Step |
Hyp |
Ref |
Expression |
0 |
|
csgrp |
⊢ Smgrp |
1 |
|
vg |
⊢ 𝑔 |
2 |
|
cmgm |
⊢ Mgm |
3 |
|
cbs |
⊢ Base |
4 |
1
|
cv |
⊢ 𝑔 |
5 |
4 3
|
cfv |
⊢ ( Base ‘ 𝑔 ) |
6 |
|
vb |
⊢ 𝑏 |
7 |
|
cplusg |
⊢ +g |
8 |
4 7
|
cfv |
⊢ ( +g ‘ 𝑔 ) |
9 |
|
vo |
⊢ 𝑜 |
10 |
|
vx |
⊢ 𝑥 |
11 |
6
|
cv |
⊢ 𝑏 |
12 |
|
vy |
⊢ 𝑦 |
13 |
|
vz |
⊢ 𝑧 |
14 |
10
|
cv |
⊢ 𝑥 |
15 |
9
|
cv |
⊢ 𝑜 |
16 |
12
|
cv |
⊢ 𝑦 |
17 |
14 16 15
|
co |
⊢ ( 𝑥 𝑜 𝑦 ) |
18 |
13
|
cv |
⊢ 𝑧 |
19 |
17 18 15
|
co |
⊢ ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) |
20 |
16 18 15
|
co |
⊢ ( 𝑦 𝑜 𝑧 ) |
21 |
14 20 15
|
co |
⊢ ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
22 |
19 21
|
wceq |
⊢ ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
23 |
22 13 11
|
wral |
⊢ ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
24 |
23 12 11
|
wral |
⊢ ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
25 |
24 10 11
|
wral |
⊢ ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
26 |
25 9 8
|
wsbc |
⊢ [ ( +g ‘ 𝑔 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
27 |
26 6 5
|
wsbc |
⊢ [ ( Base ‘ 𝑔 ) / 𝑏 ] [ ( +g ‘ 𝑔 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) |
28 |
27 1 2
|
crab |
⊢ { 𝑔 ∈ Mgm ∣ [ ( Base ‘ 𝑔 ) / 𝑏 ] [ ( +g ‘ 𝑔 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) } |
29 |
0 28
|
wceq |
⊢ Smgrp = { 𝑔 ∈ Mgm ∣ [ ( Base ‘ 𝑔 ) / 𝑏 ] [ ( +g ‘ 𝑔 ) / 𝑜 ] ∀ 𝑥 ∈ 𝑏 ∀ 𝑦 ∈ 𝑏 ∀ 𝑧 ∈ 𝑏 ( ( 𝑥 𝑜 𝑦 ) 𝑜 𝑧 ) = ( 𝑥 𝑜 ( 𝑦 𝑜 𝑧 ) ) } |