Step |
Hyp |
Ref |
Expression |
1 |
|
cpmatsrngpmat.s |
⊢ 𝑆 = ( 𝑁 ConstPolyMat 𝑅 ) |
2 |
|
cpmatsrngpmat.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
cpmatsrngpmat.c |
⊢ 𝐶 = ( 𝑁 Mat 𝑃 ) |
4 |
|
eqid |
⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 ) |
5 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
6 |
|
eqid |
⊢ ( algSc ‘ 𝑃 ) = ( algSc ‘ 𝑃 ) |
7 |
1 2 3 4 5 6
|
cpmatelimp2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑥 ∈ 𝑆 → ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑎 ∈ ( Base ‘ 𝑅 ) ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) ) ) |
8 |
2
|
ply1sca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
9 |
8
|
adantl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
11 |
10
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( Scalar ‘ 𝑃 ) = 𝑅 ) |
12 |
11
|
fveq2d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( invg ‘ ( Scalar ‘ 𝑃 ) ) = ( invg ‘ 𝑅 ) ) |
13 |
12
|
fveq1d |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) = ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) ) |
14 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
15 |
14
|
adantl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Grp ) |
16 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
17 |
5 16
|
grpinvcl |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( Base ‘ 𝑅 ) ) |
18 |
15 17
|
sylan |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) ∈ ( Base ‘ 𝑅 ) ) |
19 |
13 18
|
eqeltrd |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ∈ ( Base ‘ 𝑅 ) ) |
20 |
19
|
ad5ant14 |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ∈ ( Base ‘ 𝑅 ) ) |
21 |
|
fveq2 |
⊢ ( 𝑐 = ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) → ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) |
22 |
21
|
eqeq2d |
⊢ ( 𝑐 = ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) → ( ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ↔ ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) ) |
23 |
22
|
adantl |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) ∧ 𝑐 = ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) → ( ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ↔ ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) ) |
24 |
2
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
25 |
24
|
ad3antlr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑃 ∈ Ring ) |
26 |
|
simplr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
27 |
|
simpr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) |
28 |
25 26 27
|
3jca |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑃 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ) |
29 |
28
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ( 𝑃 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ) |
30 |
|
eqid |
⊢ ( invg ‘ 𝑃 ) = ( invg ‘ 𝑃 ) |
31 |
|
eqid |
⊢ ( invg ‘ 𝐶 ) = ( invg ‘ 𝐶 ) |
32 |
3 4 30 31
|
matinvgcell |
⊢ ( ( 𝑃 ∈ Ring ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( invg ‘ 𝑃 ) ‘ ( 𝑖 𝑥 𝑗 ) ) ) |
33 |
29 32
|
syl |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( invg ‘ 𝑃 ) ‘ ( 𝑖 𝑥 𝑗 ) ) ) |
34 |
|
fveq2 |
⊢ ( ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) → ( ( invg ‘ 𝑃 ) ‘ ( 𝑖 𝑥 𝑗 ) ) = ( ( invg ‘ 𝑃 ) ‘ ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) ) |
35 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
36 |
25
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → 𝑃 ∈ Ring ) |
37 |
2
|
ply1lmod |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ LMod ) |
38 |
37
|
ad3antlr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → 𝑃 ∈ LMod ) |
39 |
38
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → 𝑃 ∈ LMod ) |
40 |
6 35 36 39
|
asclghm |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( algSc ‘ 𝑃 ) ∈ ( ( Scalar ‘ 𝑃 ) GrpHom 𝑃 ) ) |
41 |
9
|
fveq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( Base ‘ 𝑅 ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) |
42 |
41
|
