Step |
Hyp |
Ref |
Expression |
1 |
|
carsgval.1 |
⊢ ( 𝜑 → 𝑂 ∈ 𝑉 ) |
2 |
|
carsgval.2 |
⊢ ( 𝜑 → 𝑀 : 𝒫 𝑂 ⟶ ( 0 [,] +∞ ) ) |
3 |
|
carsgsiga.1 |
⊢ ( 𝜑 → ( 𝑀 ‘ ∅ ) = 0 ) |
4 |
|
carsgsiga.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ 𝑥 ) ≤ Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
5 |
|
carsgsiga.3 |
⊢ ( ( 𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ 𝑥 ) ≤ ( 𝑀 ‘ 𝑦 ) ) |
6 |
|
carsgclctun.1 |
⊢ ( 𝜑 → 𝐴 ≼ ω ) |
7 |
|
carsgclctun.2 |
⊢ ( 𝜑 → 𝐴 ⊆ ( toCaraSiga ‘ 𝑀 ) ) |
8 |
7
|
unissd |
⊢ ( 𝜑 → ∪ 𝐴 ⊆ ∪ ( toCaraSiga ‘ 𝑀 ) ) |
9 |
1 2 3
|
carsguni |
⊢ ( 𝜑 → ∪ ( toCaraSiga ‘ 𝑀 ) = 𝑂 ) |
10 |
8 9
|
sseqtrd |
⊢ ( 𝜑 → ∪ 𝐴 ⊆ 𝑂 ) |
11 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → 𝑂 ∈ 𝑉 ) |
12 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → 𝑀 : 𝒫 𝑂 ⟶ ( 0 [,] +∞ ) ) |
13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ∅ ) = 0 ) |
14 |
4
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ 𝑥 ) ≤ Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) |
15 |
5
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ 𝑥 ) ≤ ( 𝑀 ‘ 𝑦 ) ) |
16 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → 𝐴 ≼ ω ) |
17 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → 𝐴 ⊆ ( toCaraSiga ‘ 𝑀 ) ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → 𝑒 ∈ 𝒫 𝑂 ) |
19 |
11 12 13 14 15 16 17 18
|
carsgclctunlem3 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ≤ ( 𝑀 ‘ 𝑒 ) ) |
20 |
|
inex1g |
⊢ ( 𝑒 ∈ 𝒫 𝑂 → ( 𝑒 ∩ ∪ 𝐴 ) ∈ V ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑒 ∩ ∪ 𝐴 ) ∈ V ) |
22 |
|
difexg |
⊢ ( 𝑒 ∈ 𝒫 𝑂 → ( 𝑒 ∖ ∪ 𝐴 ) ∈ V ) |
23 |
22
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑒 ∖ ∪ 𝐴 ) ∈ V ) |
24 |
|
prct |
⊢ ( ( ( 𝑒 ∩ ∪ 𝐴 ) ∈ V ∧ ( 𝑒 ∖ ∪ 𝐴 ) ∈ V ) → { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ) |
25 |
21 23 24
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ) |
26 |
18
|
elpwincl1 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑒 ∩ ∪ 𝐴 ) ∈ 𝒫 𝑂 ) |
27 |
18
|
elpwdifcl |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑒 ∖ ∪ 𝐴 ) ∈ 𝒫 𝑂 ) |
28 |
|
prssi |
⊢ ( ( ( 𝑒 ∩ ∪ 𝐴 ) ∈ 𝒫 𝑂 ∧ ( 𝑒 ∖ ∪ 𝐴 ) ∈ 𝒫 𝑂 ) → { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) |
29 |
26 27 28
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) |
30 |
|
prex |
⊢ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ∈ V |
31 |
|
breq1 |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( 𝑥 ≼ ω ↔ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ) ) |
32 |
|
sseq1 |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( 𝑥 ⊆ 𝒫 𝑂 ↔ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) ) |
33 |
31 32
|
3anbi23d |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( ( 𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂 ) ↔ ( 𝜑 ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) ) ) |
34 |
|
unieq |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ∪ 𝑥 = ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) |
35 |
34
|
fveq2d |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( 𝑀 ‘ ∪ 𝑥 ) = ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ) |
36 |
|
esumeq1 |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) = Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
37 |
35 36
|
breq12d |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( ( 𝑀 ‘ ∪ 𝑥 ) ≤ Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ↔ ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) ) |
38 |
33 37
|
imbi12d |
⊢ ( 𝑥 = { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } → ( ( ( 𝜑 ∧ 𝑥 ≼ ω ∧ 𝑥 ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ 𝑥 ) ≤ Σ* 𝑦 ∈ 𝑥 ( 𝑀 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) ) ) |
39 |
38 4
|
vtoclg |
⊢ ( { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ∈ V → ( ( 𝜑 ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) ) |
40 |
30 39
|
ax-mp |
⊢ ( ( 𝜑 ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
41 |
40
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ≼ ω ∧ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ⊆ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
42 |
25 29 41
|
mpd3an23 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) ≤ Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) ) |
43 |
|
uniprg |
