Step |
Hyp |
Ref |
Expression |
1 |
|
coe1fzgsumd.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
coe1fzgsumd.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
3 |
|
coe1fzgsumd.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
4 |
|
coe1fzgsumd.k |
⊢ ( 𝜑 → 𝐾 ∈ ℕ0 ) |
5 |
|
coe1fzgsumd.m |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 ) |
6 |
|
coe1fzgsumd.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
7 |
|
raleq |
⊢ ( 𝑛 = ∅ → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝐵 ) ) |
8 |
7
|
anbi2d |
⊢ ( 𝑛 = ∅ → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝐵 ) ) ) |
9 |
|
mpteq1 |
⊢ ( 𝑛 = ∅ → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) |
10 |
9
|
oveq2d |
⊢ ( 𝑛 = ∅ → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) |
11 |
10
|
fveq2d |
⊢ ( 𝑛 = ∅ → ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ) |
12 |
11
|
fveq1d |
⊢ ( 𝑛 = ∅ → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) ) |
13 |
|
mpteq1 |
⊢ ( 𝑛 = ∅ → ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) = ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝑛 = ∅ → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
15 |
12 14
|
eqeq12d |
⊢ ( 𝑛 = ∅ → ( ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ↔ ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
16 |
8 15
|
imbi12d |
⊢ ( 𝑛 = ∅ → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
17 |
|
raleq |
⊢ ( 𝑛 = 𝑚 → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ) ) |
18 |
17
|
anbi2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ) ) ) |
19 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) |
21 |
20
|
fveq2d |
⊢ ( 𝑛 = 𝑚 → ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ) |
22 |
21
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) ) |
23 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) = ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) |
24 |
23
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
25 |
22 24
|
eqeq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ↔ ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
26 |
18 25
|
imbi12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
27 |
|
raleq |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 ) ) |
28 |
27
|
anbi2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 ) ) ) |
29 |
|
mpteq1 |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) |
30 |
29
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) |
31 |
30
|
fveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ) |
32 |
31
|
fveq1d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) ) |
33 |
|
mpteq1 |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) = ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) |
34 |
33
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
35 |
32 34
|
eqeq12d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ↔ ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
36 |
28 35
|
imbi12d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
37 |
|
raleq |
⊢ ( 𝑛 = 𝑁 → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 ) ) |
38 |
37
|
anbi2d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 ) ) ) |
39 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) |
40 |
39
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) |
41 |
40
|
fveq2d |
⊢ ( 𝑛 = 𝑁 → ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ) |
42 |
41
|
fveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) ) |
43 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) = ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) |
44 |
43
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
45 |
42 44
|
eqeq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ↔ ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
46 |
38 45
|
imbi12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
47 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ 𝑀 ) = ∅ |
48 |
47
|
oveq2i |
⊢ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) = ( 𝑃 Σg ∅ ) |
49 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
50 |
49
|
gsum0 |
⊢ ( 𝑃 Σg ∅ ) = ( 0g ‘ 𝑃 ) |
51 |
48 50
|
eqtri |
⊢ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) = ( 0g ‘ 𝑃 ) |
52 |
51
|
fveq2i |
⊢ ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 0g ‘ 𝑃 ) ) |
53 |
52
|
a1i |
⊢ ( 𝜑 → ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) = ( coe1 ‘ ( 0g ‘ 𝑃 ) ) ) |
54 |
53
|
fveq1d |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( ( coe1 ‘ ( 0g ‘ 𝑃 ) ) ‘ 𝐾 ) ) |
55 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
56 |
1 49 55
|
coe1z |
⊢ ( 𝑅 ∈ Ring → ( coe1 ‘ ( 0g ‘ 𝑃 ) ) = ( ℕ0 × { ( 0g ‘ 𝑅 ) } ) ) |
57 |
3 56
|
syl |
⊢ ( 𝜑 → ( coe1 ‘ ( 0g ‘ 𝑃 ) ) = ( ℕ0 × { ( 0g ‘ 𝑅 ) } ) ) |
58 |
57
|
fveq1d |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 0g ‘ 𝑃 ) ) ‘ 𝐾 ) = ( ( ℕ0 × { ( 0g ‘ 𝑅 ) } ) ‘ 𝐾 ) ) |
59 |
|
fvex |
⊢ ( 0g ‘ 𝑅 ) ∈ V |
60 |
|
fvconst2g |
⊢ ( ( ( 0g ‘ 𝑅 ) ∈ V ∧ 𝐾 ∈ ℕ0 ) → ( ( ℕ0 × { ( 0g ‘ 𝑅 ) } ) ‘ 𝐾 ) = ( 0g ‘ 𝑅 ) ) |
61 |
59 4 60
|
sylancr |
⊢ ( 𝜑 → ( ( ℕ0 × { ( 0g ‘ 𝑅 ) } ) ‘ 𝐾 ) = ( 0g ‘ 𝑅 ) ) |
62 |
54 58 61
|
3eqtrd |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 0g ‘ 𝑅 ) ) |
63 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) = ∅ |
64 |
63
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 𝑅 Σg ∅ ) |
65 |
55
|
gsum0 |
⊢ ( 𝑅 Σg ∅ ) = ( 0g ‘ 𝑅 ) |
66 |
64 65
|
eqtri |
⊢ ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) = ( 0g ‘ 𝑅 ) |
67 |
62 66
|
eqtr4di |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |
69 |
1 2 3 4
|
coe1fzgsumdlem |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑 ) → ( ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
70 |
69
|
3expia |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( 𝜑 → ( ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) ) |
71 |
70
|
a2d |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( ( 𝜑 → ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) → ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) ) |
72 |
|
impexp |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
73 |
|
impexp |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ↔ ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
74 |
71 72 73
|
3imtr4g |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) → ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
75 |
16 26 36 46 68 74
|
findcard2s |
⊢ ( 𝑁 ∈ Fin → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 ) → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
76 |
75
|
expd |
⊢ ( 𝑁 ∈ Fin → ( 𝜑 → ( ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) ) |
77 |
6 76
|
mpcom |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝐵 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) ) |
78 |
5 77
|
mpd |
⊢ ( 𝜑 → ( ( coe1 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝐾 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( coe1 ‘ 𝑀 ) ‘ 𝐾 ) ) ) ) |