Step |
Hyp |
Ref |
Expression |
1 |
|
chp0mat.c |
⊢ 𝐶 = ( 𝑁 CharPlyMat 𝑅 ) |
2 |
|
chp0mat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
chp0mat.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
4 |
|
chp0mat.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
5 |
|
chp0mat.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑃 ) |
6 |
|
chp0mat.m |
⊢ ↑ = ( .g ‘ 𝐺 ) |
7 |
|
chpscmat.d |
⊢ 𝐷 = { 𝑚 ∈ ( Base ‘ 𝐴 ) ∣ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) } |
8 |
|
chpscmat.s |
⊢ 𝑆 = ( algSc ‘ 𝑃 ) |
9 |
|
chpscmat.m |
⊢ − = ( -g ‘ 𝑃 ) |
10 |
|
simpll |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝑁 ∈ Fin ) |
11 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝑅 ∈ CRing ) |
12 |
|
elrabi |
⊢ ( 𝑀 ∈ { 𝑚 ∈ ( Base ‘ 𝐴 ) ∣ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) } → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
13 |
12 7
|
eleq2s |
⊢ ( 𝑀 ∈ 𝐷 → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
15 |
14
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
16 |
|
oveq |
⊢ ( 𝑚 = 𝑀 → ( 𝑖 𝑚 𝑗 ) = ( 𝑖 𝑀 𝑗 ) ) |
17 |
16
|
eqeq1d |
⊢ ( 𝑚 = 𝑀 → ( ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ↔ ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ) ) |
18 |
17
|
2ralbidv |
⊢ ( 𝑚 = 𝑀 → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ) ) |
19 |
18
|
rexbidv |
⊢ ( 𝑚 = 𝑀 → ( ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ↔ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ) ) |
20 |
19
|
elrab |
⊢ ( 𝑀 ∈ { 𝑚 ∈ ( Base ‘ 𝐴 ) ∣ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) } ↔ ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ) ) |
21 |
|
ifnefalse |
⊢ ( 𝑖 ≠ 𝑗 → if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
22 |
21
|
eqeq2d |
⊢ ( 𝑖 ≠ 𝑗 → ( ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ↔ ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) |
23 |
22
|
biimpcd |
⊢ ( ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) |
24 |
23
|
a1i |
⊢ ( ( ( ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ 𝑐 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝑖 ∈ 𝑁 ) ∧ 𝑗 ∈ 𝑁 ) → ( ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
25 |
24
|
ralimdva |
⊢ ( ( ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ 𝑐 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝑖 ∈ 𝑁 ) → ( ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
26 |
25
|
ralimdva |
⊢ ( ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ 𝑐 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
27 |
26
|
ex |
⊢ ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ 𝑐 ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) ) |
28 |
27
|
com23 |
⊢ ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ 𝑐 ∈ ( Base ‘ 𝑅 ) ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) ) |
29 |
28
|
rexlimdva |
⊢ ( 𝑀 ∈ ( Base ‘ 𝐴 ) → ( ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) ) |
30 |
29
|
imp |
⊢ ( ( 𝑀 ∈ ( Base ‘ 𝐴 ) ∧ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑀 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
31 |
20 30
|
sylbi |
⊢ ( 𝑀 ∈ { 𝑚 ∈ ( Base ‘ 𝐴 ) ∣ ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 𝑚 𝑗 ) = if ( 𝑖 = 𝑗 , 𝑐 , ( 0g ‘ 𝑅 ) ) } → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
32 |
31 7
|
eleq2s |
⊢ ( 𝑀 ∈ 𝐷 → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
33 |
32
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) ) |
34 |
33
|
impcom |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) |
35 |
|
eqid |
⊢ ( Base ‘ 𝐴 ) = ( Base ‘ 𝐴 ) |
36 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
37 |
1 2 3 8 35 4 36 5 9
|
chpdmat |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ ( Base ‘ 𝐴 ) ) ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ( 𝑖 ≠ 𝑗 → ( 𝑖 𝑀 𝑗 ) = ( 0g ‘ 𝑅 ) ) ) → ( 𝐶 ‘ 𝑀 ) = ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) ) ) ) ) |
38 |
10 11 15 34 37
|
