Step |
Hyp |
Ref |
Expression |
1 |
|
srgz.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
srgz.t |
⊢ · = ( .r ‘ 𝑅 ) |
3 |
|
srgz.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
5 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
6 |
1 4 5 2 3
|
issrg |
⊢ ( 𝑅 ∈ SRing ↔ ( 𝑅 ∈ CMnd ∧ ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) ∧ ( ( 0 · 𝑥 ) = 0 ∧ ( 𝑥 · 0 ) = 0 ) ) ) ) |
7 |
6
|
simp3bi |
⊢ ( 𝑅 ∈ SRing → ∀ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) ∧ ( ( 0 · 𝑥 ) = 0 ∧ ( 𝑥 · 0 ) = 0 ) ) ) |
8 |
7
|
r19.21bi |
⊢ ( ( 𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐵 ∀ 𝑧 ∈ 𝐵 ( ( 𝑥 · ( 𝑦 ( +g ‘ 𝑅 ) 𝑧 ) ) = ( ( 𝑥 · 𝑦 ) ( +g ‘ 𝑅 ) ( 𝑥 · 𝑧 ) ) ∧ ( ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) · 𝑧 ) = ( ( 𝑥 · 𝑧 ) ( +g ‘ 𝑅 ) ( 𝑦 · 𝑧 ) ) ) ∧ ( ( 0 · 𝑥 ) = 0 ∧ ( 𝑥 · 0 ) = 0 ) ) ) |
9 |
8
|
simprrd |
⊢ ( ( 𝑅 ∈ SRing ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 · 0 ) = 0 ) |
10 |
9
|
ralrimiva |
⊢ ( 𝑅 ∈ SRing → ∀ 𝑥 ∈ 𝐵 ( 𝑥 · 0 ) = 0 ) |
11 |
|
oveq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 · 0 ) = ( 𝑋 · 0 ) ) |
12 |
11
|
eqeq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 · 0 ) = 0 ↔ ( 𝑋 · 0 ) = 0 ) ) |
13 |
12
|
rspcv |
⊢ ( 𝑋 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 · 0 ) = 0 → ( 𝑋 · 0 ) = 0 ) ) |
14 |
10 13
|
mpan9 |
⊢ ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 · 0 ) = 0 ) |