Step |
Hyp |
Ref |
Expression |
1 |
|
mulgnndir.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
mulgnndir.t |
⊢ · = ( .g ‘ 𝐺 ) |
3 |
|
mulgnndir.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
mndsgrp |
⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp ) |
5 |
4
|
adantr |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → 𝐺 ∈ Smgrp ) |
6 |
5
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 ∈ ℕ ) → 𝐺 ∈ Smgrp ) |
7 |
|
simplr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 ∈ ℕ ) → 𝑀 ∈ ℕ ) |
8 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 ∈ ℕ ) → 𝑁 ∈ ℕ ) |
9 |
|
simpr3 |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 ) |
10 |
9
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 ∈ ℕ ) → 𝑋 ∈ 𝐵 ) |
11 |
1 2 3
|
mulgnndir |
⊢ ( ( 𝐺 ∈ Smgrp ∧ ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
12 |
6 7 8 10 11
|
syl13anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 ∈ ℕ ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
13 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → 𝐺 ∈ Mnd ) |
14 |
|
simpr1 |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → 𝑀 ∈ ℕ0 ) |
15 |
14
|
adantr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → 𝑀 ∈ ℕ0 ) |
16 |
|
simplr3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → 𝑋 ∈ 𝐵 ) |
17 |
1 2 13 15 16
|
mulgnn0cld |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑀 · 𝑋 ) ∈ 𝐵 ) |
18 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
19 |
1 3 18
|
mndrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 · 𝑋 ) ∈ 𝐵 ) → ( ( 𝑀 · 𝑋 ) + ( 0g ‘ 𝐺 ) ) = ( 𝑀 · 𝑋 ) ) |
20 |
13 17 19
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( ( 𝑀 · 𝑋 ) + ( 0g ‘ 𝐺 ) ) = ( 𝑀 · 𝑋 ) ) |
21 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → 𝑁 = 0 ) |
22 |
21
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑁 · 𝑋 ) = ( 0 · 𝑋 ) ) |
23 |
1 18 2
|
mulg0 |
⊢ ( 𝑋 ∈ 𝐵 → ( 0 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
24 |
16 23
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 0 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
25 |
22 24
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑁 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
26 |
25
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) = ( ( 𝑀 · 𝑋 ) + ( 0g ‘ 𝐺 ) ) ) |
27 |
21
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑀 + 𝑁 ) = ( 𝑀 + 0 ) ) |
28 |
15
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → 𝑀 ∈ ℂ ) |
29 |
28
|
addridd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑀 + 0 ) = 𝑀 ) |
30 |
27 29
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( 𝑀 + 𝑁 ) = 𝑀 ) |
31 |
30
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( 𝑀 · 𝑋 ) ) |
32 |
20 26 31
|
3eqtr4rd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑁 = 0 ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
33 |
32
|
adantlr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) ∧ 𝑁 = 0 ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
34 |
|
simpr2 |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → 𝑁 ∈ ℕ0 ) |
35 |
|
elnn0 |
⊢ ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
36 |
34 35
|
sylib |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) → ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) |
38 |
12 33 37
|
mpjaodan |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 ∈ ℕ ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
39 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → 𝐺 ∈ Mnd ) |
40 |
|
simplr2 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → 𝑁 ∈ ℕ0 ) |
41 |
|
simplr3 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → 𝑋 ∈ 𝐵 ) |
42 |
1 2 39 40 41
|
mulgnn0cld |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 𝑁 · 𝑋 ) ∈ 𝐵 ) |
43 |
1 3 18
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑁 · 𝑋 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + ( 𝑁 · 𝑋 ) ) = ( 𝑁 · 𝑋 ) ) |
44 |
39 42 43
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( ( 0g ‘ 𝐺 ) + ( 𝑁 · 𝑋 ) ) = ( 𝑁 · 𝑋 ) ) |
45 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → 𝑀 = 0 ) |
46 |
45
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 𝑀 · 𝑋 ) = ( 0 · 𝑋 ) ) |
47 |
41 23
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 0 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
48 |
46 47
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 𝑀 · 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
49 |
48
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝑁 · 𝑋 ) ) ) |
50 |
45
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 𝑀 + 𝑁 ) = ( 0 + 𝑁 ) ) |
51 |
40
|
nn0cnd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → 𝑁 ∈ ℂ ) |
52 |
51
|
addlidd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 0 + 𝑁 ) = 𝑁 ) |
53 |
50 52
|
eqtrd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( 𝑀 + 𝑁 ) = 𝑁 ) |
54 |
53
|
oveq1d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( 𝑁 · 𝑋 ) ) |
55 |
44 49 54
|
3eqtr4rd |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) ∧ 𝑀 = 0 ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |
56 |
|
elnn0 |
⊢ ( 𝑀 ∈ ℕ0 ↔ ( 𝑀 ∈ ℕ ∨ 𝑀 = 0 ) ) |
57 |
14 56
|
sylib |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → ( 𝑀 ∈ ℕ ∨ 𝑀 = 0 ) ) |
58 |
38 55 57
|
mpjaodan |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝑀 + 𝑁 ) · 𝑋 ) = ( ( 𝑀 · 𝑋 ) + ( 𝑁 · 𝑋 ) ) ) |