Step |
Hyp |
Ref |
Expression |
1 |
|
elrgspn.b |
⊢ 𝐵 = ( Base ‘ 𝑅 ) |
2 |
|
elrgspn.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑅 ) |
3 |
|
elrgspn.x |
⊢ · = ( .g ‘ 𝑅 ) |
4 |
|
elrgspn.n |
⊢ 𝑁 = ( RingSpan ‘ 𝑅 ) |
5 |
|
elrgspn.f |
⊢ 𝐹 = { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } |
6 |
|
elrgspn.r |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
7 |
|
elrgspn.a |
⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 ) |
8 |
|
elrgspnlem1.1 |
⊢ 𝑆 = ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
9 |
1 2 3 4 5 6 7 8
|
elrgspnlem1 |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ) |
10 |
|
eqeq2 |
⊢ ( ( 1r ‘ 𝑅 ) = if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) → ( ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 1r ‘ 𝑅 ) ↔ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
11 |
|
eqeq2 |
⊢ ( ( 0g ‘ 𝑅 ) = if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) → ( ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ↔ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
12 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → 𝑤 = ∅ ) |
13 |
12
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) = ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ ∅ ) ) |
14 |
|
eqid |
⊢ ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑣 = ∅ ) → 𝑣 = ∅ ) |
16 |
15
|
iftrued |
⊢ ( ( 𝜑 ∧ 𝑣 = ∅ ) → if ( 𝑣 = ∅ , 1 , 0 ) = 1 ) |
17 |
|
wrd0 |
⊢ ∅ ∈ Word 𝐴 |
18 |
17
|
a1i |
⊢ ( 𝜑 → ∅ ∈ Word 𝐴 ) |
19 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
20 |
14 16 18 19
|
fvmptd2 |
⊢ ( 𝜑 → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ ∅ ) = 1 ) |
21 |
20
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ ∅ ) = 1 ) |
22 |
13 21
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) = 1 ) |
23 |
12
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( 𝑀 Σg 𝑤 ) = ( 𝑀 Σg ∅ ) ) |
24 |
|
eqid |
⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 ) |
25 |
2 24
|
ringidval |
⊢ ( 1r ‘ 𝑅 ) = ( 0g ‘ 𝑀 ) |
26 |
25
|
gsum0 |
⊢ ( 𝑀 Σg ∅ ) = ( 1r ‘ 𝑅 ) |
27 |
23 26
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( 𝑀 Σg 𝑤 ) = ( 1r ‘ 𝑅 ) ) |
28 |
22 27
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 1 · ( 1r ‘ 𝑅 ) ) ) |
29 |
1 24
|
ringidcl |
⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
30 |
6 29
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝐵 ) |
31 |
1 3
|
mulg1 |
⊢ ( ( 1r ‘ 𝑅 ) ∈ 𝐵 → ( 1 · ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
32 |
30 31
|
syl |
⊢ ( 𝜑 → ( 1 · ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
33 |
32
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( 1 · ( 1r ‘ 𝑅 ) ) = ( 1r ‘ 𝑅 ) ) |
34 |
28 33
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑤 = ∅ ) → ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 1r ‘ 𝑅 ) ) |
35 |
|
eqeq1 |
⊢ ( 𝑣 = 𝑤 → ( 𝑣 = ∅ ↔ 𝑤 = ∅ ) ) |
36 |
35
|
notbid |
⊢ ( 𝑣 = 𝑤 → ( ¬ 𝑣 = ∅ ↔ ¬ 𝑤 = ∅ ) ) |
37 |
36
|
biimparc |
⊢ ( ( ¬ 𝑤 = ∅ ∧ 𝑣 = 𝑤 ) → ¬ 𝑣 = ∅ ) |
38 |
37
|
adantll |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) ∧ 𝑣 = 𝑤 ) → ¬ 𝑣 = ∅ ) |
39 |
38
|
iffalsed |
⊢ ( ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) ∧ 𝑣 = 𝑤 ) → if ( 𝑣 = ∅ , 1 , 0 ) = 0 ) |
40 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → 𝑤 ∈ Word 𝐴 ) |
41 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → 0 ∈ ℤ ) |
42 |
14 39 40 41
|
fvmptd2 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) = 0 ) |
43 |
42
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
44 |
2
|
ringmgp |
⊢ ( 𝑅 ∈ Ring → 𝑀 ∈ Mnd ) |
45 |
6 44
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
46 |
|
sswrd |
⊢ ( 𝐴 ⊆ 𝐵 → Word 𝐴 ⊆ Word 𝐵 ) |
47 |
7 46
|
syl |
⊢ ( 𝜑 → Word 𝐴 ⊆ Word 𝐵 ) |
48 |
47
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
49 |
2 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
50 |
49
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
51 |
45 48 50
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
52 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
53 |
1 52 3
|
mulg0 |
⊢ ( ( 𝑀 Σg 𝑤 ) ∈ 𝐵 → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
54 |
51 53
|
syl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
55 |
54
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
56 |
43 55
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) ∧ ¬ 𝑤 = ∅ ) → ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
57 |
10 11 34 56
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
58 |
57
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) |
59 |
58
|
oveq2d |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) ) |
60 |
6
|
ringcmnd |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
61 |
60
|
cmnmndd |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
62 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
63 |
62
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
64 |
63 7
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
65 |
|
wrdexg |
⊢ ( 𝐴 ∈ V → Word 𝐴 ∈ V ) |
66 |
64 65
|
syl |
⊢ ( 𝜑 → Word 𝐴 ∈ V ) |
67 |
|
eqid |
⊢ ( 𝑤 ∈ Word 𝐴 ↦ if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) |
68 |
30 1
|
eleqtrdi |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ ( Base ‘ 𝑅 ) ) |
69 |
52 61 66 18 67 68
|
gsummptif1n0 |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ if ( 𝑤 = ∅ , ( 1r ‘ 𝑅 ) , ( 0g ‘ 𝑅 ) ) ) ) = ( 1r ‘ 𝑅 ) ) |
70 |
59 69
|
eqtrd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 1r ‘ 𝑅 ) ) |
71 |
|
eqid |
⊢ ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
72 |
|
fveq1 |
⊢ ( 𝑔 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( 𝑔 ‘ 𝑤 ) = ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) ) |
73 |
72
|
oveq1d |
⊢ ( 𝑔 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
74 |
73
|
mpteq2dv |
⊢ ( 𝑔 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
75 |
74
|
oveq2d |
⊢ ( 𝑔 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
76 |
75
|
eqeq2d |
⊢ ( 𝑔 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
77 |
|
breq1 |
⊢ ( 𝑓 = ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) → ( 𝑓 finSupp 0 ↔ ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) finSupp 0 ) ) |
78 |
|
zex |
⊢ ℤ ∈ V |
79 |
78
|
a1i |
⊢ ( 𝜑 → ℤ ∈ V ) |
80 |
|
1zzd |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑣 = ∅ ) → 1 ∈ ℤ ) |
81 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑣 ∈ Word 𝐴 ) ∧ ¬ 𝑣 = ∅ ) → 0 ∈ ℤ ) |
82 |
80 81
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ Word 𝐴 ) → if ( 𝑣 = ∅ , 1 , 0 ) ∈ ℤ ) |
83 |
82
|
fmpttd |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) : Word 𝐴 ⟶ ℤ ) |
84 |
79 66 83
|
elmapdd |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
85 |
66
|
mptexd |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∈ V ) |
86 |
83
|
ffund |
⊢ ( 𝜑 → Fun ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ) |
