Step |
Hyp |
Ref |
Expression |
1 |
|
srg1zr.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
srg1zr.p |
⊢ + = ( +g ‘ 𝑅 ) |
3 |
|
srg1zr.t |
⊢ ∗ = ( .r ‘ 𝑅 ) |
4 |
|
pm4.24 |
⊢ ( 𝐵 = { 𝑍 } ↔ ( 𝐵 = { 𝑍 } ∧ 𝐵 = { 𝑍 } ) ) |
5 |
|
srgmnd |
⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ Mnd ) |
6 |
5
|
3ad2ant1 |
⊢ ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) → 𝑅 ∈ Mnd ) |
7 |
6
|
adantr |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → 𝑅 ∈ Mnd ) |
8 |
|
mndmgm |
⊢ ( 𝑅 ∈ Mnd → 𝑅 ∈ Mgm ) |
9 |
7 8
|
syl |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → 𝑅 ∈ Mgm ) |
10 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → 𝑍 ∈ 𝐵 ) |
11 |
|
simpl2 |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → + Fn ( 𝐵 × 𝐵 ) ) |
12 |
1 2
|
mgmb1mgm1 |
⊢ ( ( 𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn ( 𝐵 × 𝐵 ) ) → ( 𝐵 = { 𝑍 } ↔ + = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
13 |
9 10 11 12
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ + = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
14 |
|
simpl1 |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → 𝑅 ∈ SRing ) |
15 |
|
eqid |
⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 ) |
16 |
15
|
srgmgp |
⊢ ( 𝑅 ∈ SRing → ( mulGrp ‘ 𝑅 ) ∈ Mnd ) |
17 |
|
mndmgm |
⊢ ( ( mulGrp ‘ 𝑅 ) ∈ Mnd → ( mulGrp ‘ 𝑅 ) ∈ Mgm ) |
18 |
14 16 17
|
3syl |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( mulGrp ‘ 𝑅 ) ∈ Mgm ) |
19 |
15 3
|
mgpplusg |
⊢ ∗ = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
20 |
19
|
fneq1i |
⊢ ( ∗ Fn ( 𝐵 × 𝐵 ) ↔ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) Fn ( 𝐵 × 𝐵 ) ) |
21 |
20
|
biimpi |
⊢ ( ∗ Fn ( 𝐵 × 𝐵 ) → ( +g ‘ ( mulGrp ‘ 𝑅 ) ) Fn ( 𝐵 × 𝐵 ) ) |
22 |
21
|
3ad2ant3 |
⊢ ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) → ( +g ‘ ( mulGrp ‘ 𝑅 ) ) Fn ( 𝐵 × 𝐵 ) ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( +g ‘ ( mulGrp ‘ 𝑅 ) ) Fn ( 𝐵 × 𝐵 ) ) |
24 |
15 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) ) |
25 |
|
eqid |
⊢ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = ( +g ‘ ( mulGrp ‘ 𝑅 ) ) |
26 |
24 25
|
mgmb1mgm1 |
⊢ ( ( ( mulGrp ‘ 𝑅 ) ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) Fn ( 𝐵 × 𝐵 ) ) → ( 𝐵 = { 𝑍 } ↔ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
27 |
18 10 23 26
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
28 |
19
|
eqcomi |
⊢ ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = ∗ |
29 |
28
|
a1i |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = ∗ ) |
30 |
29
|
eqeq1d |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( ( +g ‘ ( mulGrp ‘ 𝑅 ) ) = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ↔ ∗ = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
31 |
27 30
|
bitrd |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ∗ = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) |
32 |
13 31
|
anbi12d |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( ( 𝐵 = { 𝑍 } ∧ 𝐵 = { 𝑍 } ) ↔ ( + = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ∧ ∗ = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) ) |
33 |
4 32
|
syl5bb |
⊢ ( ( ( 𝑅 ∈ SRing ∧ + Fn ( 𝐵 × 𝐵 ) ∧ ∗ Fn ( 𝐵 × 𝐵 ) ) ∧ 𝑍 ∈ 𝐵 ) → ( 𝐵 = { 𝑍 } ↔ ( + = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ∧ ∗ = { 〈 〈 𝑍 , 𝑍 〉 , 𝑍 〉 } ) ) ) |