Step |
Hyp |
Ref |
Expression |
1 |
|
esumrnmpt2.1 |
⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐷 ) |
2 |
|
esumrnmpt2.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
esumrnmpt2.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
4 |
|
esumrnmpt2.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑊 ) |
5 |
|
esumrnmpt2.5 |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝐷 = 0 ) |
6 |
|
esumrnmpt2.6 |
⊢ ( 𝜑 → Disj 𝑘 ∈ 𝐴 𝐵 ) |
7 |
|
nfrab1 |
⊢ Ⅎ 𝑘 { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } |
8 |
|
ssrab2 |
⊢ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴 |
9 |
8
|
a1i |
⊢ ( 𝜑 → { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴 ) |
10 |
2 9
|
ssexd |
⊢ ( 𝜑 → { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V ) |
11 |
9
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝑘 ∈ 𝐴 ) |
12 |
11 3
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
13 |
11 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐵 ∈ 𝑊 ) |
14 |
|
rabid |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↔ ( 𝑘 ∈ 𝐴 ∧ ¬ 𝐵 = ∅ ) ) |
15 |
14
|
simprbi |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } → ¬ 𝐵 = ∅ ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → ¬ 𝐵 = ∅ ) |
17 |
|
elsng |
⊢ ( 𝐵 ∈ 𝑊 → ( 𝐵 ∈ { ∅ } ↔ 𝐵 = ∅ ) ) |
18 |
13 17
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → ( 𝐵 ∈ { ∅ } ↔ 𝐵 = ∅ ) ) |
19 |
16 18
|
mtbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → ¬ 𝐵 ∈ { ∅ } ) |
20 |
13 19
|
eldifd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝐵 ∈ ( 𝑊 ∖ { ∅ } ) ) |
21 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐴 |
22 |
7 21
|
disjss1f |
⊢ ( { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ⊆ 𝐴 → ( Disj 𝑘 ∈ 𝐴 𝐵 → Disj 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐵 ) ) |
23 |
9 6 22
|
sylc |
⊢ ( 𝜑 → Disj 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐵 ) |
24 |
7 1 10 12 20 23
|
esumrnmpt |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) |
25 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
26 |
|
snex |
⊢ { ∅ } ∈ V |
27 |
26
|
a1i |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → { ∅ } ∈ V ) |
28 |
|
velsn |
⊢ ( 𝑦 ∈ { ∅ } ↔ 𝑦 = ∅ ) |
29 |
28
|
biimpi |
⊢ ( 𝑦 ∈ { ∅ } → 𝑦 = ∅ ) |
30 |
29
|
adantl |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝑦 = ∅ ) |
31 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
32 |
|
nfre1 |
⊢ Ⅎ 𝑘 ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ |
33 |
31 32
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
34 |
|
nfv |
⊢ Ⅎ 𝑘 𝑦 = ∅ |
35 |
33 34
|
nfan |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) |
36 |
|
nfv |
⊢ Ⅎ 𝑘 𝐶 = 0 |
37 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝑦 = ∅ ) |
38 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝐵 = ∅ ) |
39 |
37 38
|
eqtr4d |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝑦 = 𝐵 ) |
40 |
39 1
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝐶 = 𝐷 ) |
41 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝜑 ) |
42 |
|
simplr |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝑘 ∈ 𝐴 ) |
43 |
41 42 38 5
|
syl21anc |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝐷 = 0 ) |
44 |
40 43
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) ∧ 𝑘 ∈ 𝐴 ) ∧ 𝐵 = ∅ ) → 𝐶 = 0 ) |
45 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) → ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
46 |
35 36 44 45
|
r19.29af2 |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 = ∅ ) → 𝐶 = 0 ) |
47 |
30 46
|
syldan |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝐶 = 0 ) |
48 |
|
0e0iccpnf |
⊢ 0 ∈ ( 0 [,] +∞ ) |
49 |
47 48
|
eqeltrdi |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) ∧ 𝑦 ∈ { ∅ } ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
50 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑦 |
51 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) |
52 |
51
|
nfrn |
⊢ Ⅎ 𝑘 ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) |
53 |
50 52
|
nfel |
⊢ Ⅎ 𝑘 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) |
54 |
31 53
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) |
55 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 ) |
56 |
|
rabid |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↔ ( 𝑘 ∈ 𝐴 ∧ 𝐵 = ∅ ) ) |
57 |
56
|
simprbi |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } → 𝐵 = ∅ ) |
58 |
57
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐵 = ∅ ) |
59 |
55 58
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 = ∅ ) |
60 |
59 28
