| Step |
Hyp |
Ref |
Expression |
| 1 |
|
madufval.a |
⊢ 𝐴 = ( 𝑁 Mat 𝑅 ) |
| 2 |
|
madufval.d |
⊢ 𝐷 = ( 𝑁 maDet 𝑅 ) |
| 3 |
|
madufval.j |
⊢ 𝐽 = ( 𝑁 maAdju 𝑅 ) |
| 4 |
|
madufval.b |
⊢ 𝐵 = ( Base ‘ 𝐴 ) |
| 5 |
|
madufval.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
| 6 |
|
madufval.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
| 7 |
|
eleq2 |
⊢ ( 𝑚 = ∅ → ( 𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ∅ ) ) |
| 8 |
7
|
ifbid |
⊢ ( 𝑚 = ∅ → if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 9 |
8
|
ifeq2d |
⊢ ( 𝑚 = ∅ → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 10 |
9
|
mpoeq3dv |
⊢ ( 𝑚 = ∅ → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 11 |
10
|
fveq2d |
⊢ ( 𝑚 = ∅ → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 12 |
11
|
eqeq2d |
⊢ ( 𝑚 = ∅ → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ↔ ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 13 |
|
eleq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝑘 ∈ 𝑚 ↔ 𝑘 ∈ 𝑛 ) ) |
| 14 |
13
|
ifbid |
⊢ ( 𝑚 = 𝑛 → if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 15 |
14
|
ifeq2d |
⊢ ( 𝑚 = 𝑛 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 16 |
15
|
mpoeq3dv |
⊢ ( 𝑚 = 𝑛 → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 17 |
16
|
fveq2d |
⊢ ( 𝑚 = 𝑛 → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 18 |
17
|
eqeq2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ↔ ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 19 |
|
eleq2 |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → ( 𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ) ) |
| 20 |
19
|
ifbid |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 21 |
20
|
ifeq2d |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 22 |
21
|
mpoeq3dv |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 23 |
22
|
fveq2d |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 24 |
23
|
eqeq2d |
⊢ ( 𝑚 = ( 𝑛 ∪ { 𝑟 } ) → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ↔ ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 25 |
|
eleq2 |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → ( 𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) ) ) |
| 26 |
25
|
ifbid |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 27 |
26
|
ifeq2d |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 28 |
27
|
mpoeq3dv |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 29 |
28
|
fveq2d |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 30 |
29
|
eqeq2d |
⊢ ( 𝑚 = ( 𝑁 ∖ { 𝐻 } ) → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑚 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ↔ ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 31 |
1 2 3 4 5 6
|
maducoeval |
⊢ ( ( 𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 32 |
31
|
3adant1l |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 33 |
|
noel |
⊢ ¬ 𝑘 ∈ ∅ |
| 34 |
|
iffalse |
⊢ ( ¬ 𝑘 ∈ ∅ → if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = ( 𝑘 𝑀 𝑙 ) ) |
| 35 |
33 34
|
mp1i |
⊢ ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = ( 𝑘 𝑀 𝑙 ) ) |
| 36 |
35
|
ifeq2d |
⊢ ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 37 |
36
|
mpoeq3ia |
⊢ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 38 |
37
|
fveq2i |
⊢ ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 39 |
32 38
|
eqtr4di |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ∅ , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 40 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
| 41 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
| 42 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
| 43 |
|
simpl1l |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑅 ∈ CRing ) |