87 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
88 |
|
snfi |
⊢ { ∅ } ∈ Fin |
89 |
88
|
a1i |
⊢ ( 𝜑 → { ∅ } ∈ Fin ) |
90 |
|
eldifsni |
⊢ ( 𝑣 ∈ ( Word 𝐴 ∖ { ∅ } ) → 𝑣 ≠ ∅ ) |
91 |
90
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Word 𝐴 ∖ { ∅ } ) ) → 𝑣 ≠ ∅ ) |
92 |
91
|
neneqd |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Word 𝐴 ∖ { ∅ } ) ) → ¬ 𝑣 = ∅ ) |
93 |
92
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑣 ∈ ( Word 𝐴 ∖ { ∅ } ) ) → if ( 𝑣 = ∅ , 1 , 0 ) = 0 ) |
94 |
93 66
|
suppss2 |
⊢ ( 𝜑 → ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) supp 0 ) ⊆ { ∅ } ) |
95 |
|
suppssfifsupp |
⊢ ( ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∈ V ∧ Fun ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∧ 0 ∈ ℤ ) ∧ ( { ∅ } ∈ Fin ∧ ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) supp 0 ) ⊆ { ∅ } ) ) → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) finSupp 0 ) |
96 |
85 86 87 89 94 95
|
syl32anc |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) finSupp 0 ) |
97 |
77 84 96
|
elrabd |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
98 |
97 5
|
eleqtrrdi |
⊢ ( 𝜑 → ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ∈ 𝐹 ) |
99 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
100 |
76 98 99
|
rspcedvdw |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐹 ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
101 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ V ) |
102 |
71 100 101
|
elrnmptd |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
103 |
102 8
|
eleqtrrdi |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ if ( 𝑣 = ∅ , 1 , 0 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑆 ) |
104 |
70 103
|
eqeltrrd |
⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝑆 ) |
105 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
106 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
107 |
105 106
|
oveq12d |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
108 |
|
eqid |
⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 ) |
109 |
66
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → Word 𝐴 ∈ V ) |
110 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑅 ∈ Ring ) |
111 |
6
|
ringgrpd |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
112 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
113 |
5
|
ssrab3 |
⊢ 𝐹 ⊆ ( ℤ ↑m Word 𝐴 ) |
114 |
113
|
a1i |
⊢ ( 𝜑 → 𝐹 ⊆ ( ℤ ↑m Word 𝐴 ) ) |
115 |
114
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
116 |
79 66
|
elmapd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑔 : Word 𝐴 ⟶ ℤ ) ) |
117 |
116
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑔 : Word 𝐴 ⟶ ℤ ) ) |
118 |
115 117
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
119 |
118
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
120 |
51
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
121 |
1 3 112 119 120
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
122 |
121
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
123 |
122
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∀ 𝑤 ∈ Word 𝐴 ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
124 |
|
fveq2 |
⊢ ( 𝑢 = 𝑤 → ( 𝑔 ‘ 𝑢 ) = ( 𝑔 ‘ 𝑤 ) ) |
125 |
|
oveq2 |
⊢ ( 𝑢 = 𝑤 → ( 𝑀 Σg 𝑢 ) = ( 𝑀 Σg 𝑤 ) ) |
126 |
124 125
|
oveq12d |
⊢ ( 𝑢 = 𝑤 → ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
127 |
126
|
eleq1d |
⊢ ( 𝑢 = 𝑤 → ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ∈ 𝐵 ↔ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) ) |
128 |
127
|
cbvralvw |
⊢ ( ∀ 𝑢 ∈ Word 𝐴 ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ∈ 𝐵 ↔ ∀ 𝑤 ∈ Word 𝐴 ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
129 |
123 128
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∀ 𝑢 ∈ Word 𝐴 ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ∈ 𝐵 ) |
130 |
129
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑢 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ∈ 𝐵 ) |
131 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
132 |
|
breq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 finSupp 0 ↔ 𝑖 finSupp 0 ) ) |
133 |
132 5
|
elrab2 |
⊢ ( 𝑖 ∈ 𝐹 ↔ ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑖 finSupp 0 ) ) |
134 |
133
|
simplbi |
⊢ ( 𝑖 ∈ 𝐹 → 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) |
135 |
134
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) |
136 |
79 66
|
elmapd |
⊢ ( 𝜑 → ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑖 : Word 𝐴 ⟶ ℤ ) ) |
137 |
136
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑖 : Word 𝐴 ⟶ ℤ ) ) |
138 |
135 137
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
139 |
138
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑤 ) ∈ ℤ ) |
140 |
51
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
141 |
1 3 131 139 140
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
142 |
141
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
143 |
142
|
ralrimiva |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∀ 𝑤 ∈ Word 𝐴 ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
144 |
|
fveq2 |
⊢ ( 𝑣 = 𝑤 → ( 𝑖 ‘ 𝑣 ) = ( 𝑖 ‘ 𝑤 ) ) |
145 |
|
oveq2 |
⊢ ( 𝑣 = 𝑤 → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg 𝑤 ) ) |
146 |
144 145
|
oveq12d |
⊢ ( 𝑣 = 𝑤 → ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
147 |
146
|
eleq1d |
⊢ ( 𝑣 = 𝑤 → ( ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ∈ 𝐵 ↔ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) ) |
148 |
147
|
cbvralvw |
⊢ ( ∀ 𝑣 ∈ Word 𝐴 ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ∈ 𝐵 ↔ ∀ 𝑤 ∈ Word 𝐴 ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
149 |
143 148
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∀ 𝑣 ∈ Word 𝐴 ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ∈ 𝐵 ) |
150 |
149
|
r19.21bi |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ∈ 𝐵 ) |
151 |
126
|
cbvmptv |
⊢ ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
152 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 0g ‘ 𝑅 ) ∈ V ) |
153 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 0 ∈ ℤ ) |
154 |
66
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ∈ V ) |
155 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ⊆ Word 𝐴 ) |
156 |
|
breq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 finSupp 0 ↔ 𝑔 finSupp 0 ) ) |
157 |
156 5
|
elrab2 |
⊢ ( 𝑔 ∈ 𝐹 ↔ ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑔 finSupp 0 ) ) |
158 |
157
|
simprbi |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 finSupp 0 ) |
159 |
158
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 finSupp 0 ) |
160 |
1 52 3
|
mulg0 |
⊢ ( 𝑦 ∈ 𝐵 → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
161 |
160
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
162 |
152 153 154 155 120 118 159 161
|
