Step |
Hyp |
Ref |
Expression |
1 |
|
mgm2nsgrp.s |
⊢ 𝑆 = { 𝐴 , 𝐵 } |
2 |
|
mgm2nsgrp.b |
⊢ ( Base ‘ 𝑀 ) = 𝑆 |
3 |
|
sgrp2nmnd.o |
⊢ ( +g ‘ 𝑀 ) = ( 𝑥 ∈ 𝑆 , 𝑦 ∈ 𝑆 ↦ if ( 𝑥 = 𝐴 , 𝐴 , 𝐵 ) ) |
4 |
1
|
hashprdifel |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ) |
5 |
|
eqid |
⊢ ( +g ‘ 𝑀 ) = ( +g ‘ 𝑀 ) |
6 |
1 2 3 5
|
sgrp2nmndlem2 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) = 𝐴 ) |
7 |
6
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) = 𝐴 ) |
8 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ≠ 𝐵 ) |
9 |
7 8
|
eqnetrd |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) |
10 |
9
|
olcd |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) |
11 |
|
oveq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ) |
12 |
|
id |
⊢ ( 𝑦 = 𝐴 → 𝑦 = 𝐴 ) |
13 |
11 12
|
neeq12d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ) ) |
14 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ) |
15 |
|
id |
⊢ ( 𝑦 = 𝐵 → 𝑦 = 𝐵 ) |
16 |
14 15
|
neeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) |
17 |
13 16
|
rexprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) ) |
18 |
17
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐴 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐴 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) ) |
19 |
10 18
|
mpbird |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) |
20 |
1 2 3 5
|
sgrp2nmndlem3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐵 ) |
21 |
|
necom |
⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴 ) |
22 |
|
df-ne |
⊢ ( 𝐵 ≠ 𝐴 ↔ ¬ 𝐵 = 𝐴 ) |
23 |
21 22
|
sylbb |
⊢ ( 𝐴 ≠ 𝐵 → ¬ 𝐵 = 𝐴 ) |
24 |
23
|
3ad2ant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ¬ 𝐵 = 𝐴 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐵 ) → ¬ 𝐵 = 𝐴 ) |
26 |
|
eqeq1 |
⊢ ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐵 → ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐴 ↔ 𝐵 = 𝐴 ) ) |
27 |
26
|
adantl |
⊢ ( ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐵 ) → ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐴 ↔ 𝐵 = 𝐴 ) ) |
28 |
25 27
|
mtbird |
⊢ ( ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐵 ) → ¬ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐴 ) |
29 |
20 28
|
mpdan |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ¬ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) = 𝐴 ) |
30 |
29
|
neqned |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ) |
31 |
30
|
orcd |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐵 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) |
32 |
|
oveq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ) |
33 |
32 12
|
neeq12d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ) ) |
34 |
|
oveq2 |
⊢ ( 𝑦 = 𝐵 → ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐵 ( +g ‘ 𝑀 ) 𝐵 ) ) |
35 |
34 15
|
neeq12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐵 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) |
36 |
33 35
|
rexprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐵 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) ) |
37 |
36
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ( 𝐵 ( +g ‘ 𝑀 ) 𝐴 ) ≠ 𝐴 ∨ ( 𝐵 ( +g ‘ 𝑀 ) 𝐵 ) ≠ 𝐵 ) ) ) |
38 |
31 37
|
mpbird |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) |
39 |
|
oveq1 |
⊢ ( 𝑥 = 𝐴 → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ) |
40 |
39
|
neeq1d |
⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) |
41 |
40
|
rexbidv |
⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) |
42 |
|
oveq1 |
⊢ ( 𝑥 = 𝐵 → ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) = ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ) |
43 |
42
|
neeq1d |
⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) |
44 |
43
|
rexbidv |
⊢ ( 𝑥 = 𝐵 → ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) |
45 |
41 44
|
ralprg |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ∧ ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) ) |
46 |
45
|
3adant3 |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ↔ ( ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐴 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ∧ ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝐵 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) ) ) |
47 |
19 38 46
|
mpbir2and |
⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵 ) → ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 ) |
48 |
2 1
|
eqtr2i |
⊢ { 𝐴 , 𝐵 } = ( Base ‘ 𝑀 ) |
49 |
48 5
|
isnmnd |
⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ∃ 𝑦 ∈ { 𝐴 , 𝐵 } ( 𝑥 ( +g ‘ 𝑀 ) 𝑦 ) ≠ 𝑦 → 𝑀 ∉ Mnd ) |
50 |
4 47 49
|
3syl |
⊢ ( ( ♯ ‘ 𝑆 ) = 2 → 𝑀 ∉ Mnd ) |