| 44 |
|
simp1r |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → 𝑀 ∈ 𝐵 ) |
| 45 |
1 4
|
matrcl |
⊢ ( 𝑀 ∈ 𝐵 → ( 𝑁 ∈ Fin ∧ 𝑅 ∈ V ) ) |
| 46 |
45
|
simpld |
⊢ ( 𝑀 ∈ 𝐵 → 𝑁 ∈ Fin ) |
| 47 |
44 46
|
syl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → 𝑁 ∈ Fin ) |
| 48 |
47
|
adantr |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑁 ∈ Fin ) |
| 49 |
|
simp1l |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → 𝑅 ∈ CRing ) |
| 50 |
49
|
ad2antrr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑅 ∈ CRing ) |
| 51 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
| 52 |
50 51
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑅 ∈ Ring ) |
| 53 |
40 6
|
ring0cl |
⊢ ( 𝑅 ∈ Ring → 0 ∈ ( Base ‘ 𝑅 ) ) |
| 54 |
52 53
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 0 ∈ ( Base ‘ 𝑅 ) ) |
| 55 |
|
simpl1r |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑀 ∈ 𝐵 ) |
| 56 |
1 40 4
|
matbas2i |
⊢ ( 𝑀 ∈ 𝐵 → 𝑀 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) ) |
| 57 |
|
elmapi |
⊢ ( 𝑀 ∈ ( ( Base ‘ 𝑅 ) ↑m ( 𝑁 × 𝑁 ) ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
| 58 |
55 56 57
|
3syl |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
| 59 |
58
|
adantr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑀 : ( 𝑁 × 𝑁 ) ⟶ ( Base ‘ 𝑅 ) ) |
| 60 |
|
eldifi |
⊢ ( 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) → 𝑟 ∈ ( 𝑁 ∖ { 𝐻 } ) ) |
| 61 |
60
|
ad2antll |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑟 ∈ ( 𝑁 ∖ { 𝐻 } ) ) |
| 62 |
61
|
eldifad |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑟 ∈ 𝑁 ) |
| 63 |
62
|
adantr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑟 ∈ 𝑁 ) |
| 64 |
|
simpr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑙 ∈ 𝑁 ) |
| 65 |
59 63 64
|
fovrnd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( 𝑟 𝑀 𝑙 ) ∈ ( Base ‘ 𝑅 ) ) |
| 66 |
54 65
|
ifcld |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ∈ ( Base ‘ 𝑅 ) ) |
| 67 |
40 5
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) ) |
| 68 |
52 67
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 1 ∈ ( Base ‘ 𝑅 ) ) |
| 69 |
68 54
|
ifcld |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , 1 , 0 ) ∈ ( Base ‘ 𝑅 ) ) |
| 70 |
54
|
3adant2 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → 0 ∈ ( Base ‘ 𝑅 ) ) |
| 71 |
58
|
fovrnda |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ) → ( 𝑘 𝑀 𝑙 ) ∈ ( Base ‘ 𝑅 ) ) |
| 72 |
71
|
3impb |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑘 𝑀 𝑙 ) ∈ ( Base ‘ 𝑅 ) ) |
| 73 |
70 72
|
ifcld |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) ∈ ( Base ‘ 𝑅 ) ) |
| 74 |
73 72
|
ifcld |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ∈ ( Base ‘ 𝑅 ) ) |
| 75 |
|
simpl2 |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝐼 ∈ 𝑁 ) |
| 76 |
58 62 75
|
fovrnd |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝑟 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) |
| 77 |
|
simpl3 |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝐻 ∈ 𝑁 ) |
| 78 |
|
eldifsni |
⊢ ( 𝑟 ∈ ( 𝑁 ∖ { 𝐻 } ) → 𝑟 ≠ 𝐻 ) |
| 79 |
61 78
|
syl |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → 𝑟 ≠ 𝐻 ) |
| 80 |
2 40 41 42 43 48 66 69 74 76 62 77 79
|
mdetero |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 81 |
|
ifnot |
⊢ if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) = if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) |
| 82 |
81
|
eqcomi |
⊢ if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) = if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) |
| 83 |
82
|
a1i |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) = if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) |
| 84 |