fisuppov1 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
163 |
162
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
164 |
151 163
|
eqbrtrid |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
165 |
146
|
cbvmptv |
⊢ ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
166 |
162
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
167 |
|
fveq1 |
⊢ ( 𝑔 = 𝑖 → ( 𝑔 ‘ 𝑤 ) = ( 𝑖 ‘ 𝑤 ) ) |
168 |
167
|
oveq1d |
⊢ ( 𝑔 = 𝑖 → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
169 |
168
|
mpteq2dv |
⊢ ( 𝑔 = 𝑖 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
170 |
169
|
breq1d |
⊢ ( 𝑔 = 𝑖 → ( ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ↔ ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) ) |
171 |
170
|
cbvralvw |
⊢ ( ∀ 𝑔 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ↔ ∀ 𝑖 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
172 |
166 171
|
sylib |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
173 |
172
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
174 |
173
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
175 |
165 174
|
eqbrtrid |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
176 |
1 108 52 109 109 110 130 150 164 175
|
gsumdixp |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 , 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) ) |
177 |
151
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
178 |
165
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
179 |
177 178
|
oveq12i |
⊢ ( ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
180 |
179
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
181 |
110
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → 𝑅 ∈ Ring ) |
182 |
122
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
183 |
111
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
184 |
138
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
185 |
184
|
ffvelcdmda |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑓 ) ∈ ℤ ) |
186 |
185
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑓 ) ∈ ℤ ) |
187 |
45
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → 𝑀 ∈ Mnd ) |
188 |
47
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ⊆ Word 𝐵 ) |
189 |
188
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → Word 𝐴 ⊆ Word 𝐵 ) |
190 |
189
|
sselda |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → 𝑓 ∈ Word 𝐵 ) |
191 |
49
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑓 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑓 ) ∈ 𝐵 ) |
192 |
187 190 191
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑓 ) ∈ 𝐵 ) |
193 |
1 3 183 186 192
|
mulgcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ∈ 𝐵 ) |
194 |
1 108 181 182 193
|
ringcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ∈ 𝐵 ) |
195 |
194
|
anasss |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ ( 𝑤 ∈ Word 𝐴 ∧ 𝑓 ∈ Word 𝐴 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ∈ 𝐵 ) |
196 |
195
|
ralrimivva |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∀ 𝑤 ∈ Word 𝐴 ∀ 𝑓 ∈ Word 𝐴 ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ∈ 𝐵 ) |
197 |
|
eqid |
⊢ ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) = ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) |
198 |
197
|
fmpo |
⊢ ( ∀ 𝑤 ∈ Word 𝐴 ∀ 𝑓 ∈ Word 𝐴 ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ∈ 𝐵 ↔ ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) : ( Word 𝐴 × Word 𝐴 ) ⟶ 𝐵 ) |
199 |
196 198
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) : ( Word 𝐴 × Word 𝐴 ) ⟶ 𝐵 ) |
200 |
|
vex |
⊢ 𝑤 ∈ V |
201 |
|
vex |
⊢ 𝑓 ∈ V |
202 |
200 201
|
op1std |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 1st ‘ 𝑎 ) = 𝑤 ) |
203 |
202
|
fveq2d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) = ( 𝑔 ‘ 𝑤 ) ) |
204 |
202
|
oveq2d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) = ( 𝑀 Σg 𝑤 ) ) |
205 |
203 204
|
oveq12d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
206 |
200 201
|
op2ndd |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 2nd ‘ 𝑎 ) = 𝑓 ) |
207 |
206
|
fveq2d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) = ( 𝑖 ‘ 𝑓 ) ) |
208 |
206
|
oveq2d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) = ( 𝑀 Σg 𝑓 ) ) |
209 |
207 208
|
oveq12d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) = ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) |
210 |
205 209
|
oveq12d |
⊢ ( 𝑎 = 〈 𝑤 , 𝑓 〉 → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) |
211 |
210
|
mpompt |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) = ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) |
212 |
66 66
|
xpexd |
⊢ ( 𝜑 → ( Word 𝐴 × Word 𝐴 ) ∈ V ) |
213 |
212
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( Word 𝐴 × Word 𝐴 ) ∈ V ) |
214 |
213
|
mptexd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) ∈ V ) |
215 |
|
fvexd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 0g ‘ 𝑅 ) ∈ V ) |
216 |
110
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑅 ∈ Ring ) |
217 |
111
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑅 ∈ Grp ) |
218 |
118
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
219 |
|
xp1st |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
220 |
219
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
221 |
218 220
|
ffvelcdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) ∈ ℤ ) |
222 |
216 44
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑀 ∈ Mnd ) |
223 |
188
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
224 |
223 220
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐵 ) |
225 |
49
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 1st ‘ 𝑎 ) ∈ Word 𝐵 ) → ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ∈ 𝐵 ) |
226 |
222 224 225
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ∈ 𝐵 ) |
227 |
1 3 217 221 226
|
mulgcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ∈ 𝐵 ) |
228 |
184
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
229 |
|
xp2nd |
⊢ ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
230 |
229
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
231 |
228 230
|
ffvelcdmd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) ∈ ℤ ) |
232 |
223 230
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐵 ) |
233 |
49
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ ( 2nd ‘ 𝑎 ) ∈ Word 𝐵 ) → ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ∈ 𝐵 ) |
234 |
222 232 233
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ∈ 𝐵 ) |
235 |
1 3 217 231 234
|
mulgcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ∈ 𝐵 ) |
236 |
1 108 216 227 235
|
ringcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ∈ 𝐵 ) |
237 |
236
|
fmpttd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) : ( Word 𝐴 × Word 𝐴 ) ⟶ 𝐵 ) |
238 |
237
|
