Step |
Hyp |
Ref |
Expression |
1 |
|
fsumo1.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
fsumo1.2 |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
3 |
|
fsumo1.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) → 𝐶 ∈ 𝑉 ) |
4 |
|
fsumo1.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |
5 |
|
ssid |
⊢ 𝐵 ⊆ 𝐵 |
6 |
|
sseq1 |
⊢ ( 𝑤 = ∅ → ( 𝑤 ⊆ 𝐵 ↔ ∅ ⊆ 𝐵 ) ) |
7 |
|
sumeq1 |
⊢ ( 𝑤 = ∅ → Σ 𝑘 ∈ 𝑤 𝐶 = Σ 𝑘 ∈ ∅ 𝐶 ) |
8 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ 𝐶 = 0 |
9 |
7 8
|
eqtrdi |
⊢ ( 𝑤 = ∅ → Σ 𝑘 ∈ 𝑤 𝐶 = 0 ) |
10 |
9
|
mpteq2dv |
⊢ ( 𝑤 = ∅ → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 0 ) ) |
11 |
10
|
eleq1d |
⊢ ( 𝑤 = ∅ → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) ) |
12 |
6 11
|
imbi12d |
⊢ ( 𝑤 = ∅ → ( ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ↔ ( ∅ ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑤 = ∅ → ( ( 𝜑 → ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ) ↔ ( 𝜑 → ( ∅ ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) ) ) ) |
14 |
|
sseq1 |
⊢ ( 𝑤 = 𝑦 → ( 𝑤 ⊆ 𝐵 ↔ 𝑦 ⊆ 𝐵 ) ) |
15 |
|
sumeq1 |
⊢ ( 𝑤 = 𝑦 → Σ 𝑘 ∈ 𝑤 𝐶 = Σ 𝑘 ∈ 𝑦 𝐶 ) |
16 |
15
|
mpteq2dv |
⊢ ( 𝑤 = 𝑦 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ) |
17 |
16
|
eleq1d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) |
18 |
14 17
|
imbi12d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ↔ ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑤 = 𝑦 → ( ( 𝜑 → ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ) ↔ ( 𝜑 → ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
20 |
|
sseq1 |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑤 ⊆ 𝐵 ↔ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) |
21 |
|
sumeq1 |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → Σ 𝑘 ∈ 𝑤 𝐶 = Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) |
22 |
21
|
mpteq2dv |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ) |
23 |
22
|
eleq1d |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) |
24 |
20 23
|
imbi12d |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ↔ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) |
25 |
24
|
imbi2d |
⊢ ( 𝑤 = ( 𝑦 ∪ { 𝑧 } ) → ( ( 𝜑 → ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ) ↔ ( 𝜑 → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
26 |
|
sseq1 |
⊢ ( 𝑤 = 𝐵 → ( 𝑤 ⊆ 𝐵 ↔ 𝐵 ⊆ 𝐵 ) ) |
27 |
|
sumeq1 |
⊢ ( 𝑤 = 𝐵 → Σ 𝑘 ∈ 𝑤 𝐶 = Σ 𝑘 ∈ 𝐵 𝐶 ) |
28 |
27
|
mpteq2dv |
⊢ ( 𝑤 = 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ) |
29 |
28
|
eleq1d |
⊢ ( 𝑤 = 𝐵 → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) ) |
30 |
26 29
|
imbi12d |
⊢ ( 𝑤 = 𝐵 → ( ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ↔ ( 𝐵 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) ) ) |
31 |
30
|
imbi2d |
⊢ ( 𝑤 = 𝐵 → ( ( 𝜑 → ( 𝑤 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑤 𝐶 ) ∈ 𝑂(1) ) ) ↔ ( 𝜑 → ( 𝐵 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
32 |
|
0cn |
⊢ 0 ∈ ℂ |
33 |
|
o1const |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 0 ∈ ℂ ) → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) |
34 |
1 32 33
|
sylancl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) |
35 |
34
|
a1d |
⊢ ( 𝜑 → ( ∅ ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ 0 ) ∈ 𝑂(1) ) ) |
36 |
|
ssun1 |
⊢ 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) |
37 |
|
sstr |
⊢ ( ( 𝑦 ⊆ ( 𝑦 ∪ { 𝑧 } ) ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) → 𝑦 ⊆ 𝐵 ) |
38 |
36 37
|
mpan |
⊢ ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → 𝑦 ⊆ 𝐵 ) |
39 |
38
|
imim1i |
⊢ ( ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) |