|
ovif2 |
⊢ ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) = if ( 𝑙 = 𝐼 , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) ) |
| 85 |
76
|
adantr |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( 𝑟 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) |
| 86 |
40 42 5
|
ringridm |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑟 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑟 𝑀 𝐼 ) ) |
| 87 |
52 85 86
|
syl2anc |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑟 𝑀 𝐼 ) ) |
| 88 |
87
|
adantr |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑙 = 𝐼 ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑟 𝑀 𝐼 ) ) |
| 89 |
|
oveq2 |
⊢ ( 𝑙 = 𝐼 → ( 𝑟 𝑀 𝑙 ) = ( 𝑟 𝑀 𝐼 ) ) |
| 90 |
89
|
adantl |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑙 = 𝐼 ) → ( 𝑟 𝑀 𝑙 ) = ( 𝑟 𝑀 𝐼 ) ) |
| 91 |
88 90
|
eqtr4d |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑙 = 𝐼 ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) = ( 𝑟 𝑀 𝑙 ) ) |
| 92 |
91
|
ifeq1da |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) ) = if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) ) ) |
| 93 |
40 42 6
|
ringrz |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑟 𝑀 𝐼 ) ∈ ( Base ‘ 𝑅 ) ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
| 94 |
52 85 93
|
syl2anc |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) = 0 ) |
| 95 |
94
|
ifeq2d |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) ) = if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) |
| 96 |
92 95
|
eqtrd |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑙 = 𝐼 , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 1 ) , ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) 0 ) ) = if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) |
| 97 |
84 96
|
syl5eq |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) = if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) |
| 98 |
83 97
|
oveq12d |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ( +g ‘ 𝑅 ) if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) ) |
| 99 |
|
ringmnd |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Mnd ) |
| 100 |
52 99
|
syl |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → 𝑅 ∈ Mnd ) |
| 101 |
|
id |
⊢ ( ¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼 ) |
| 102 |
|
imnan |
⊢ ( ( ¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼 ) ↔ ¬ ( ¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼 ) ) |
| 103 |
101 102
|
mpbi |
⊢ ¬ ( ¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼 ) |
| 104 |
103
|
a1i |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ¬ ( ¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼 ) ) |
| 105 |
40 6 41
|
mndifsplit |
⊢ ( ( 𝑅 ∈ Mnd ∧ ( 𝑟 𝑀 𝑙 ) ∈ ( Base ‘ 𝑅 ) ∧ ¬ ( ¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼 ) ) → if ( ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) , ( 𝑟 𝑀 𝑙 ) , 0 ) = ( if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ( +g ‘ 𝑅 ) if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) ) |
| 106 |
100 65 104 105
|
syl3anc |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) , ( 𝑟 𝑀 𝑙 ) , 0 ) = ( if ( ¬ 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ( +g ‘ 𝑅 ) if ( 𝑙 = 𝐼 , ( 𝑟 𝑀 𝑙 ) , 0 ) ) ) |
| 107 |
|
pm2.1 |
⊢ ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) |
| 108 |
|
iftrue |
⊢ ( ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) → if ( ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) , ( 𝑟 𝑀 𝑙 ) , 0 ) = ( 𝑟 𝑀 𝑙 ) ) |
| 109 |
107 108
|
mp1i |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → if ( ( ¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼 ) , ( 𝑟 𝑀 𝑙 ) , 0 ) = ( 𝑟 𝑀 𝑙 ) ) |