ffund |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → Fun ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) ) |
239 |
159
|
fsuppimpd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑔 supp 0 ) ∈ Fin ) |
240 |
133
|
simprbi |
⊢ ( 𝑖 ∈ 𝐹 → 𝑖 finSupp 0 ) |
241 |
240
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 finSupp 0 ) |
242 |
241
|
fsuppimpd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑖 supp 0 ) ∈ Fin ) |
243 |
|
xpfi |
⊢ ( ( ( 𝑔 supp 0 ) ∈ Fin ∧ ( 𝑖 supp 0 ) ∈ Fin ) → ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ∈ Fin ) |
244 |
239 242 243
|
syl2an2r |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ∈ Fin ) |
245 |
118
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 Fn Word 𝐴 ) |
246 |
245
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 Fn Word 𝐴 ) |
247 |
246
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 𝑔 Fn Word 𝐴 ) |
248 |
109
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → Word 𝐴 ∈ V ) |
249 |
|
0zd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 0 ∈ ℤ ) |
250 |
|
xp1st |
⊢ ( 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) → ( 1st ‘ 𝑎 ) ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) |
251 |
250
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) |
252 |
247 248 249 251
|
fvdifsupp |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) = 0 ) |
253 |
252
|
oveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) = ( 0 · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ) |
254 |
45
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 𝑀 ∈ Mnd ) |
255 |
188
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
256 |
251
|
eldifad |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
257 |
255 256
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐵 ) |
258 |
254 257 225
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ∈ 𝐵 ) |
259 |
1 52 3
|
mulg0 |
⊢ ( ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ∈ 𝐵 → ( 0 · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
260 |
258 259
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 0 · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
261 |
253 260
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
262 |
261
|
oveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) |
263 |
110
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 𝑅 ∈ Ring ) |
264 |
111
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 𝑅 ∈ Grp ) |
265 |
184
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
266 |
|
xp2nd |
⊢ ( 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
267 |
266
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
268 |
265 267
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) ∈ ℤ ) |
269 |
255 267
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐵 ) |
270 |
254 269 233
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ∈ 𝐵 ) |
271 |
1 3 264 268 270
|
mulgcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ∈ 𝐵 ) |
272 |
1 108 52 263 271
|
ringlzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( 0g ‘ 𝑅 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
273 |
262 272
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
274 |
138
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 Fn Word 𝐴 ) |
275 |
274
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 Fn Word 𝐴 ) |
276 |
275
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 𝑖 Fn Word 𝐴 ) |
277 |
109
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → Word 𝐴 ∈ V ) |
278 |
|
0zd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 0 ∈ ℤ ) |
279 |
|
xp2nd |
⊢ ( 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) → ( 2nd ‘ 𝑎 ) ∈ ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) |
280 |
279
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 2nd ‘ 𝑎 ) ∈ ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) |
281 |
276 277 278 280
|
fvdifsupp |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) = 0 ) |
282 |
281
|
oveq1d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) = ( 0 · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) |
283 |
45
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 𝑀 ∈ Mnd ) |
284 |
188
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → Word 𝐴 ⊆ Word 𝐵 ) |
285 |
280
|
eldifad |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐴 ) |
286 |
284 285
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 2nd ‘ 𝑎 ) ∈ Word 𝐵 ) |
287 |
283 286 233
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ∈ 𝐵 ) |
288 |
1 52 3
|
mulg0 |
⊢ ( ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ∈ 𝐵 → ( 0 · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
289 |
287 288
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 0 · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
290 |
282 289
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) = ( 0g ‘ 𝑅 ) ) |
291 |
290
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) ) |
292 |
110
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 𝑅 ∈ Ring ) |
293 |
111
|
ad4antr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 𝑅 ∈ Grp ) |
294 |
118
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
295 |
294
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
296 |
|
xp1st |
⊢ ( 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
297 |
296
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐴 ) |
298 |
295 297
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) ∈ ℤ ) |
299 |
284 297
|
sseldd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 1st ‘ 𝑎 ) ∈ Word 𝐵 ) |
300 |
283 299 225
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ∈ 𝐵 ) |
301 |
1 3 293 298 300
|
mulgcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ∈ 𝐵 ) |
302 |
1 108 52 292 301
|
ringrzd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) ) |
303 |
291 302
|
eqtrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ∧ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
304 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) → 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
305 |
|
difxp |
⊢ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) = ( ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ∪ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) |
306 |
304 305
|
eleqtrdi |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) → 𝑎 ∈ ( ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ∪ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) ) |
307 |
|
elun |
⊢ ( 𝑎 ∈ ( ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ∪ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) ↔ ( 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ∨ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) ) |
308 |
306 307
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) → ( 𝑎 ∈ ( ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) × Word 𝐴 ) ∨ 𝑎 ∈ ( Word 𝐴 × ( Word 𝐴 ∖ ( 𝑖 supp 0 ) ) ) ) ) |