⊢ ( ( ( 𝑒 ∩ ∪ 𝐴 ) ∈ 𝒫 𝑂 ∧ ( 𝑒 ∖ ∪ 𝐴 ) ∈ 𝒫 𝑂 ) → ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } = ( ( 𝑒 ∩ ∪ 𝐴 ) ∪ ( 𝑒 ∖ ∪ 𝐴 ) ) ) |
44 |
26 27 43
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } = ( ( 𝑒 ∩ ∪ 𝐴 ) ∪ ( 𝑒 ∖ ∪ 𝐴 ) ) ) |
45 |
|
inundif |
⊢ ( ( 𝑒 ∩ ∪ 𝐴 ) ∪ ( 𝑒 ∖ ∪ 𝐴 ) ) = 𝑒 |
46 |
44 45
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } = 𝑒 ) |
47 |
46
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ∪ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ) = ( 𝑀 ‘ 𝑒 ) ) |
48 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑦 = ( 𝑒 ∩ ∪ 𝐴 ) ) → 𝑦 = ( 𝑒 ∩ ∪ 𝐴 ) ) |
49 |
48
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑦 = ( 𝑒 ∩ ∪ 𝐴 ) ) → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) ) |
50 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑦 = ( 𝑒 ∖ ∪ 𝐴 ) ) → 𝑦 = ( 𝑒 ∖ ∪ 𝐴 ) ) |
51 |
50
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ 𝑦 = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( 𝑀 ‘ 𝑦 ) = ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) |
52 |
12 26
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) ∈ ( 0 [,] +∞ ) ) |
53 |
12 27
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ∈ ( 0 [,] +∞ ) ) |
54 |
|
ineq2 |
⊢ ( ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) → ( ( 𝑒 ∩ ∪ 𝐴 ) ∩ ( 𝑒 ∩ ∪ 𝐴 ) ) = ( ( 𝑒 ∩ ∪ 𝐴 ) ∩ ( 𝑒 ∖ ∪ 𝐴 ) ) ) |
55 |
|
inidm |
⊢ ( ( 𝑒 ∩ ∪ 𝐴 ) ∩ ( 𝑒 ∩ ∪ 𝐴 ) ) = ( 𝑒 ∩ ∪ 𝐴 ) |
56 |
|
inindif |
⊢ ( ( 𝑒 ∩ ∪ 𝐴 ) ∩ ( 𝑒 ∖ ∪ 𝐴 ) ) = ∅ |
57 |
54 55 56
|
3eqtr3g |
⊢ ( ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) → ( 𝑒 ∩ ∪ 𝐴 ) = ∅ ) |
58 |
57
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( 𝑒 ∩ ∪ 𝐴 ) = ∅ ) |
59 |
58
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = ( 𝑀 ‘ ∅ ) ) |
60 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( 𝑀 ‘ ∅ ) = 0 ) |
61 |
59 60
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = 0 ) |
62 |
61
|
orcd |
⊢ ( ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) ∧ ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) ) → ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = 0 ∨ ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = +∞ ) ) |
63 |
62
|
ex |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( 𝑒 ∩ ∪ 𝐴 ) = ( 𝑒 ∖ ∪ 𝐴 ) → ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = 0 ∨ ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) = +∞ ) ) ) |
64 |
49 51 26 27 52 53 63
|
esumpr2 |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → Σ* 𝑦 ∈ { ( 𝑒 ∩ ∪ 𝐴 ) , ( 𝑒 ∖ ∪ 𝐴 ) } ( 𝑀 ‘ 𝑦 ) = ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ) |
65 |
42 47 64
|
3brtr3d |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ 𝑒 ) ≤ ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ) |
66 |
19 65
|
jca |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ≤ ( 𝑀 ‘ 𝑒 ) ∧ ( 𝑀 ‘ 𝑒 ) ≤ ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ) ) |
67 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
68 |
67 52
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) ∈ ℝ* ) |
69 |
67 53
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ∈ ℝ* ) |
70 |
68 69
|
xaddcld |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ∈ ℝ* ) |
71 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ 𝑒 ) ∈ ( 0 [,] +∞ ) ) |
72 |
67 71
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( 𝑀 ‘ 𝑒 ) ∈ ℝ* ) |
73 |
|
xrletri3 |
⊢ ( ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ∈ ℝ* ∧ ( 𝑀 ‘ 𝑒 ) ∈ ℝ* ) → ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ↔ ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ≤ ( 𝑀 ‘ 𝑒 ) ∧ ( 𝑀 ‘ 𝑒 ) ≤ ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ) ) ) |
74 |
70 72 73
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ↔ ( ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ≤ ( 𝑀 ‘ 𝑒 ) ∧ ( 𝑀 ‘ 𝑒 ) ≤ ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) ) ) ) |
75 |
66 74
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑒 ∈ 𝒫 𝑂 ) → ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) |
76 |
75
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) |
77 |
1 2
|
elcarsg |
⊢ ( 𝜑 → ( ∪ 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ↔ ( ∪ 𝐴 ⊆ 𝑂 ∧ ∀ 𝑒 ∈ 𝒫 𝑂 ( ( 𝑀 ‘ ( 𝑒 ∩ ∪ 𝐴 ) ) +𝑒 ( 𝑀 ‘ ( 𝑒 ∖ ∪ 𝐴 ) ) ) = ( 𝑀 ‘ 𝑒 ) ) ) ) |
78 |
10 76 77
|
mpbir2and |
⊢ ( 𝜑 → ∪ 𝐴 ∈ ( toCaraSiga ‘ 𝑀 ) ) |