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 |
6
|
ringgrpd |
⊢ ( 𝜑 → 𝑅 ∈ Grp ) |
10 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
11 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
12 |
6
|
ringcmnd |
⊢ ( 𝜑 → 𝑅 ∈ CMnd ) |
13 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑅 ∈ CMnd ) |
14 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
15 |
14
|
a1i |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
16 |
15 7
|
ssexd |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
17 |
|
wrdexg |
⊢ ( 𝐴 ∈ V → Word 𝐴 ∈ V ) |
18 |
16 17
|
syl |
⊢ ( 𝜑 → Word 𝐴 ∈ V ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ∈ V ) |
20 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
21 |
5
|
ssrab3 |
⊢ 𝐹 ⊆ ( ℤ ↑m Word 𝐴 ) |
22 |
21
|
a1i |
⊢ ( 𝜑 → 𝐹 ⊆ ( ℤ ↑m Word 𝐴 ) ) |
23 |
22
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
24 |
|
zex |
⊢ ℤ ∈ V |
25 |
24
|
a1i |
⊢ ( 𝜑 → ℤ ∈ V ) |
26 |
25 18
|
elmapd |
⊢ ( 𝜑 → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑔 : Word 𝐴 ⟶ ℤ ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑔 : Word 𝐴 ⟶ ℤ ) ) |
28 |
23 27
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
29 |
28
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
30 |
2
|
ringmgp |
⊢ ( 𝑅 ∈ Ring → 𝑀 ∈ Mnd ) |
31 |
6 30
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ Mnd ) |
32 |
|
sswrd |
⊢ ( 𝐴 ⊆ 𝐵 → Word 𝐴 ⊆ Word 𝐵 ) |
33 |
7 32
|
syl |
⊢ ( 𝜑 → Word 𝐴 ⊆ Word 𝐵 ) |
34 |
33
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐵 ) |
35 |
2 1
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
36 |
35
|
gsumwcl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑤 ∈ Word 𝐵 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
37 |
31 34 36
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
38 |
37
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
39 |
1 3 20 29 38
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
40 |
39
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) : Word 𝐴 ⟶ 𝐵 ) |
41 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 0g ‘ 𝑅 ) ∈ V ) |
42 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 0 ∈ ℤ ) |
43 |
|
ssidd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Word 𝐴 ⊆ Word 𝐴 ) |
44 |
|
breq1 |
⊢ ( 𝑓 = 𝑔 → ( 𝑓 finSupp 0 ↔ 𝑔 finSupp 0 ) ) |
45 |
44 5
|
elrab2 |
⊢ ( 𝑔 ∈ 𝐹 ↔ ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑔 finSupp 0 ) ) |
46 |
45
|
simprbi |
⊢ ( 𝑔 ∈ 𝐹 → 𝑔 finSupp 0 ) |
47 |
46
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 finSupp 0 ) |
48 |
1 11 3
|
mulg0 |
⊢ ( 𝑦 ∈ 𝐵 → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 0 · 𝑦 ) = ( 0g ‘ 𝑅 ) ) |
50 |
41 42 19 43 38 28 47 49
|
fisuppov1 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
51 |
1 11 13 19 40 50
|
gsumcl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
52 |
51
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
53 |
10 52
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑥 ∈ 𝐵 ) |
54 |
8
|
eleq2i |
⊢ ( 𝑥 ∈ 𝑆 ↔ 𝑥 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
55 |
|
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 ) ) ) ) ) |
56 |
55
|
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 ) ) ) ) ) ) |
57 |
56
|
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 ) ) ) ) ) |
58 |
54 57
|
sylbb |
⊢ ( 𝑥 ∈ 𝑆 → ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
59 |
58
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
60 |
53 59
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ 𝐵 ) |
61 |
60 1
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → 𝑥 ∈ ( Base ‘ 𝑅 ) ) |
62 |
61
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑆 → 𝑥 ∈ ( Base ‘ 𝑅 ) ) ) |
63 |
62
|
ssrdv |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑅 ) ) |
64 |
63 1
|
sseqtrrdi |
⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 ) |
65 |
|
breq1 |
⊢ ( 𝑓 = ( Word 𝐴 × { 0 } ) → ( 𝑓 finSupp 0 ↔ ( Word 𝐴 × { 0 } ) finSupp 0 ) ) |
66 |
|
0z |
⊢ 0 ∈ ℤ |
67 |
66
|
fconst6 |
⊢ ( Word 𝐴 × { 0 } ) : Word 𝐴 ⟶ ℤ |
68 |
67
|
a1i |
⊢ ( 𝜑 → ( Word 𝐴 × { 0 } ) : Word 𝐴 ⟶ ℤ ) |
69 |
25 18 68
|
elmapdd |
⊢ ( 𝜑 → ( Word 𝐴 × { 0 } ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
70 |
|
c0ex |
⊢ 0 ∈ V |
71 |
70
|
a1i |
⊢ ( 𝜑 → 0 ∈ V ) |
72 |
18 71
|
fczfsuppd |
⊢ ( 𝜑 → ( Word 𝐴 × { 0 } ) finSupp 0 ) |
73 |
65 69 72
