Step |
Hyp |
Ref |
Expression |
1 |
|
evl1gsumd.q |
⊢ 𝑂 = ( eval1 ‘ 𝑅 ) |
2 |
|
evl1gsumd.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
evl1gsumd.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
4 |
|
evl1gsumd.u |
⊢ 𝑈 = ( Base ‘ 𝑃 ) |
5 |
|
evl1gsumd.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
6 |
|
evl1gsumd.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
7 |
|
evl1gsumd.m |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 ) |
8 |
|
evl1gsumd.n |
⊢ ( 𝜑 → 𝑁 ∈ Fin ) |
9 |
|
raleq |
⊢ ( 𝑛 = ∅ → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝑈 ) ) |
10 |
9
|
anbi2d |
⊢ ( 𝑛 = ∅ → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝑈 ) ) ) |
11 |
|
mpteq1 |
⊢ ( 𝑛 = ∅ → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝑛 = ∅ → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝑛 = ∅ → ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ) |
14 |
13
|
fveq1d |
⊢ ( 𝑛 = ∅ → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) ) |
15 |
|
mpteq1 |
⊢ ( 𝑛 = ∅ → ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) = ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) |
16 |
15
|
oveq2d |
⊢ ( 𝑛 = ∅ → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
17 |
14 16
|
eqeq12d |
⊢ ( 𝑛 = ∅ → ( ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ↔ ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
18 |
10 17
|
imbi12d |
⊢ ( 𝑛 = ∅ → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
19 |
|
raleq |
⊢ ( 𝑛 = 𝑚 → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 ) ) |
20 |
19
|
anbi2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 ) ) ) |
21 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) |
23 |
22
|
fveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ) |
24 |
23
|
fveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) ) |
25 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) = ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑛 = 𝑚 → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
27 |
24 26
|
eqeq12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ↔ ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
28 |
20 27
|
imbi12d |
⊢ ( 𝑛 = 𝑚 → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
29 |
|
raleq |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 ) ) |
30 |
29
|
anbi2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 ) ) ) |
31 |
|
mpteq1 |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) |
32 |
31
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) |
33 |
32
|
fveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ) |
34 |
33
|
fveq1d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) ) |
35 |
|
mpteq1 |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) = ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
37 |
34 36
|
eqeq12d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ↔ ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
38 |
30 37
|
imbi12d |
⊢ ( 𝑛 = ( 𝑚 ∪ { 𝑎 } ) → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
39 |
|
raleq |
⊢ ( 𝑛 = 𝑁 → ( ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ↔ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 ) ) |
40 |
39
|
anbi2d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) ↔ ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 ) ) ) |
41 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) = ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) |
42 |
41
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) |
43 |
42
|
fveq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ) |
44 |
43
|
fveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) ) |
45 |
|
mpteq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) = ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) |
46 |
45
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
47 |
44 46
|
eqeq12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ↔ ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
48 |
40 47
|
imbi12d |
⊢ ( 𝑛 = 𝑁 → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑛 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑛 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑛 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
49 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ 𝑀 ) = ∅ |
50 |
49
|
oveq2i |
⊢ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) = ( 𝑃 Σg ∅ ) |
51 |
|
eqid |
⊢ ( 0g ‘ 𝑃 ) = ( 0g ‘ 𝑃 ) |
52 |
51
|
gsum0 |
⊢ ( 𝑃 Σg ∅ ) = ( 0g ‘ 𝑃 ) |
53 |
50 52
|
eqtri |
⊢ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) = ( 0g ‘ 𝑃 ) |
54 |
53
|
fveq2i |
⊢ ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( 0g ‘ 𝑃 ) ) |
55 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
56 |
5 55
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
57 |
|
eqid |
⊢ ( algSc ‘ 𝑃 ) = ( algSc ‘ 𝑃 ) |
58 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
59 |
2 57 58 51
|
ply1scl0 |
⊢ ( 𝑅 ∈ Ring → ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑃 ) ) |
60 |
56 59
|
syl |
⊢ ( 𝜑 → ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑃 ) ) |
61 |
60
|
eqcomd |
⊢ ( 𝜑 → ( 0g ‘ 𝑃 ) = ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) |
62 |
61
|
fveq2d |
⊢ ( 𝜑 → ( 𝑂 ‘ ( 0g ‘ 𝑃 ) ) = ( 𝑂 ‘ ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) ) |
63 |
54 62
|
syl5eq |
⊢ ( 𝜑 → ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) = ( 𝑂 ‘ ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) ) |
64 |
63
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( ( 𝑂 ‘ ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) ‘ 𝑌 ) ) |
65 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
66 |
56 65
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
67 |
3 58
|
grpidcl |
⊢ ( 𝑅 ∈ Grp → ( 0g ‘ 𝑅 ) ∈ 𝐵 ) |
68 |
66 67
|
syl |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ 𝐵 ) |
69 |
1 2 3 57 4 5 68 6
|
evl1scad |
⊢ ( 𝜑 → ( ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) ‘ 𝑌 ) = ( 0g ‘ 𝑅 ) ) ) |
70 |
69
|
simprd |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( ( algSc ‘ 𝑃 ) ‘ ( 0g ‘ 𝑅 ) ) ) ‘ 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
71 |
64 70
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 0g ‘ 𝑅 ) ) |
72 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) = ∅ |
73 |
72
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 𝑅 Σg ∅ ) |
74 |
58
|
gsum0 |
⊢ ( 𝑅 Σg ∅ ) = ( 0g ‘ 𝑅 ) |
75 |
73 74
|
eqtri |
⊢ ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) = ( 0g ‘ 𝑅 ) |
76 |
71 75
|
eqtr4di |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
77 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ ∅ 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ∅ ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ∅ ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |
78 |
1 2 3 4 5 6
|
evl1gsumdlem |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑 ) → ( ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
79 |
78
|
3expia |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( 𝜑 → ( ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) ) |
80 |
79
|
a2d |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( ( 𝜑 → ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) → ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) ) |
81 |
|
impexp |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
82 |
|
impexp |
⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ↔ ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
83 |
80 81 82
|
3imtr4g |
⊢ ( ( 𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ) → ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑚 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑚 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑚 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) → ( ( 𝜑 ∧ ∀ 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ ( 𝑚 ∪ { 𝑎 } ) ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
84 |
18 28 38 48 77 83
|
findcard2s |
⊢ ( 𝑁 ∈ Fin → ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 ) → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
85 |
84
|
expd |
⊢ ( 𝑁 ∈ Fin → ( 𝜑 → ( ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) ) |
86 |
8 85
|
mpcom |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝑁 𝑀 ∈ 𝑈 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) ) |
87 |
7 86
|
mpd |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑀 ) ) ) ‘ 𝑌 ) = ( 𝑅 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) ) ) ) |