40 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ¬ 𝑧 ∈ 𝑦 ) |
41 |
|
disjsn |
⊢ ( ( 𝑦 ∩ { 𝑧 } ) = ∅ ↔ ¬ 𝑧 ∈ 𝑦 ) |
42 |
40 41
|
sylibr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑦 ∩ { 𝑧 } ) = ∅ ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∩ { 𝑧 } ) = ∅ ) |
44 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∪ { 𝑧 } ) = ( 𝑦 ∪ { 𝑧 } ) ) |
45 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → 𝐵 ∈ Fin ) |
46 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) |
47 |
45 46
|
ssfid |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑦 ∪ { 𝑧 } ) ∈ Fin ) |
49 |
46
|
sselda |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → 𝑘 ∈ 𝐵 ) |
50 |
49
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → 𝑘 ∈ 𝐵 ) |
51 |
3
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 ) |
52 |
51 4
|
o1mptrcl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
53 |
52
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ℂ ) |
54 |
53
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ℂ ) |
55 |
50 54
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) ) → 𝐶 ∈ ℂ ) |
56 |
43 44 48 55
|
fsumsplit |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 = ( Σ 𝑘 ∈ 𝑦 𝐶 + Σ 𝑘 ∈ { 𝑧 } 𝐶 ) ) |
57 |
|
nfcv |
⊢ Ⅎ 𝑤 𝐶 |
58 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑤 / 𝑘 ⦌ 𝐶 |
59 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑤 → 𝐶 = ⦋ 𝑤 / 𝑘 ⦌ 𝐶 ) |
60 |
57 58 59
|
cbvsumi |
⊢ Σ 𝑘 ∈ { 𝑧 } 𝐶 = Σ 𝑤 ∈ { 𝑧 } ⦋ 𝑤 / 𝑘 ⦌ 𝐶 |
61 |
46
|
unssbd |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → { 𝑧 } ⊆ 𝐵 ) |
62 |
|
vex |
⊢ 𝑧 ∈ V |
63 |
62
|
snss |
⊢ ( 𝑧 ∈ 𝐵 ↔ { 𝑧 } ⊆ 𝐵 ) |
64 |
61 63
|
sylibr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → 𝑧 ∈ 𝐵 ) |
65 |
64
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑧 ∈ 𝐵 ) |
66 |
54
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ℂ ) |
67 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑧 / 𝑘 ⦌ 𝐶 |
68 |
67
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ∈ ℂ |
69 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑧 → 𝐶 = ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
70 |
69
|
eleq1d |
⊢ ( 𝑘 = 𝑧 → ( 𝐶 ∈ ℂ ↔ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ∈ ℂ ) ) |
71 |
68 70
|
rspc |
⊢ ( 𝑧 ∈ 𝐵 → ( ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ∈ ℂ ) ) |
72 |
65 66 71
|
sylc |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ∈ ℂ ) |
73 |
|
csbeq1 |
⊢ ( 𝑤 = 𝑧 → ⦋ 𝑤 / 𝑘 ⦌ 𝐶 = ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
74 |
73
|
sumsn |
⊢ ( ( 𝑧 ∈ 𝐵 ∧ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ∈ ℂ ) → Σ 𝑤 ∈ { 𝑧 } ⦋ 𝑤 / 𝑘 ⦌ 𝐶 = ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
75 |
65 72 74
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → Σ 𝑤 ∈ { 𝑧 } ⦋ 𝑤 / 𝑘 ⦌ 𝐶 = ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
76 |
60 75
|
eqtrid |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → Σ 𝑘 ∈ { 𝑧 } 𝐶 = ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
77 |
76
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( Σ 𝑘 ∈ 𝑦 𝐶 + Σ 𝑘 ∈ { 𝑧 } 𝐶 ) = ( Σ 𝑘 ∈ 𝑦 𝐶 + ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) |
78 |
56 77
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 = ( Σ 𝑘 ∈ 𝑦 𝐶 + ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) |
79 |
78
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ ( Σ 𝑘 ∈ 𝑦 𝐶 + ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ) |
80 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → 𝐴 ⊆ ℝ ) |
81 |
|
reex |
⊢ ℝ ∈ V |
82 |
81
|
ssex |
⊢ ( 𝐴 ⊆ ℝ → 𝐴 ∈ V ) |
83 |
80 82
|