syl31anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝐶 ‘ 𝑀 ) = ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) ) ) ) ) |
39 |
|
id |
⊢ ( 𝑛 = 𝑘 → 𝑛 = 𝑘 ) |
40 |
39 39
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 𝑀 𝑛 ) = ( 𝑘 𝑀 𝑘 ) ) |
41 |
40
|
eqeq1d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝑛 𝑀 𝑛 ) = 𝐸 ↔ ( 𝑘 𝑀 𝑘 ) = 𝐸 ) ) |
42 |
41
|
rspccv |
⊢ ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑘 ∈ 𝑁 → ( 𝑘 𝑀 𝑘 ) = 𝐸 ) ) |
43 |
42
|
3ad2ant3 |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) → ( 𝑘 ∈ 𝑁 → ( 𝑘 𝑀 𝑘 ) = 𝐸 ) ) |
44 |
43
|
adantl |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝑘 ∈ 𝑁 → ( 𝑘 𝑀 𝑘 ) = 𝐸 ) ) |
45 |
44
|
imp |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) ∧ 𝑘 ∈ 𝑁 ) → ( 𝑘 𝑀 𝑘 ) = 𝐸 ) |
46 |
45
|
fveq2d |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) ∧ 𝑘 ∈ 𝑁 ) → ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) = ( 𝑆 ‘ 𝐸 ) ) |
47 |
46
|
oveq2d |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) ∧ 𝑘 ∈ 𝑁 ) → ( 𝑋 − ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) ) = ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) |
48 |
47
|
mpteq2dva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) ) ) = ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) |
49 |
48
|
oveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ ( 𝑘 𝑀 𝑘 ) ) ) ) ) = ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) ) |
50 |
2
|
ply1crng |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ CRing ) |
51 |
5
|
crngmgp |
⊢ ( 𝑃 ∈ CRing → 𝐺 ∈ CMnd ) |
52 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
53 |
50 51 52
|
3syl |
⊢ ( 𝑅 ∈ CRing → 𝐺 ∈ Mnd ) |
54 |
53
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝐺 ∈ Mnd ) |
55 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
56 |
2
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
57 |
55 56
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ Ring ) |
58 |
|
ringgrp |
⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ Grp ) |
59 |
57 58
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ Grp ) |
60 |
59
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝑃 ∈ Grp ) |
61 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
62 |
4 2 61
|
vr1cl |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
63 |
55 62
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
64 |
63
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → 𝑋 ∈ ( Base ‘ 𝑃 ) ) |
65 |
|
simpr |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝐼 ∈ 𝑁 ) |
66 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
67 |
57
|
ad2antll |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → 𝑃 ∈ Ring ) |
68 |
67
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝑃 ∈ Ring ) |
69 |
2
|
ply1lmod |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ LMod ) |
70 |
55 69
|
syl |
⊢ ( 𝑅 ∈ CRing → 𝑃 ∈ LMod ) |
71 |
70
|
ad2antll |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → 𝑃 ∈ LMod ) |
72 |
71
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝑃 ∈ LMod ) |
73 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) |
74 |
8 66 68 72 73 61
|
asclf |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝑆 : ( Base ‘ ( Scalar ‘ 𝑃 ) ) ⟶ ( Base ‘ 𝑃 ) ) |
75 |
13
|
adantr |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
76 |
75
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝑀 ∈ ( Base ‘ 𝐴 ) ) |
77 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
78 |
3 77
|
matecl |
⊢ ( ( 𝐼 ∈ 𝑁 ∧ 𝐼 ∈ 𝑁 ∧ 𝑀 ∈ ( Base ‘ 𝐴 ) ) → ( 𝐼 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) |
79 |
65 65 76 78
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( 𝐼 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) |
80 |
2
|
ply1sca |
⊢ ( 𝑅 ∈ CRing → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
81 |
80
|
ad2antll |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