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ { ∅ } ) |
61 |
|
vex |
⊢ 𝑦 ∈ V |
62 |
|
eqid |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) |
63 |
62
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝑦 = 𝐵 ) ) |
64 |
61 63
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
65 |
64
|
biimpi |
⊢ ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) → ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
66 |
65
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) → ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
67 |
54 60 66
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) → 𝑦 ∈ { ∅ } ) |
68 |
67
|
ex |
⊢ ( 𝜑 → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) → 𝑦 ∈ { ∅ } ) ) |
69 |
68
|
ssrdv |
⊢ ( 𝜑 → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ⊆ { ∅ } ) |
70 |
69
|
adantr |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ⊆ { ∅ } ) |
71 |
25 27 49 70
|
esummono |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ Σ* 𝑦 ∈ { ∅ } 𝐶 ) |
72 |
|
0ex |
⊢ ∅ ∈ V |
73 |
72
|
a1i |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → ∅ ∈ V ) |
74 |
48
|
a1i |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → 0 ∈ ( 0 [,] +∞ ) ) |
75 |
46 73 74
|
esumsn |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ { ∅ } 𝐶 = 0 ) |
76 |
71 75
|
breqtrd |
⊢ ( ( 𝜑 ∧ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ) |
77 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
78 |
|
nfv |
⊢ Ⅎ 𝑦 ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ |
79 |
32
|
nfn |
⊢ Ⅎ 𝑘 ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ |
80 |
|
rabn0 |
⊢ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ≠ ∅ ↔ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
81 |
80
|
biimpi |
⊢ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ≠ ∅ → ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) |
82 |
81
|
necon1bi |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } = ∅ ) |
83 |
|
eqid |
⊢ 𝐵 = 𝐵 |
84 |
83
|
a1i |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → 𝐵 = 𝐵 ) |
85 |
79 82 84
|
mpteq12df |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ ∅ ↦ 𝐵 ) ) |
86 |
|
mpt0 |
⊢ ( 𝑘 ∈ ∅ ↦ 𝐵 ) = ∅ |
87 |
85 86
|
eqtrdi |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) = ∅ ) |
88 |
87
|
rneqd |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) = ran ∅ ) |
89 |
|
rn0 |
⊢ ran ∅ = ∅ |
90 |
88 89
|
eqtrdi |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) = ∅ ) |
91 |
78 90
|
esumeq1d |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = Σ* 𝑦 ∈ ∅ 𝐶 ) |
92 |
|
esumnul |
⊢ Σ* 𝑦 ∈ ∅ 𝐶 = 0 |
93 |
91 92
|
eqtrdi |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ) |
94 |
|
0le0 |
⊢ 0 ≤ 0 |
95 |
93 94
|
eqbrtrdi |
⊢ ( ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ) |
96 |
77 95
|
syl |
⊢ ( ( 𝜑 ∧ ¬ ∃ 𝑘 ∈ 𝐴 𝐵 = ∅ ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ) |
97 |
76 96
|
pm2.61dan |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ) |
98 |
|
ssrab2 |
⊢ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ⊆ 𝐴 |
99 |
98
|
a1i |
⊢ ( 𝜑 → { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ⊆ 𝐴 ) |
100 |
2 99
|
ssexd |
⊢ ( 𝜑 → { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∈ V ) |
101 |
|
nfrab1 |
⊢ Ⅎ 𝑘 { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } |
102 |
101
|
mptexgf |
⊢ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∈ V → ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
103 |
|
rnexg |
⊢ ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
104 |
100 102 103
|
3syl |
⊢ ( 𝜑 → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
105 |
1
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐷 ) |
106 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝜑 ) |
107 |
99
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝑘 ∈ 𝐴 ) |
108 |
107
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝑘 ∈ 𝐴 ) |
109 |
108
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑘 ∈ 𝐴 ) |
110 |
106 109 3
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
111 |
105 110
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
112 |
54 111 66
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
113 |
112
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
114 |
|
nfcv |
⊢ Ⅎ 𝑦 ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) |
115 |
114
|
esumcl |
⊢ ( ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ∧ ∀ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
116 |
104 113 115
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ) |
117 |
|