| 110 |
98 106 109
|
3eqtr2d |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑙 ∈ 𝑁 ) → ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑟 𝑀 𝑙 ) ) |
| 111 |
110
|
3adant2 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑟 𝑀 𝑙 ) ) |
| 112 |
|
oveq1 |
⊢ ( 𝑘 = 𝑟 → ( 𝑘 𝑀 𝑙 ) = ( 𝑟 𝑀 𝑙 ) ) |
| 113 |
112
|
eqeq2d |
⊢ ( 𝑘 = 𝑟 → ( ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑘 𝑀 𝑙 ) ↔ ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑟 𝑀 𝑙 ) ) ) |
| 114 |
111 113
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑘 = 𝑟 → ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑘 𝑀 𝑙 ) ) ) |
| 115 |
114
|
imp |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) = ( 𝑘 𝑀 𝑙 ) ) |
| 116 |
|
iftrue |
⊢ ( 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) ) |
| 117 |
116
|
adantl |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) ) |
| 118 |
79
|
neneqd |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ¬ 𝑟 = 𝐻 ) |
| 119 |
118
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ¬ 𝑟 = 𝐻 ) |
| 120 |
|
eqeq1 |
⊢ ( 𝑘 = 𝑟 → ( 𝑘 = 𝐻 ↔ 𝑟 = 𝐻 ) ) |
| 121 |
120
|
notbid |
⊢ ( 𝑘 = 𝑟 → ( ¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻 ) ) |
| 122 |
119 121
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑘 = 𝑟 → ¬ 𝑘 = 𝐻 ) ) |
| 123 |
122
|
imp |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → ¬ 𝑘 = 𝐻 ) |
| 124 |
123
|
iffalsed |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 125 |
|
eldifn |
⊢ ( 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) → ¬ 𝑟 ∈ 𝑛 ) |
| 126 |
125
|
ad2antll |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ¬ 𝑟 ∈ 𝑛 ) |
| 127 |
126
|
3ad2ant1 |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ¬ 𝑟 ∈ 𝑛 ) |
| 128 |
|
eleq1w |
⊢ ( 𝑘 = 𝑟 → ( 𝑘 ∈ 𝑛 ↔ 𝑟 ∈ 𝑛 ) ) |
| 129 |
128
|
notbid |
⊢ ( 𝑘 = 𝑟 → ( ¬ 𝑘 ∈ 𝑛 ↔ ¬ 𝑟 ∈ 𝑛 ) ) |
| 130 |
127 129
|
syl5ibrcom |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → ( 𝑘 = 𝑟 → ¬ 𝑘 ∈ 𝑛 ) ) |
| 131 |
130
|
imp |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → ¬ 𝑘 ∈ 𝑛 ) |
| 132 |
131
|
iffalsed |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = ( 𝑘 𝑀 𝑙 ) ) |
| 133 |
124 132
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = ( 𝑘 𝑀 𝑙 ) ) |
| 134 |
115 117 133
|
3eqtr4d |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 135 |
|
iffalse |
⊢ ( ¬ 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 136 |
135
|
adantl |
⊢ ( ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 137 |
134 136
|
pm2.61dan |
⊢ ( ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 138 |
137
|
mpoeq3dva |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 139 |
138
|
fveq2d |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , ( if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ( +g ‘ 𝑅 ) ( ( 𝑟 𝑀 𝐼 ) ( .r ‘ 𝑅 ) if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 140 |
|
neeq2 |
⊢ ( 𝑘 = 𝐻 → ( 𝑟 ≠ 𝑘 ↔ 𝑟 ≠ 𝐻 ) ) |
| 141 |
140
|
biimparc |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → 𝑟 ≠ 𝑘 ) |
| 142 |
141
|
necomd |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → 𝑘 ≠ 𝑟 ) |
| 143 |
142
|
neneqd |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → ¬ 𝑘 = 𝑟 ) |
| 144 |
143
|
iffalsed |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑙 = 𝐼 , 1 , 0 ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 145 |
|
iftrue |
⊢ ( 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 146 |
145
|
adantl |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 147 |
146
|
ifeq2d |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑙 = 𝐼 , 1 , 0 ) ) ) |
| 148 |
|
iftrue |
⊢ ( 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 149 |
148
|
adantl |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 150 |