82 |
81
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
83 |
82
|
eqcomd |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( Scalar ‘ 𝑃 ) = 𝑅 ) |
84 |
83
|
fveq2d |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ 𝑅 ) ) |
85 |
79 84
|
eleqtrrd |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( 𝐼 𝑀 𝐼 ) ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) |
86 |
74 85
|
ffvelrnd |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( 𝑆 ‘ ( 𝐼 𝑀 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) |
87 |
|
fveq2 |
⊢ ( 𝐸 = ( 𝐼 𝑀 𝐼 ) → ( 𝑆 ‘ 𝐸 ) = ( 𝑆 ‘ ( 𝐼 𝑀 𝐼 ) ) ) |
88 |
87
|
eqcoms |
⊢ ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) = ( 𝑆 ‘ ( 𝐼 𝑀 𝐼 ) ) ) |
89 |
88
|
eleq1d |
⊢ ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ↔ ( 𝑆 ‘ ( 𝐼 𝑀 𝐼 ) ) ∈ ( Base ‘ 𝑃 ) ) ) |
90 |
86 89
|
syl5ibrcom |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) |
91 |
90
|
adantr |
⊢ ( ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) ∧ 𝑛 = 𝐼 ) → ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) |
92 |
|
id |
⊢ ( 𝑛 = 𝐼 → 𝑛 = 𝐼 ) |
93 |
92 92
|
oveq12d |
⊢ ( 𝑛 = 𝐼 → ( 𝑛 𝑀 𝑛 ) = ( 𝐼 𝑀 𝐼 ) ) |
94 |
93
|
eqeq1d |
⊢ ( 𝑛 = 𝐼 → ( ( 𝑛 𝑀 𝑛 ) = 𝐸 ↔ ( 𝐼 𝑀 𝐼 ) = 𝐸 ) ) |
95 |
94
|
imbi1d |
⊢ ( 𝑛 = 𝐼 → ( ( ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ↔ ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) |
96 |
95
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) ∧ 𝑛 = 𝐼 ) → ( ( ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ↔ ( ( 𝐼 𝑀 𝐼 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) |
97 |
91 96
|
mpbird |
⊢ ( ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) ∧ 𝑛 = 𝐼 ) → ( ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) |
98 |
65 97
|
rspcimdv |
⊢ ( ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) ∧ 𝐼 ∈ 𝑁 ) → ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) |
99 |
98
|
ex |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → ( 𝐼 ∈ 𝑁 → ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) |
100 |
99
|
com23 |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ) → ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝐼 ∈ 𝑁 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) |
101 |
100
|
ex |
⊢ ( 𝑀 ∈ 𝐷 → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( 𝐼 ∈ 𝑁 → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) ) |
102 |
101
|
com24 |
⊢ ( 𝑀 ∈ 𝐷 → ( 𝐼 ∈ 𝑁 → ( ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) ) ) |
103 |
102
|
3imp |
⊢ ( ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) → ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) ) |
104 |
103
|
impcom |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) |
105 |
61 9
|
grpsubcl |
⊢ ( ( 𝑃 ∈ Grp ∧ 𝑋 ∈ ( Base ‘ 𝑃 ) ∧ ( 𝑆 ‘ 𝐸 ) ∈ ( Base ‘ 𝑃 ) ) → ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ∈ ( Base ‘ 𝑃 ) ) |
106 |
60 64 104 105
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ∈ ( Base ‘ 𝑃 ) ) |
107 |
5 61
|
mgpbas |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝐺 ) |
108 |
106 107
|
eleqtrdi |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ∈ ( Base ‘ 𝐺 ) ) |
109 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
110 |
109 6
|
gsumconst |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑁 ∈ Fin ∧ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ∈ ( Base ‘ 𝐺 ) ) → ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) = ( ( ♯ ‘ 𝑁 ) ↑ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) |
111 |
54 10 108 110
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝐺 Σg ( 𝑘 ∈ 𝑁 ↦ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) = ( ( ♯ ‘ 𝑁 ) ↑ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) |
112 |
38 49 111
|
3eqtrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ) ∧ ( 𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀ 𝑛 ∈ 𝑁 ( 𝑛 𝑀 𝑛 ) = 𝐸 ) ) → ( 𝐶 ‘ 𝑀 ) = ( ( ♯ ‘ 𝑁 ) ↑ ( 𝑋 − ( 𝑆 ‘ 𝐸 ) ) ) ) |