309 |
273 303 308
|
mpjaodan |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑎 ∈ ( ( Word 𝐴 × Word 𝐴 ) ∖ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) → ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
310 |
309 213
|
suppss2 |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ⊆ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) |
311 |
244 310
|
ssfid |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) supp ( 0g ‘ 𝑅 ) ) ∈ Fin ) |
312 |
214 215 238 311
|
isfsuppd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑎 ∈ ( Word 𝐴 × Word 𝐴 ) ↦ ( ( ( 𝑔 ‘ ( 1st ‘ 𝑎 ) ) · ( 𝑀 Σg ( 1st ‘ 𝑎 ) ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ ( 2nd ‘ 𝑎 ) ) · ( 𝑀 Σg ( 2nd ‘ 𝑎 ) ) ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
313 |
211 312
|
eqbrtrrid |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
314 |
60
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑅 ∈ CMnd ) |
315 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝐴 ⊆ 𝐵 ) |
316 |
1 52 199 313 314 315
|
gsumwrd2dccat |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ) = ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) ) ) ) ) |
317 |
126
|
oveq1d |
⊢ ( 𝑢 = 𝑤 → ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) |
318 |
|
fveq2 |
⊢ ( 𝑣 = 𝑓 → ( 𝑖 ‘ 𝑣 ) = ( 𝑖 ‘ 𝑓 ) ) |
319 |
|
oveq2 |
⊢ ( 𝑣 = 𝑓 → ( 𝑀 Σg 𝑣 ) = ( 𝑀 Σg 𝑓 ) ) |
320 |
318 319
|
oveq12d |
⊢ ( 𝑣 = 𝑓 → ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) |
321 |
320
|
oveq2d |
⊢ ( 𝑣 = 𝑓 → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) |
322 |
317 321
|
cbvmpov |
⊢ ( 𝑢 ∈ Word 𝐴 , 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) |
323 |
322
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 , 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ) |
324 |
323
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 , 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ) ) |
325 |
|
pfxcctswrd |
⊢ ( ( 𝑣 ∈ Word 𝐴 ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) = 𝑣 ) |
326 |
325
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) = 𝑣 ) |
327 |
326
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) = ( 𝑀 Σg 𝑣 ) ) |
328 |
327
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) = ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) |
329 |
328
|
mpteq2dva |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) = ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) ) |
330 |
329
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
331 |
|
df-ov |
⊢ ( ( 𝑣 prefix 𝑗 ) ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) = ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) |
332 |
188
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
333 |
332
|
ad4ant13 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
334 |
187 333 50
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
335 |
1 3 108
|
mulgass3 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑖 ‘ 𝑓 ) ∈ ℤ ∧ ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ∧ ( 𝑀 Σg 𝑓 ) ∈ 𝐵 ) ) → ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑖 ‘ 𝑓 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) |
336 |
181 186 334 192 335
|
syl13anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑖 ‘ 𝑓 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) |
337 |
336
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑖 ‘ 𝑓 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) ) |
338 |
119
|
ad4ant13 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
339 |
1 3 108
|
mulgass2 |
⊢ ( ( 𝑅 ∈ Ring ∧ ( ( 𝑔 ‘ 𝑤 ) ∈ ℤ ∧ ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ∧ ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ∈ 𝐵 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ) |
340 |
181 338 334 193 339
|
syl13anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ) |
341 |
1 108 181 334 192
|
ringcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ∈ 𝐵 ) |
342 |
1 3
|
mulgass |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑔 ‘ 𝑤 ) ∈ ℤ ∧ ( 𝑖 ‘ 𝑓 ) ∈ ℤ ∧ ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ∈ 𝐵 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑖 ‘ 𝑓 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) ) |
343 |
183 338 186 341 342
|
syl13anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) = ( ( 𝑔 ‘ 𝑤 ) · ( ( 𝑖 ‘ 𝑓 ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) ) |
344 |
337 340 343
|
3eqtr4d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) |
345 |
2 108
|
mgpplusg |
⊢ ( .r ‘ 𝑅 ) = ( +g ‘ 𝑀 ) |
346 |
49 345
|
gsumccat |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵 ∧ 𝑓 ∈ Word 𝐵 ) → ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) = ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) |
347 |
187 333 190 346
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) = ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) |
348 |
347
|
oveq2d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( ( 𝑀 Σg 𝑤 ) ( .r ‘ 𝑅 ) ( 𝑀 Σg 𝑓 ) ) ) ) |
349 |
344 348
|
eqtr4d |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) ) |
350 |
349
|
adantllr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) ) |
351 |
350
|
adantllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) ∧ 𝑤 ∈ Word 𝐴 ) ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) ) |
352 |
351
|
3impa |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) ∧ 𝑤 ∈ Word 𝐴 ∧ 𝑓 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) ) |
353 |
352
|
mpoeq3dva |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) = ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) ) ) |
354 |
|
fveq2 |
⊢ ( 𝑤 = ( 𝑣 prefix 𝑗 ) → ( 𝑔 ‘ 𝑤 ) = ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) ) |
355 |
|
fveq2 |
⊢ ( 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) → ( 𝑖 ‘ 𝑓 ) = ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) |
356 |
354 355
|
oveqan12d |
⊢ ( ( 𝑤 = ( 𝑣 prefix 𝑗 ) ∧ 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) = ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
357 |
|
oveq12 |
⊢ ( ( 𝑤 = ( 𝑣 prefix 𝑗 ) ∧ 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) → ( 𝑤 ++ 𝑓 ) = ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) |
358 |
357
|
oveq2d |
⊢ ( ( 𝑤 = ( 𝑣 prefix 𝑗 ) ∧ 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) → ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) = ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
359 |
356 358
|
oveq12d |
⊢ ( ( 𝑤 = ( 𝑣 prefix 𝑗 ) ∧ 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) = ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) |
360 |
359
|
adantl |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) ∧ ( 𝑤 = ( 𝑣 prefix 𝑗 ) ∧ 𝑓 = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑖 ‘ 𝑓 ) ) · ( 𝑀 Σg ( 𝑤 ++ 𝑓 ) ) ) = ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) |
