Step |
Hyp |
Ref |
Expression |
1 |
|
gsumccat.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumccat.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
oveq1 |
⊢ ( 𝑊 = ∅ → ( 𝑊 ++ 𝑋 ) = ( ∅ ++ 𝑋 ) ) |
4 |
3
|
oveq2d |
⊢ ( 𝑊 = ∅ → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( 𝐺 Σg ( ∅ ++ 𝑋 ) ) ) |
5 |
|
oveq2 |
⊢ ( 𝑊 = ∅ → ( 𝐺 Σg 𝑊 ) = ( 𝐺 Σg ∅ ) ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
7 |
6
|
gsum0 |
⊢ ( 𝐺 Σg ∅ ) = ( 0g ‘ 𝐺 ) |
8 |
5 7
|
eqtrdi |
⊢ ( 𝑊 = ∅ → ( 𝐺 Σg 𝑊 ) = ( 0g ‘ 𝐺 ) ) |
9 |
8
|
oveq1d |
⊢ ( 𝑊 = ∅ → ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝐺 Σg 𝑋 ) ) ) |
10 |
4 9
|
eqeq12d |
⊢ ( 𝑊 = ∅ → ( ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ↔ ( 𝐺 Σg ( ∅ ++ 𝑋 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝐺 Σg 𝑋 ) ) ) ) |
11 |
|
oveq2 |
⊢ ( 𝑋 = ∅ → ( 𝑊 ++ 𝑋 ) = ( 𝑊 ++ ∅ ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝑋 = ∅ → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( 𝐺 Σg ( 𝑊 ++ ∅ ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑋 = ∅ → ( 𝐺 Σg 𝑋 ) = ( 𝐺 Σg ∅ ) ) |
14 |
13 7
|
eqtrdi |
⊢ ( 𝑋 = ∅ → ( 𝐺 Σg 𝑋 ) = ( 0g ‘ 𝐺 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑋 = ∅ → ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 0g ‘ 𝐺 ) ) ) |
16 |
12 15
|
eqeq12d |
⊢ ( 𝑋 = ∅ → ( ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ↔ ( 𝐺 Σg ( 𝑊 ++ ∅ ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 0g ‘ 𝐺 ) ) ) ) |
17 |
|
mndsgrp |
⊢ ( 𝐺 ∈ Mnd → 𝐺 ∈ Smgrp ) |
18 |
17
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → 𝐺 ∈ Smgrp ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑋 ≠ ∅ ) → 𝐺 ∈ Smgrp ) |
20 |
|
3simpc |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑋 ≠ ∅ ) → ( 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ) |
22 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → 𝑊 ≠ ∅ ) |
23 |
22
|
anim1i |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑋 ≠ ∅ ) → ( 𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅ ) ) |
24 |
1 2
|
gsumsgrpccat |
⊢ ( ( 𝐺 ∈ Smgrp ∧ ( 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ ( 𝑊 ≠ ∅ ∧ 𝑋 ≠ ∅ ) ) → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ) |
25 |
19 21 23 24
|
syl3anc |
⊢ ( ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) ∧ 𝑋 ≠ ∅ ) → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ) |
26 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → 𝑊 ∈ Word 𝐵 ) |
27 |
|
ccatrid |
⊢ ( 𝑊 ∈ Word 𝐵 → ( 𝑊 ++ ∅ ) = 𝑊 ) |
28 |
26 27
|
syl |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( 𝑊 ++ ∅ ) = 𝑊 ) |
29 |
28
|
oveq2d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( 𝐺 Σg ( 𝑊 ++ ∅ ) ) = ( 𝐺 Σg 𝑊 ) ) |
30 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → 𝐺 ∈ Mnd ) |
31 |
1
|
gsumwcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ) |
32 |
31
|
3adant3 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ) |
33 |
32
|
adantr |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ) |
34 |
1 2 6
|
mndrid |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 Σg 𝑊 ) ∈ 𝐵 ) → ( ( 𝐺 Σg 𝑊 ) + ( 0g ‘ 𝐺 ) ) = ( 𝐺 Σg 𝑊 ) ) |
35 |
30 33 34
|
syl2anc |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( ( 𝐺 Σg 𝑊 ) + ( 0g ‘ 𝐺 ) ) = ( 𝐺 Σg 𝑊 ) ) |
36 |
29 35
|
eqtr4d |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( 𝐺 Σg ( 𝑊 ++ ∅ ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 0g ‘ 𝐺 ) ) ) |
37 |
16 25 36
|
pm2.61ne |
⊢ ( ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) ∧ 𝑊 ≠ ∅ ) → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ) |
38 |
|
ccatlid |
⊢ ( 𝑋 ∈ Word 𝐵 → ( ∅ ++ 𝑋 ) = 𝑋 ) |
39 |
38
|
3ad2ant3 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( ∅ ++ 𝑋 ) = 𝑋 ) |
40 |
39
|
oveq2d |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝐺 Σg ( ∅ ++ 𝑋 ) ) = ( 𝐺 Σg 𝑋 ) ) |
41 |
|
simp1 |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → 𝐺 ∈ Mnd ) |
42 |
1
|
gsumwcl |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝐺 Σg 𝑋 ) ∈ 𝐵 ) |
43 |
1 2 6
|
mndlid |
⊢ ( ( 𝐺 ∈ Mnd ∧ ( 𝐺 Σg 𝑋 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝐺 ) + ( 𝐺 Σg 𝑋 ) ) = ( 𝐺 Σg 𝑋 ) ) |
44 |
41 42 43
|
3imp3i2an |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( ( 0g ‘ 𝐺 ) + ( 𝐺 Σg 𝑋 ) ) = ( 𝐺 Σg 𝑋 ) ) |
45 |
40 44
|
eqtr4d |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝐺 Σg ( ∅ ++ 𝑋 ) ) = ( ( 0g ‘ 𝐺 ) + ( 𝐺 Σg 𝑋 ) ) ) |
46 |
10 37 45
|
pm2.61ne |
⊢ ( ( 𝐺 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵 ∧ 𝑋 ∈ Word 𝐵 ) → ( 𝐺 Σg ( 𝑊 ++ 𝑋 ) ) = ( ( 𝐺 Σg 𝑊 ) + ( 𝐺 Σg 𝑋 ) ) ) |