eleq2d |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑎 ∈ ( Base ‘ 𝑅 ) ↔ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) ) |
43 |
42
|
biimpd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑎 ∈ ( Base ‘ 𝑅 ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( 𝑎 ∈ ( Base ‘ 𝑅 ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) ) |
45 |
44
|
imp |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) |
46 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ 𝑃 ) ) = ( Base ‘ ( Scalar ‘ 𝑃 ) ) |
47 |
|
eqid |
⊢ ( invg ‘ ( Scalar ‘ 𝑃 ) ) = ( invg ‘ ( Scalar ‘ 𝑃 ) ) |
48 |
46 47 30
|
ghminv |
⊢ ( ( ( algSc ‘ 𝑃 ) ∈ ( ( Scalar ‘ 𝑃 ) GrpHom 𝑃 ) ∧ 𝑎 ∈ ( Base ‘ ( Scalar ‘ 𝑃 ) ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) = ( ( invg ‘ 𝑃 ) ‘ ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) ) |
49 |
40 45 48
|
syl2anc |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) = ( ( invg ‘ 𝑃 ) ‘ ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) ) |
50 |
49
|
eqcomd |
⊢ ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) → ( ( invg ‘ 𝑃 ) ‘ ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) |
51 |
34 50
|
sylan9eqr |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ( ( invg ‘ 𝑃 ) ‘ ( 𝑖 𝑥 𝑗 ) ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) |
52 |
33 51
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ ( ( invg ‘ ( Scalar ‘ 𝑃 ) ) ‘ 𝑎 ) ) ) |
53 |
20 23 52
|
rspcedvd |
⊢ ( ( ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) ∧ 𝑎 ∈ ( Base ‘ 𝑅 ) ) ∧ ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) |
54 |
53
|
rexlimdva2 |
⊢ ( ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) ∧ ( 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ) ) → ( ∃ 𝑎 ∈ ( Base ‘ 𝑅 ) ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) → ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
55 |
54
|
ralimdvva |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑎 ∈ ( Base ‘ 𝑅 ) ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
56 |
55
|
expimpd |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( ( 𝑥 ∈ ( Base ‘ 𝐶 ) ∧ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑎 ∈ ( Base ‘ 𝑅 ) ( 𝑖 𝑥 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑎 ) ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
57 |
7 56
|
syld |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ( 𝑥 ∈ 𝑆 → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
58 |
57
|
imp |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) |
59 |
|
simpll |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → 𝑁 ∈ Fin ) |
60 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → 𝑅 ∈ Ring ) |
61 |
2 3
|
pmatring |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Ring ) |
62 |
|
ringgrp |
⊢ ( 𝐶 ∈ Ring → 𝐶 ∈ Grp ) |
63 |
61 62
|
syl |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → 𝐶 ∈ Grp ) |
64 |
63
|
adantr |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → 𝐶 ∈ Grp ) |
65 |
1 2 3 4
|
cpmatpmat |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
66 |
65
|
3expa |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( Base ‘ 𝐶 ) ) |
67 |
4 31
|
grpinvcl |
⊢ ( ( 𝐶 ∈ Grp ∧ 𝑥 ∈ ( Base ‘ 𝐶 ) ) → ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐶 ) ) |
68 |
64 66 67
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐶 ) ) |
69 |
1 2 3 4 5 6
|
cpmatel2 |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐶 ) ) → ( ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
70 |
59 60 68 69
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → ( ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ 𝑆 ↔ ∀ 𝑖 ∈ 𝑁 ∀ 𝑗 ∈ 𝑁 ∃ 𝑐 ∈ ( Base ‘ 𝑅 ) ( 𝑖 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) 𝑗 ) = ( ( algSc ‘ 𝑃 ) ‘ 𝑐 ) ) ) |
71 |
58 70
|
mpbird |
⊢ ( ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) ∧ 𝑥 ∈ 𝑆 ) → ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ 𝑆 ) |
72 |
71
|
ralrimiva |
⊢ ( ( 𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ) → ∀ 𝑥 ∈ 𝑆 ( ( invg ‘ 𝐶 ) ‘ 𝑥 ) ∈ 𝑆 ) |