361 |
|
pfxcl |
⊢ ( 𝑣 ∈ Word 𝐴 → ( 𝑣 prefix 𝑗 ) ∈ Word 𝐴 ) |
362 |
361
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑣 prefix 𝑗 ) ∈ Word 𝐴 ) |
363 |
|
swrdcl |
⊢ ( 𝑣 ∈ Word 𝐴 → ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ∈ Word 𝐴 ) |
364 |
363
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ∈ Word 𝐴 ) |
365 |
|
ovexd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ∈ V ) |
366 |
353 360 362 364 365
|
ovmpod |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑣 prefix 𝑗 ) ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) = ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) |
367 |
331 366
|
eqtr3id |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) = ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) |
368 |
367
|
mpteq2dva |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) = ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) ) |
369 |
368
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg ( ( 𝑣 prefix 𝑗 ) ++ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) ) ) ) |
370 |
|
eqid |
⊢ ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) |
371 |
|
fveq2 |
⊢ ( 𝑡 = 𝑣 → ( ♯ ‘ 𝑡 ) = ( ♯ ‘ 𝑣 ) ) |
372 |
371
|
oveq2d |
⊢ ( 𝑡 = 𝑣 → ( 0 ... ( ♯ ‘ 𝑡 ) ) = ( 0 ... ( ♯ ‘ 𝑣 ) ) ) |
373 |
|
fvoveq1 |
⊢ ( 𝑡 = 𝑣 → ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) = ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) ) |
374 |
|
id |
⊢ ( 𝑡 = 𝑣 → 𝑡 = 𝑣 ) |
375 |
371
|
opeq2d |
⊢ ( 𝑡 = 𝑣 → 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 = 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) |
376 |
374 375
|
oveq12d |
⊢ ( 𝑡 = 𝑣 → ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) = ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) |
377 |
376
|
fveq2d |
⊢ ( 𝑡 = 𝑣 → ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) = ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) |
378 |
373 377
|
oveq12d |
⊢ ( 𝑡 = 𝑣 → ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
379 |
378
|
adantr |
⊢ ( ( 𝑡 = 𝑣 ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
380 |
372 379
|
sumeq12dv |
⊢ ( 𝑡 = 𝑣 → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
381 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → 𝑣 ∈ Word 𝐴 ) |
382 |
|
fzfid |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 0 ... ( ♯ ‘ 𝑣 ) ) ∈ Fin ) |
383 |
294
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
384 |
383 362
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) ∈ ℤ ) |
385 |
184
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
386 |
385 364
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ∈ ℤ ) |
387 |
384 386
|
zmulcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ∈ ℤ ) |
388 |
387
|
zcnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ) → ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ∈ ℂ ) |
389 |
382 388
|
fsumcl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ∈ ℂ ) |
390 |
370 380 381 389
|
fvmptd3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) = Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) ) |
391 |
390
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) |
392 |
111
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
393 |
45
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → 𝑀 ∈ Mnd ) |
394 |
315 46
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → Word 𝐴 ⊆ Word 𝐵 ) |
395 |
394
|
sselda |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → 𝑣 ∈ Word 𝐵 ) |
396 |
49
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑣 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑣 ) ∈ 𝐵 ) |
397 |
393 395 396
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑣 ) ∈ 𝐵 ) |
398 |
1 3 392 382 397 387
|
gsummulgc2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) |
399 |
391 398
|
eqtr4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( ( 𝑔 ‘ ( 𝑣 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) ) ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
400 |
330 369 399
|
3eqtr4rd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) ) ) |
401 |
400
|
mpteq2dva |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) = ( 𝑣 ∈ Word 𝐴 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) ) ) ) |
402 |
401
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( 𝑅 Σg ( 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑣 ) ) ↦ ( ( 𝑤 ∈ Word 𝐴 , 𝑓 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑓 ) · ( 𝑀 Σg 𝑓 ) ) ) ) ‘ 〈 ( 𝑣 prefix 𝑗 ) , ( 𝑣 substr 〈 𝑗 , ( ♯ ‘ 𝑣 ) 〉 ) 〉 ) ) ) ) ) ) |
403 |
316 324 402
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑢 ∈ Word 𝐴 , 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑢 ) · ( 𝑀 Σg 𝑢 ) ) ( .r ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) = ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
404 |
176 180 403
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ) |
405 |
|
fveq1 |
⊢ ( 𝑔 = ℎ → ( 𝑔 ‘ 𝑤 ) = ( ℎ ‘ 𝑤 ) ) |
406 |
405
|
oveq1d |
⊢ ( 𝑔 = ℎ → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
407 |
406
|
mpteq2dv |
⊢ ( 𝑔 = ℎ → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
408 |
407
|
oveq2d |
⊢ ( 𝑔 = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
409 |
408
|
cbvmptv |
⊢ ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( ℎ ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
410 |
|
fveq1 |
⊢ ( ℎ = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( ℎ ‘ 𝑤 ) = ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) ) |
411 |
410
|
oveq1d |
⊢ ( ℎ = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
412 |
411
|
mpteq2dv |
⊢ ( ℎ = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
413 |
412
|
oveq2d |
⊢ ( ℎ = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
414 |
413
|
eqeq2d |
⊢ ( ℎ = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
415 |
|
breq1 |
⊢ ( 𝑓 = ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) → ( 𝑓 finSupp 0 ↔ ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) finSupp 0 ) ) |
416 |
78
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ℤ ∈ V ) |
417 |
|
fzfid |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) → ( 0 ... ( ♯ ‘ 𝑡 ) ) ∈ Fin ) |
418 |
294
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
419 |
|
pfxcl |
⊢ ( 𝑡 ∈ Word 𝐴 → ( 𝑡 prefix 𝑗 ) ∈ Word 𝐴 ) |
420 |
419
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( 𝑡 prefix 𝑗 ) ∈ Word 𝐴 ) |
421 |
418 420
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) ∈ ℤ ) |
422 |
184
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
423 |
|
swrdcl |
⊢ ( 𝑡 ∈ Word 𝐴 → ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ∈ Word 𝐴 ) |
424 |
423
|
ad2antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ∈ Word 𝐴 ) |
425 |
422 424
