Step |
Hyp |
Ref |
Expression |
1 |
|
marep01ma.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
2 |
|
marep01ma.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
3 |
|
marep01ma.r |
⊢ 𝑅 ∈ CRing |
4 |
|
marep01ma.0 |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
marep01ma.1 |
⊢ 1 = ( 1r ‘ 𝑅 ) |
6 |
|
smadiadetlem.p |
⊢ 𝑃 = ( Base ‘ ( SymGrp ‘ 𝑁 ) ) |
7 |
|
smadiadetlem.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑅 ) |
8 |
3
|
a1i |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → 𝑅 ∈ CRing ) |
9 |
1 2
|
matrcl |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
10 |
9
|
simpld |
⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin ) |
11 |
10
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → 𝑁 ∈ Fin ) |
12 |
11
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → 𝑁 ∈ Fin ) |
13 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
14 |
3 13
|
mp1i |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → 𝑅 ∈ Ring ) |
15 |
|
eldifi |
⊢ ( 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) → 𝑄 ∈ 𝑃 ) |
16 |
15
|
adantl |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → 𝑄 ∈ 𝑃 ) |
17 |
1 2 3 4 5
|
marep01ma |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 ) |
19 |
18
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 ) |
20 |
1 2 6
|
matepm2cl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ∈ 𝐵 ) → ∀ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ∈ ( Base ‘ 𝑅 ) ) |
21 |
14 16 19 20
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → ∀ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ∈ ( Base ‘ 𝑅 ) ) |
22 |
|
id |
⊢ ( 𝑚 = 𝑛 → 𝑚 = 𝑛 ) |
23 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝑄 ‘ 𝑚 ) = ( 𝑄 ‘ 𝑛 ) ) |
24 |
22 23
|
oveq12d |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) = ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ) |
25 |
24
|
eleq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ∈ ( Base ‘ 𝑅 ) ↔ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) ) |
26 |
25
|
rspccv |
⊢ ( ∀ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ∈ ( Base ‘ 𝑅 ) → ( 𝑛 ∈ 𝑁 → ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) ) |
27 |
21 26
|
syl |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → ( 𝑛 ∈ 𝑁 → ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) ) |
28 |
27
|
imp |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) ∧ 𝑛 ∈ 𝑁 ) → ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ∈ ( Base ‘ 𝑅 ) ) |
29 |
|
id |
⊢ ( 𝑛 = 𝑚 → 𝑛 = 𝑚 ) |
30 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝑄 ‘ 𝑛 ) = ( 𝑄 ‘ 𝑚 ) ) |
31 |
29 30
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) = ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ) |
32 |
31
|
adantl |
⊢ ( ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) ∧ 𝑛 = 𝑚 ) → ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) = ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) ) |
33 |
6 4 5
|
symgmatr01 |
⊢ ( ( 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) → ∃ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) = 0 ) ) |
34 |
33
|
3adant1 |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) → ∃ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) = 0 ) ) |
35 |
34
|
imp |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → ∃ 𝑚 ∈ 𝑁 ( 𝑚 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑚 ) ) = 0 ) |
36 |
7 4 8 12 28 32 35
|
gsummgp0 |
⊢ ( ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) ∧ 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) ) → ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ) ) = 0 ) |
37 |
36
|
ex |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁 ) → ( 𝑄 ∈ ( 𝑃 ∖ { 𝑞 ∈ 𝑃 ∣ ( 𝑞 ‘ 𝐾 ) = 𝐿 } ) → ( 𝐺 Σg ( 𝑛 ∈ 𝑁 ↦ ( 𝑛 ( 𝑖 ∈ 𝑁 , 𝑗 ∈ 𝑁 ↦ if ( 𝑖 = 𝐾 , if ( 𝑗 = 𝐿 , 1 , 0 ) , ( 𝑖 𝑀 𝑗 ) ) ) ( 𝑄 ‘ 𝑛 ) ) ) ) = 0 ) ) |