) |
84 |
55 83
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) ∧ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ) ∧ 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → 𝑎 ∈ ( 0 [,] +∞ ) ) |
85 |
|
vex |
⊢ 𝑎 ∈ V |
86 |
46
|
elrnmpt |
⊢ ( 𝑎 ∈ V → ( 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) |
87 |
85 86
|
ax-mp |
⊢ ( 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
88 |
87
|
biimpi |
⊢ ( 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
89 |
88
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → ∃ 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } 𝑎 = Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) |
90 |
54 84 89
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → 𝑎 ∈ ( 0 [,] +∞ ) ) |
91 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
92 |
|
iccgelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑎 ∈ ( 0 [,] +∞ ) ) → 0 ≤ 𝑎 ) |
93 |
45 91 92
|
mp3an12 |
⊢ ( 𝑎 ∈ ( 0 [,] +∞ ) → 0 ≤ 𝑎 ) |
94 |
90 93
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → 0 ≤ 𝑎 ) |
95 |
13 90
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → 𝑎 ∈ ℝ* ) |
96 |
|
xrlenlt |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ) → ( 0 ≤ 𝑎 ↔ ¬ 𝑎 < 0 ) ) |
97 |
96
|
bicomd |
⊢ ( ( 0 ∈ ℝ* ∧ 𝑎 ∈ ℝ* ) → ( ¬ 𝑎 < 0 ↔ 0 ≤ 𝑎 ) ) |
98 |
45 95 97
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → ( ¬ 𝑎 < 0 ↔ 0 ≤ 𝑎 ) ) |
99 |
94 98
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) ) → ¬ 𝑎 < 0 ) |
100 |
17 19 49 99
|
infmin |
⊢ ( 𝜑 → inf ( ran ( 𝑥 ∈ { 𝑧 ∈ 𝒫 dom 𝑅 ∣ ( ∅ ⊆ ∪ 𝑧 ∧ 𝑧 ≼ ω ) } ↦ Σ* 𝑦 ∈ 𝑥 ( 𝑅 ‘ 𝑦 ) ) , ( 0 [,] +∞ ) , < ) = 0 ) |
101 |
12 100
|
eqtrd |
⊢ ( 𝜑 → ( ( toOMeas ‘ 𝑅 ) ‘ ∅ ) = 0 ) |
102 |
6 101
|
syl5eq |
⊢ ( 𝜑 → ( 𝑀 ‘ ∅ ) = 0 ) |