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ∈ ℤ ) |
426 |
421 425
|
zmulcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ∈ ℤ ) |
427 |
417 426
|
fsumzcl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑡 ∈ Word 𝐴 ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ∈ ℤ ) |
428 |
427
|
fmpttd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) : Word 𝐴 ⟶ ℤ ) |
429 |
416 109 428
|
elmapdd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
430 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 0 ∈ ℤ ) |
431 |
428
|
ffund |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → Fun ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ) |
432 |
|
ccatfn |
⊢ ++ Fn ( V × V ) |
433 |
|
fnfun |
⊢ ( ++ Fn ( V × V ) → Fun ++ ) |
434 |
432 433
|
ax-mp |
⊢ Fun ++ |
435 |
|
imafi |
⊢ ( ( Fun ++ ∧ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ∈ Fin ) → ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ∈ Fin ) |
436 |
434 244 435
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ∈ Fin ) |
437 |
|
fveq2 |
⊢ ( 𝑡 = 𝑤 → ( ♯ ‘ 𝑡 ) = ( ♯ ‘ 𝑤 ) ) |
438 |
437
|
oveq2d |
⊢ ( 𝑡 = 𝑤 → ( 0 ... ( ♯ ‘ 𝑡 ) ) = ( 0 ... ( ♯ ‘ 𝑤 ) ) ) |
439 |
|
fvoveq1 |
⊢ ( 𝑡 = 𝑤 → ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) = ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ) |
440 |
|
id |
⊢ ( 𝑡 = 𝑤 → 𝑡 = 𝑤 ) |
441 |
437
|
opeq2d |
⊢ ( 𝑡 = 𝑤 → 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 = 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) |
442 |
440 441
|
oveq12d |
⊢ ( 𝑡 = 𝑤 → ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) = ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) |
443 |
442
|
fveq2d |
⊢ ( 𝑡 = 𝑤 → ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) = ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) |
444 |
439 443
|
oveq12d |
⊢ ( 𝑡 = 𝑤 → ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) ) |
445 |
444
|
adantr |
⊢ ( ( 𝑡 = 𝑤 ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ) → ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) ) |
446 |
438 445
|
sumeq12dv |
⊢ ( 𝑡 = 𝑤 → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) ) |
447 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝑤 prefix 𝑗 ) → ( 𝑢 ++ 𝑣 ) = ( ( 𝑤 prefix 𝑗 ) ++ 𝑣 ) ) |
448 |
447
|
eqeq2d |
⊢ ( 𝑢 = ( 𝑤 prefix 𝑗 ) → ( 𝑤 = ( 𝑢 ++ 𝑣 ) ↔ 𝑤 = ( ( 𝑤 prefix 𝑗 ) ++ 𝑣 ) ) ) |
449 |
|
oveq2 |
⊢ ( 𝑣 = ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) → ( ( 𝑤 prefix 𝑗 ) ++ 𝑣 ) = ( ( 𝑤 prefix 𝑗 ) ++ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) |
450 |
449
|
eqeq2d |
⊢ ( 𝑣 = ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) → ( 𝑤 = ( ( 𝑤 prefix 𝑗 ) ++ 𝑣 ) ↔ 𝑤 = ( ( 𝑤 prefix 𝑗 ) ++ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) ) |
451 |
246
|
ad4antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑔 Fn Word 𝐴 ) |
452 |
109
|
ad4antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → Word 𝐴 ∈ V ) |
453 |
|
0zd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 0 ∈ ℤ ) |
454 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) |
455 |
454
|
eldifad |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → 𝑤 ∈ Word 𝐴 ) |
456 |
455
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → 𝑤 ∈ Word 𝐴 ) |
457 |
|
pfxcl |
⊢ ( 𝑤 ∈ Word 𝐴 → ( 𝑤 prefix 𝑗 ) ∈ Word 𝐴 ) |
458 |
456 457
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑤 prefix 𝑗 ) ∈ Word 𝐴 ) |
459 |
458
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑤 prefix 𝑗 ) ∈ Word 𝐴 ) |
460 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) |
461 |
451 452 453 459 460
|
elsuppfnd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑤 prefix 𝑗 ) ∈ ( 𝑔 supp 0 ) ) |
462 |
275
|
ad4antr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑖 Fn Word 𝐴 ) |
463 |
|
swrdcl |
⊢ ( 𝑤 ∈ Word 𝐴 → ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ∈ Word 𝐴 ) |
464 |
456 463
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ∈ Word 𝐴 ) |
465 |
464
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ∈ Word 𝐴 ) |
466 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) |
467 |
462 452 453 465 466
|
elsuppfnd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ∈ ( 𝑖 supp 0 ) ) |
468 |
456
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑤 ∈ Word 𝐴 ) |
469 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) |
470 |
|
pfxcctswrd |
⊢ ( ( 𝑤 ∈ Word 𝐴 ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( ( 𝑤 prefix 𝑗 ) ++ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 𝑤 ) |
471 |
468 469 470
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ( ( 𝑤 prefix 𝑗 ) ++ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 𝑤 ) |
472 |
471
|
eqcomd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑤 = ( ( 𝑤 prefix 𝑗 ) ++ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) |
473 |
448 450 461 467 472
|
2rspcedvdw |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ∃ 𝑢 ∈ ( 𝑔 supp 0 ) ∃ 𝑣 ∈ ( 𝑖 supp 0 ) 𝑤 = ( 𝑢 ++ 𝑣 ) ) |
474 |
|
fnov |
⊢ ( ++ Fn ( V × V ) ↔ ++ = ( 𝑢 ∈ V , 𝑣 ∈ V ↦ ( 𝑢 ++ 𝑣 ) ) ) |
475 |
432 474
|
mpbi |
⊢ ++ = ( 𝑢 ∈ V , 𝑣 ∈ V ↦ ( 𝑢 ++ 𝑣 ) ) |
476 |
200
|
a1i |
⊢ ( ⊤ → 𝑤 ∈ V ) |
477 |
|
ssv |
⊢ ( 𝑔 supp 0 ) ⊆ V |
478 |
477
|
a1i |
⊢ ( ⊤ → ( 𝑔 supp 0 ) ⊆ V ) |
479 |
|
ssv |
⊢ ( 𝑖 supp 0 ) ⊆ V |
480 |
479
|
a1i |
⊢ ( ⊤ → ( 𝑖 supp 0 ) ⊆ V ) |
481 |
475 476 478 480
|
elimampo |
⊢ ( ⊤ → ( 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ↔ ∃ 𝑢 ∈ ( 𝑔 supp 0 ) ∃ 𝑣 ∈ ( 𝑖 supp 0 ) 𝑤 = ( 𝑢 ++ 𝑣 ) ) ) |
482 |
481
|
mptru |
⊢ ( 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ↔ ∃ 𝑢 ∈ ( 𝑔 supp 0 ) ∃ 𝑣 ∈ ( 𝑖 supp 0 ) 𝑤 = ( 𝑢 ++ 𝑣 ) ) |
483 |
473 482
|
sylibr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
484 |
483
|
anasss |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ) → 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
485 |
454
|
ad3antrrr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) |
486 |
485
|
eldifbd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ) ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) → ¬ 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
487 |
486
|
anasss |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) ∧ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ) → ¬ 𝑤 ∈ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
488 |
484 487
|
pm2.65da |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ¬ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ) |
489 |
|
df-ne |
⊢ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ↔ ¬ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ) |
490 |
|
df-ne |
⊢ ( ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ↔ ¬ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) |
491 |
489 490
|
anbi12i |
⊢ ( ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ↔ ( ¬ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∧ ¬ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ) |