elxrge0 |
⊢ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) |
118 |
117
|
simprbi |
⊢ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ( 0 [,] +∞ ) → 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) |
119 |
116 118
|
syl |
⊢ ( 𝜑 → 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) |
120 |
97 119
|
jca |
⊢ ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) |
121 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
122 |
121 116
|
sselid |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ) |
123 |
121 48
|
sselii |
⊢ 0 ∈ ℝ* |
124 |
123
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ* ) |
125 |
|
xrletri3 |
⊢ ( ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) ) |
126 |
122 124 125
|
syl2anc |
⊢ ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ↔ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ≤ 0 ∧ 0 ≤ Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) ) |
127 |
120 126
|
mpbird |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 = 0 ) |
128 |
127
|
oveq1d |
⊢ ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) |
129 |
12
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) ) |
130 |
7
|
esumcl |
⊢ ( ( { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V ∧ ∀ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) ) → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) ) |
131 |
10 129 130
|
syl2anc |
⊢ ( 𝜑 → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ( 0 [,] +∞ ) ) |
132 |
121 131
|
sselid |
⊢ ( 𝜑 → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ℝ* ) |
133 |
24 132
|
eqeltrd |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* ) |
134 |
|
xaddid2 |
⊢ ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ∈ ℝ* → ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) |
135 |
133 134
|
syl |
⊢ ( 𝜑 → ( 0 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) |
136 |
128 135
|
eqtrd |
⊢ ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) |
137 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝜑 ) |
138 |
57
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝐵 = ∅ ) |
139 |
137 107 138 5
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝐷 = 0 ) |
140 |
139
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 = 0 ) |
141 |
31 140
|
esumeq2d |
⊢ ( 𝜑 → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 = Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 0 ) |
142 |
101
|
esum0 |
⊢ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∈ V → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 0 = 0 ) |
143 |
100 142
|
syl |
⊢ ( 𝜑 → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 0 = 0 ) |
144 |
141 143
|
eqtrd |
⊢ ( 𝜑 → Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 = 0 ) |
145 |
144
|
oveq1d |
⊢ ( 𝜑 → ( Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) ) |
146 |
|
xaddid2 |
⊢ ( Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ∈ ℝ* → ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) |
147 |
132 146
|
syl |
⊢ ( 𝜑 → ( 0 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) |
148 |
145 147
|
eqtrd |
⊢ ( 𝜑 → ( Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) = Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) |
149 |
24 136 148
|
3eqtr4d |
⊢ ( 𝜑 → ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) = ( Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) ) |
150 |
|
nfv |
⊢ Ⅎ 𝑦 𝜑 |
151 |
|
nfcv |
⊢ Ⅎ 𝑦 ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
152 |
7
|
mptexgf |
⊢ ( { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∈ V → ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
153 |
|
rnexg |
⊢ ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
154 |
10 152 153
|
3syl |
⊢ ( 𝜑 → ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∈ V ) |
155 |
69
|
ssrind |
⊢ ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ) |
156 |
|
incom |
⊢ ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
157 |
15
|
neqned |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } → 𝐵 ≠ ∅ ) |
158 |
157
|
necomd |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } → ∅ ≠ 𝐵 ) |
159 |
158
|
neneqd |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } → ¬ ∅ = 𝐵 ) |
160 |
159
|
nrex |
⊢ ¬ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵 |
161 |
|
eqid |
⊢ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) = ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
162 |
161
|
elrnmpt |
⊢ ( ∅ ∈ V → ( ∅ ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵 ) ) |
163 |
72 162
|
ax-mp |
