Step |
Hyp |
Ref |
Expression |
1 |
|
psrring.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
psrring.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑉 ) |
3 |
|
psrring.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
4 |
|
psr1.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
5 |
|
psr1.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
6 |
|
psr1.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
7 |
|
psr1.u |
⊢ 𝑈 = ( 1r ‘ 𝑆 ) |
8 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
10 |
1 2 3 4 5 6 8 9
|
psr1cl |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ∈ ( Base ‘ 𝑆 ) ) |
11 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝐼 ∈ 𝑉 ) |
12 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑅 ∈ Ring ) |
13 |
|
eqid |
⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) ) |
15 |
1 11 12 4 5 6 8 9 13 14
|
psrlidm |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ( .r ‘ 𝑆 ) 𝑦 ) = 𝑦 ) |
16 |
1 11 12 4 5 6 8 9 13 14
|
psrridm |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) = 𝑦 ) |
17 |
15 16
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ( .r ‘ 𝑆 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) = 𝑦 ) ) |
18 |
17
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ( .r ‘ 𝑆 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) = 𝑦 ) ) |
19 |
1 2 3
|
psrring |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
20 |
9 13 7
|
isringid |
⊢ ( 𝑆 ∈ Ring → ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ∈ ( Base ‘ 𝑆 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ( .r ‘ 𝑆 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) = 𝑦 ) ) ↔ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ∈ ( Base ‘ 𝑆 ) ∧ ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ( .r ‘ 𝑆 ) 𝑦 ) = 𝑦 ∧ ( 𝑦 ( .r ‘ 𝑆 ) ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) = 𝑦 ) ) ↔ 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) ) |
22 |
10 18 21
|
mpbi2and |
⊢ ( 𝜑 → 𝑈 = ( 𝑥 ∈ 𝐷 ↦ if ( 𝑥 = ( 𝐼 × { 0 } ) , 1 , 0 ) ) ) |