492 |
491
|
notbii |
⊢ ( ¬ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ↔ ¬ ( ¬ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∧ ¬ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ) |
493 |
|
pm4.57 |
⊢ ( ¬ ( ¬ ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∧ ¬ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ↔ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∨ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ) |
494 |
492 493
|
bitr2i |
⊢ ( ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∨ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ↔ ¬ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ≠ 0 ∧ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ≠ 0 ) ) |
495 |
488 494
|
sylibr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∨ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ) |
496 |
294
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
497 |
496 458
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ∈ ℤ ) |
498 |
497
|
zcnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) ∈ ℂ ) |
499 |
184
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
500 |
499 464
|
ffvelcdmd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ∈ ℤ ) |
501 |
500
|
zcnd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ∈ ℂ ) |
502 |
498 501
|
mul0ord |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) = 0 ↔ ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) = 0 ∨ ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) = 0 ) ) ) |
503 |
495 502
|
mpbird |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ) → ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) = 0 ) |
504 |
503
|
sumeq2dv |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) = Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 0 ) |
505 |
|
fzssuz |
⊢ ( 0 ... ( ♯ ‘ 𝑤 ) ) ⊆ ( ℤ≥ ‘ 0 ) |
506 |
|
sumz |
⊢ ( ( ( 0 ... ( ♯ ‘ 𝑤 ) ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... ( ♯ ‘ 𝑤 ) ) ∈ Fin ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 0 = 0 ) |
507 |
506
|
orcs |
⊢ ( ( 0 ... ( ♯ ‘ 𝑤 ) ) ⊆ ( ℤ≥ ‘ 0 ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 0 = 0 ) |
508 |
505 507
|
mp1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) 0 = 0 ) |
509 |
504 508
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑤 ) ) ( ( 𝑔 ‘ ( 𝑤 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑤 substr 〈 𝑗 , ( ♯ ‘ 𝑤 ) 〉 ) ) ) = 0 ) |
510 |
446 509
|
sylan9eqr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) ∧ 𝑡 = 𝑤 ) → Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) = 0 ) |
511 |
|
0zd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → 0 ∈ ℤ ) |
512 |
370 510 455 511
|
fvmptd2 |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ ( Word 𝐴 ∖ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) ) → ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) = 0 ) |
513 |
428 512
|
suppss |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) supp 0 ) ⊆ ( ++ “ ( ( 𝑔 supp 0 ) × ( 𝑖 supp 0 ) ) ) ) |
514 |
436 513
|
ssfid |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) supp 0 ) ∈ Fin ) |
515 |
429 430 431 514
|
isfsuppd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) finSupp 0 ) |
516 |
415 429 515
|
elrabd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
517 |
516 5
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ∈ 𝐹 ) |
518 |
|
fveq2 |
⊢ ( 𝑣 = 𝑤 → ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) = ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) ) |
519 |
518 145
|
oveq12d |
⊢ ( 𝑣 = 𝑤 → ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) = ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
520 |
519
|
cbvmptv |
⊢ ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
521 |
520
|
oveq2i |
⊢ ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
522 |
521
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
523 |
414 517 522
|
rspcedvdw |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∃ ℎ ∈ 𝐹 ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
524 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ∈ V ) |
525 |
409 523 524
|
elrnmptd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
526 |
525 8
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑣 ∈ Word 𝐴 ↦ ( ( ( 𝑡 ∈ Word 𝐴 ↦ Σ 𝑗 ∈ ( 0 ... ( ♯ ‘ 𝑡 ) ) ( ( 𝑔 ‘ ( 𝑡 prefix 𝑗 ) ) · ( 𝑖 ‘ ( 𝑡 substr 〈 𝑗 , ( ♯ ‘ 𝑡 ) 〉 ) ) ) ) ‘ 𝑣 ) · ( 𝑀 Σg 𝑣 ) ) ) ) ∈ 𝑆 ) |
527 |
404 526
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
528 |
527
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
529 |
528
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
530 |
529
|
adantlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
531 |
530
|
adantr |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( .r ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
532 |
107 531
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
533 |
8
|
eleq2i |
⊢ ( 𝑦 ∈ 𝑆 ↔ 𝑦 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
534 |
169
|
oveq2d |
⊢ ( 𝑔 = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
535 |
534
|
cbvmptv |
⊢ ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑖 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
536 |
535
|
elrnmpt |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
537 |
536
|
elv |
⊢ ( 𝑦 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
538 |
533 537
|
sylbb |
⊢ ( 𝑦 ∈ 𝑆 → ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
539 |
538
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
540 |
539
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
541 |
532 540
|
r19.29a |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
542 |
8
|
eleq2i |
⊢ ( 𝑥 ∈ 𝑆 ↔ 𝑥 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
543 |
71
|
elrnmpt |
⊢ ( 𝑥 ∈ V → ( 𝑥 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
544 |
543
|
elv |
⊢ ( 𝑥 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ↔ ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
545 |
542 544
|
sylbb |
⊢ ( 𝑥 ∈ 𝑆 → ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
546 |
545
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
547 |
541 546
|
r19.29a |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
548 |
547
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
549 |
548
|
ralrimivva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
550 |
1 24 108
|
issubrg2 |
⊢ ( 𝑅 ∈ Ring → ( 𝑆 ∈ ( SubRing ‘ 𝑅 ) ↔ ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) ) ) |
551 |
550
|
biimpar |
⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ∧ ( 1r ‘ 𝑅 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( .r ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) ) → 𝑆 ∈ ( SubRing ‘ 𝑅 ) ) |
552 |
6 9 104 549 551
|
syl13anc |
⊢ ( 𝜑 → 𝑆 ∈ ( SubRing ‘ 𝑅 ) ) |