⊢ ( ∅ ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ∅ = 𝐵 ) |
164 |
160 163
|
mtbir |
⊢ ¬ ∅ ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
165 |
|
disjsn |
⊢ ( ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ∅ ↔ ¬ ∅ ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
166 |
164 165
|
mpbir |
⊢ ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ∩ { ∅ } ) = ∅ |
167 |
156 166
|
eqtr3i |
⊢ ( { ∅ } ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅ |
168 |
155 167
|
sseqtrdi |
⊢ ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ∅ ) |
169 |
|
ss0 |
⊢ ( ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ⊆ ∅ → ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅ ) |
170 |
168 169
|
syl |
⊢ ( 𝜑 → ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∩ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ∅ ) |
171 |
|
nfmpt1 |
⊢ Ⅎ 𝑘 ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
172 |
171
|
nfrn |
⊢ Ⅎ 𝑘 ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
173 |
50 172
|
nfel |
⊢ Ⅎ 𝑘 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) |
174 |
31 173
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
175 |
1
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 = 𝐷 ) |
176 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝜑 ) |
177 |
11
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) → 𝑘 ∈ 𝐴 ) |
178 |
177
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝑘 ∈ 𝐴 ) |
179 |
176 178 3
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
180 |
175 179
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ∧ 𝑦 = 𝐵 ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
181 |
161
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 ) ) |
182 |
61 181
|
ax-mp |
⊢ ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ↔ ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
183 |
182
|
biimpi |
⊢ ( 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) → ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
184 |
183
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) → ∃ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝑦 = 𝐵 ) |
185 |
174 180 184
|
r19.29af |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) → 𝐶 ∈ ( 0 [,] +∞ ) ) |
186 |
150 114 151 104 154 170 112 185
|
esumsplit |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 = ( Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 +𝑒 Σ* 𝑦 ∈ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) 𝐶 ) ) |
187 |
|
rabnc |
⊢ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∩ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) = ∅ |
188 |
187
|
a1i |
⊢ ( 𝜑 → ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∩ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) = ∅ ) |
189 |
107 3
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
190 |
31 101 7 100 10 188 189 12
|
esumsplit |
⊢ ( 𝜑 → Σ* 𝑘 ∈ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 = ( Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } 𝐷 +𝑒 Σ* 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } 𝐷 ) ) |
191 |
149 186 190
|
3eqtr4d |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 = Σ* 𝑘 ∈ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 ) |
192 |
|
rabxm |
⊢ 𝐴 = ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) |
193 |
192 83
|
mpteq12i |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ↦ 𝐵 ) |
194 |
|
mptun |
⊢ ( 𝑘 ∈ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ↦ 𝐵 ) = ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
195 |
193 194
|
eqtri |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
196 |
195
|
rneqi |
⊢ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ran ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
197 |
|
rnun |
⊢ ran ( ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) = ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
198 |
196 197
|
eqtri |
⊢ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) |
199 |
198
|
a1i |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) ) |
200 |
150 199
|
esumeq1d |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) 𝐶 = Σ* 𝑦 ∈ ( ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ↦ 𝐵 ) ∪ ran ( 𝑘 ∈ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ↦ 𝐵 ) ) 𝐶 ) |
201 |
192
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) ) |
202 |
31 201
|
esumeq1d |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐷 = Σ* 𝑘 ∈ ( { 𝑘 ∈ 𝐴 ∣ 𝐵 = ∅ } ∪ { 𝑘 ∈ 𝐴 ∣ ¬ 𝐵 = ∅ } ) 𝐷 ) |
203 |
191 200 202
|
3eqtr4d |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) 𝐶 = Σ* 𝑘 ∈ 𝐴 𝐷 ) |