syl |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → 𝐴 ∈ V ) |
84 |
|
sumex |
⊢ Σ 𝑘 ∈ 𝑦 𝐶 ∈ V |
85 |
84
|
a1i |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ 𝑥 ∈ 𝐴 ) → Σ 𝑘 ∈ 𝑦 𝐶 ∈ V ) |
86 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ) |
87 |
|
eqidd |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) |
88 |
83 85 72 86 87
|
offval2 |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∘f + ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( Σ 𝑘 ∈ 𝑦 𝐶 + ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ) |
89 |
79 88
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) = ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∘f + ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ) |
90 |
89
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) = ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∘f + ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ) |
91 |
|
id |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) |
92 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐵 ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |
93 |
92
|
adantr |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ∀ 𝑘 ∈ 𝐵 ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ) |
94 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐴 |
95 |
94 67
|
nfmpt |
⊢ Ⅎ 𝑘 ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) |
96 |
95
|
nfel1 |
⊢ Ⅎ 𝑘 ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ∈ 𝑂(1) |
97 |
69
|
mpteq2dv |
⊢ ( 𝑘 = 𝑧 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) |
98 |
97
|
eleq1d |
⊢ ( 𝑘 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ∈ 𝑂(1) ) ) |
99 |
96 98
|
rspc |
⊢ ( 𝑧 ∈ 𝐵 → ( ∀ 𝑘 ∈ 𝐵 ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ 𝑂(1) → ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ∈ 𝑂(1) ) ) |
100 |
64 93 99
|
sylc |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ∈ 𝑂(1) ) |
101 |
|
o1add |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∘f + ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ∈ 𝑂(1) ) |
102 |
91 100 101
|
syl2anr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∘f + ( 𝑥 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑘 ⦌ 𝐶 ) ) ∈ 𝑂(1) ) |
103 |
90 102
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) ∧ ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) |
104 |
103
|
ex |
⊢ ( ( 𝜑 ∧ ( ¬ 𝑧 ∈ 𝑦 ∧ ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 ) ) → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) |
105 |
104
|
expr |
⊢ ( ( 𝜑 ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) |
106 |
105
|
a2d |
⊢ ( ( 𝜑 ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) |
107 |
39 106
|
syl5 |
⊢ ( ( 𝜑 ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) |
108 |
107
|
expcom |
⊢ ( ¬ 𝑧 ∈ 𝑦 → ( 𝜑 → ( ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
109 |
108
|
a2d |
⊢ ( ¬ 𝑧 ∈ 𝑦 → ( ( 𝜑 → ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) → ( 𝜑 → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
110 |
109
|
adantl |
⊢ ( ( 𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦 ) → ( ( 𝜑 → ( 𝑦 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝑦 𝐶 ) ∈ 𝑂(1) ) ) → ( 𝜑 → ( ( 𝑦 ∪ { 𝑧 } ) ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ ( 𝑦 ∪ { 𝑧 } ) 𝐶 ) ∈ 𝑂(1) ) ) ) ) |
111 |
13 19 25 31 35 110
|
findcard2s |
⊢ ( 𝐵 ∈ Fin → ( 𝜑 → ( 𝐵 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) ) ) |
112 |
2 111
|
mpcom |
⊢ ( 𝜑 → ( 𝐵 ⊆ 𝐵 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) ) |
113 |
5 112
|
mpi |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ Σ 𝑘 ∈ 𝐵 𝐶 ) ∈ 𝑂(1) ) |