Step |
Hyp |
Ref |
Expression |
1 |
|
rrgval.e |
⊢ 𝐸 = ( RLReg ‘ 𝑅 ) |
2 |
|
rrgval.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
3 |
|
rrgval.t |
⊢ · = ( .r ‘ 𝑅 ) |
4 |
|
rrgval.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = ( Base ‘ 𝑅 ) ) |
6 |
5 2
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( Base ‘ 𝑟 ) = 𝐵 ) |
7 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = ( .r ‘ 𝑅 ) ) |
8 |
7 3
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( .r ‘ 𝑟 ) = · ) |
9 |
8
|
oveqd |
⊢ ( 𝑟 = 𝑅 → ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 𝑥 · 𝑦 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = ( 0g ‘ 𝑅 ) ) |
11 |
10 4
|
eqtr4di |
⊢ ( 𝑟 = 𝑅 → ( 0g ‘ 𝑟 ) = 0 ) |
12 |
9 11
|
eqeq12d |
⊢ ( 𝑟 = 𝑅 → ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 0g ‘ 𝑟 ) ↔ ( 𝑥 · 𝑦 ) = 0 ) ) |
13 |
11
|
eqeq2d |
⊢ ( 𝑟 = 𝑅 → ( 𝑦 = ( 0g ‘ 𝑟 ) ↔ 𝑦 = 0 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑟 = 𝑅 → ( ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 0g ‘ 𝑟 ) → 𝑦 = ( 0g ‘ 𝑟 ) ) ↔ ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) ) ) |
15 |
6 14
|
raleqbidv |
⊢ ( 𝑟 = 𝑅 → ( ∀ 𝑦 ∈ ( Base ‘ 𝑟 ) ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 0g ‘ 𝑟 ) → 𝑦 = ( 0g ‘ 𝑟 ) ) ↔ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) ) ) |
16 |
6 15
|
rabeqbidv |
⊢ ( 𝑟 = 𝑅 → { 𝑥 ∈ ( Base ‘ 𝑟 ) ∣ ∀ 𝑦 ∈ ( Base ‘ 𝑟 ) ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 0g ‘ 𝑟 ) → 𝑦 = ( 0g ‘ 𝑟 ) ) } = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } ) |
17 |
|
df-rlreg |
⊢ RLReg = ( 𝑟 ∈ V ↦ { 𝑥 ∈ ( Base ‘ 𝑟 ) ∣ ∀ 𝑦 ∈ ( Base ‘ 𝑟 ) ( ( 𝑥 ( .r ‘ 𝑟 ) 𝑦 ) = ( 0g ‘ 𝑟 ) → 𝑦 = ( 0g ‘ 𝑟 ) ) } ) |
18 |
2
|
fvexi |
⊢ 𝐵 ∈ V |
19 |
18
|
rabex |
⊢ { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } ∈ V |
20 |
16 17 19
|
fvmpt |
⊢ ( 𝑅 ∈ V → ( RLReg ‘ 𝑅 ) = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } ) |
21 |
|
fvprc |
⊢ ( ¬ 𝑅 ∈ V → ( RLReg ‘ 𝑅 ) = ∅ ) |
22 |
|
fvprc |
⊢ ( ¬ 𝑅 ∈ V → ( Base ‘ 𝑅 ) = ∅ ) |
23 |
2 22
|
eqtrid |
⊢ ( ¬ 𝑅 ∈ V → 𝐵 = ∅ ) |
24 |
23
|
rabeqdv |
⊢ ( ¬ 𝑅 ∈ V → { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } = { 𝑥 ∈ ∅ ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } ) |
25 |
|
rab0 |
⊢ { 𝑥 ∈ ∅ ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } = ∅ |
26 |
24 25
|
eqtrdi |
⊢ ( ¬ 𝑅 ∈ V → { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } = ∅ ) |
27 |
21 26
|
eqtr4d |
⊢ ( ¬ 𝑅 ∈ V → ( RLReg ‘ 𝑅 ) = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } ) |
28 |
20 27
|
pm2.61i |
⊢ ( RLReg ‘ 𝑅 ) = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } |
29 |
1 28
|
eqtri |
⊢ 𝐸 = { 𝑥 ∈ 𝐵 ∣ ∀ 𝑦 ∈ 𝐵 ( ( 𝑥 · 𝑦 ) = 0 → 𝑦 = 0 ) } |