144 147 149
|
3eqtr4d |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 151 |
112
|
ifeq2d |
⊢ ( 𝑘 = 𝑟 → if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ) |
| 152 |
|
vsnid |
⊢ 𝑟 ∈ { 𝑟 } |
| 153 |
|
elun2 |
⊢ ( 𝑟 ∈ { 𝑟 } → 𝑟 ∈ ( 𝑛 ∪ { 𝑟 } ) ) |
| 154 |
152 153
|
ax-mp |
⊢ 𝑟 ∈ ( 𝑛 ∪ { 𝑟 } ) |
| 155 |
|
eleq1w |
⊢ ( 𝑘 = 𝑟 → ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ↔ 𝑟 ∈ ( 𝑛 ∪ { 𝑟 } ) ) ) |
| 156 |
154 155
|
mpbiri |
⊢ ( 𝑘 = 𝑟 → 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ) |
| 157 |
156
|
iftrued |
⊢ ( 𝑘 = 𝑟 → if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) ) |
| 158 |
|
iftrue |
⊢ ( 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) ) |
| 159 |
151 157 158
|
3eqtr4rd |
⊢ ( 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 160 |
159
|
adantl |
⊢ ( ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) ∧ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 161 |
|
iffalse |
⊢ ( ¬ 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 162 |
|
orc |
⊢ ( 𝑘 ∈ 𝑛 → ( 𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟 ) ) |
| 163 |
|
orel2 |
⊢ ( ¬ 𝑘 = 𝑟 → ( ( 𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟 ) → 𝑘 ∈ 𝑛 ) ) |
| 164 |
162 163
|
impbid2 |
⊢ ( ¬ 𝑘 = 𝑟 → ( 𝑘 ∈ 𝑛 ↔ ( 𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟 ) ) ) |
| 165 |
|
elun |
⊢ ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ↔ ( 𝑘 ∈ 𝑛 ∨ 𝑘 ∈ { 𝑟 } ) ) |
| 166 |
|
velsn |
⊢ ( 𝑘 ∈ { 𝑟 } ↔ 𝑘 = 𝑟 ) |
| 167 |
166
|
orbi2i |
⊢ ( ( 𝑘 ∈ 𝑛 ∨ 𝑘 ∈ { 𝑟 } ) ↔ ( 𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟 ) ) |
| 168 |
165 167
|
bitr2i |
⊢ ( ( 𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟 ) ↔ 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ) |
| 169 |
164 168
|
bitrdi |
⊢ ( ¬ 𝑘 = 𝑟 → ( 𝑘 ∈ 𝑛 ↔ 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) ) ) |
| 170 |
169
|
ifbid |
⊢ ( ¬ 𝑘 = 𝑟 → if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 171 |
161 170
|
eqtrd |
⊢ ( ¬ 𝑘 = 𝑟 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 172 |
171
|
adantl |
⊢ ( ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) ∧ ¬ 𝑘 = 𝑟 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 173 |
160 172
|
pm2.61dan |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 174 |
|
iffalse |
⊢ ( ¬ 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 175 |
174
|
ifeq2d |
⊢ ( ¬ 𝑘 = 𝐻 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 176 |
175
|
adantl |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 177 |
|
iffalse |
⊢ ( ¬ 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 178 |
177
|
adantl |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 179 |
173 176 178
|
3eqtr4d |
⊢ ( ( 𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 180 |
150 179
|
pm2.61dan |
⊢ ( 𝑟 ≠ 𝐻 → if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 181 |
180
|
mpoeq3dv |
⊢ ( 𝑟 ≠ 𝐻 → ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |
| 182 |
181
|
fveq2d |
⊢ ( 𝑟 ≠ 𝐻 → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 183 |
79 182
|
syl |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝑟 , if ( 𝑙 = 𝐼 , 0 , ( 𝑟 𝑀 𝑙 ) ) , if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 184 |
80 139 183
|
3eqtr3d |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 185 |
184
|
eqeq2d |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ↔ ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 186 |
185
|
biimpd |
⊢ ( ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) ∧ ( 𝑛 ⊆ ( 𝑁 ∖ { 𝐻 } ) ∧ 𝑟 ∈ ( ( 𝑁 ∖ { 𝐻 } ) ∖ 𝑛 ) ) ) → ( ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ 𝑛 , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑛 ∪ { 𝑟 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) ) |
| 187 |
|
difss |
⊢ ( 𝑁 ∖ { 𝐻 } ) ⊆ 𝑁 |
| 188 |
|
ssfi |
⊢ ( ( 𝑁 ∈ Fin ∧ ( 𝑁 ∖ { 𝐻 } ) ⊆ 𝑁 ) → ( 𝑁 ∖ { 𝐻 } ) ∈ Fin ) |
| 189 |
47 187 188
|
sylancl |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝑁 ∖ { 𝐻 } ) ∈ Fin ) |
| 190 |
12 18 24 30 39 186 189
|
findcard2d |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) ) |
| 191 |
|
iba |
⊢ ( 𝑘 = 𝐻 → ( 𝑙 = 𝐼 ↔ ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) ) ) |
| 192 |
191
|
ifbid |
⊢ ( 𝑘 = 𝐻 → if ( 𝑙 = 𝐼 , 1 , 0 ) = if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) ) |
| 193 |
|
iftrue |
⊢ ( 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑙 = 𝐼 , 1 , 0 ) ) |
| 194 |
|
iftrue |
⊢ ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) → if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) = if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) ) |
| 195 |
194
|
orcs |
⊢ ( 𝑘 = 𝐻 → if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) = if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) ) |
| 196 |
192 193 195
|
3eqtr4d |
⊢ ( 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 197 |
196
|
adantl |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 198 |
|
iffalse |
⊢ ( ¬ 𝑘 = 𝐻 → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 199 |
198
|
adantl |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 200 |
|
neqne |
⊢ ( ¬ 𝑘 = 𝐻 → 𝑘 ≠ 𝐻 ) |
| 201 |
200
|
anim2i |
⊢ ( ( 𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐻 ) → ( 𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻 ) ) |
| 202 |
201
|
adantlr |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → ( 𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻 ) ) |
| 203 |
|
eldifsn |
⊢ ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) ↔ ( 𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻 ) ) |
| 204 |
202 203
|
sylibr |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) ) |
| 205 |
204
|
iftrued |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) = if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) ) |
| 206 |
|
biorf |
⊢ ( ¬ 𝑘 = 𝐻 → ( 𝑙 = 𝐼 ↔ ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) ) ) |
| 207 |
|
id |
⊢ ( ¬ 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻 ) |
| 208 |
207
|
intnand |
⊢ ( ¬ 𝑘 = 𝐻 → ¬ ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) ) |
| 209 |
208
|
iffalsed |
⊢ ( ¬ 𝑘 = 𝐻 → if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) = 0 ) |
| 210 |
209
|
eqcomd |
⊢ ( ¬ 𝑘 = 𝐻 → 0 = if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) ) |
| 211 |
206 210
|
ifbieq1d |
⊢ ( ¬ 𝑘 = 𝐻 → if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 212 |
211
|
adantl |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 213 |
199 205 212
|
3eqtrd |
⊢ ( ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) ∧ ¬ 𝑘 = 𝐻 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 214 |
197 213
|
pm2.61dan |
⊢ ( ( 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁 ) → if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) = if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 215 |
214
|
mpoeq3ia |
⊢ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) = ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) |
| 216 |
215
|
fveq2i |
⊢ ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( 𝑘 = 𝐻 , if ( 𝑙 = 𝐼 , 1 , 0 ) , if ( 𝑘 ∈ ( 𝑁 ∖ { 𝐻 } ) , if ( 𝑙 = 𝐼 , 0 , ( 𝑘 𝑀 𝑙 ) ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) |
| 217 |
190 216
|
eqtrdi |
⊢ ( ( ( 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁 ) → ( 𝐼 ( 𝐽 ‘ 𝑀 ) 𝐻 ) = ( 𝐷 ‘ ( 𝑘 ∈ 𝑁 , 𝑙 ∈ 𝑁 ↦ if ( ( 𝑘 = 𝐻 ∨ 𝑙 = 𝐼 ) , if ( ( 𝑙 = 𝐼 ∧ 𝑘 = 𝐻 ) , 1 , 0 ) , ( 𝑘 𝑀 𝑙 ) ) ) ) ) |