|
elrabd |
⊢ ( 𝜑 → ( Word 𝐴 × { 0 } ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
74 |
73 5
|
eleqtrrdi |
⊢ ( 𝜑 → ( Word 𝐴 × { 0 } ) ∈ 𝐹 ) |
75 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑔 = ( Word 𝐴 × { 0 } ) ) |
76 |
75
|
fveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) = ( ( Word 𝐴 × { 0 } ) ‘ 𝑤 ) ) |
77 |
70
|
fconst |
⊢ ( Word 𝐴 × { 0 } ) : Word 𝐴 ⟶ { 0 } |
78 |
77
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( Word 𝐴 × { 0 } ) : Word 𝐴 ⟶ { 0 } ) |
79 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐴 ) |
80 |
|
fvconst |
⊢ ( ( ( Word 𝐴 × { 0 } ) : Word 𝐴 ⟶ { 0 } ∧ 𝑤 ∈ Word 𝐴 ) → ( ( Word 𝐴 × { 0 } ) ‘ 𝑤 ) = 0 ) |
81 |
78 79 80
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( Word 𝐴 × { 0 } ) ‘ 𝑤 ) = 0 ) |
82 |
76 81
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) = 0 ) |
83 |
82
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
84 |
37
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
85 |
1 11 3
|
mulg0 |
⊢ ( ( 𝑀 Σg 𝑤 ) ∈ 𝐵 → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
86 |
84 85
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
87 |
83 86
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
88 |
87
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) |
89 |
88
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) ) |
90 |
12
|
cmnmndd |
⊢ ( 𝜑 → 𝑅 ∈ Mnd ) |
91 |
11
|
gsumz |
⊢ ( ( 𝑅 ∈ Mnd ∧ Word 𝐴 ∈ V ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
92 |
90 18 91
|
syl2anc |
⊢ ( 𝜑 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
93 |
92
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
94 |
89 93
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
95 |
94
|
eqeq2d |
⊢ ( ( 𝜑 ∧ 𝑔 = ( Word 𝐴 × { 0 } ) ) → ( ( 0g ‘ 𝑅 ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) ) ) |
96 |
|
eqidd |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) ) |
97 |
74 95 96
|
rspcedvd |
⊢ ( 𝜑 → ∃ 𝑔 ∈ 𝐹 ( 0g ‘ 𝑅 ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
98 |
|
fvexd |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ V ) |
99 |
55 97 98
|
elrnmptd |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
100 |
99 8
|
eleqtrrdi |
⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ 𝑆 ) |
101 |
100
|
ne0d |
⊢ ( 𝜑 → 𝑆 ≠ ∅ ) |
102 |
|
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 ) ) ) ) ) |
103 |
|
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 ) ) ) ) ) |
104 |
102 103
|
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 ( +g ‘ 𝑅 ) 𝑦 ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
105 |
|
eqid |
⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 ) |
106 |
13
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑅 ∈ CMnd ) |
107 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → Word 𝐴 ∈ V ) |
108 |
39
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
109 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
110 |
|
breq1 |
⊢ ( 𝑓 = 𝑖 → ( 𝑓 finSupp 0 ↔ 𝑖 finSupp 0 ) ) |
111 |
110 5
|
elrab2 |
⊢ ( 𝑖 ∈ 𝐹 ↔ ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑖 finSupp 0 ) ) |
112 |
111
|
simplbi |
⊢ ( 𝑖 ∈ 𝐹 → 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) |
113 |
112
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) |
114 |
25 18
|
elmapd |
⊢ ( 𝜑 → ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑖 : Word 𝐴 ⟶ ℤ ) ) |
115 |
114
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → ( 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ↔ 𝑖 : Word 𝐴 ⟶ ℤ ) ) |
116 |
113 115
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
117 |
116
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑤 ) ∈ ℤ ) |
118 |
37
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
119 |
1 3 109 117 118
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
120 |
119
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
121 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
122 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
123 |
50
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
124 |
50
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑔 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
125 |
|
fveq1 |
⊢ ( 𝑔 = 𝑖 → ( 𝑔 ‘ 𝑤 ) = ( 𝑖 ‘ 𝑤 ) ) |
126 |
125
|
oveq1d |
⊢ ( 𝑔 = 𝑖 → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
127 |
126
|
mpteq2dv |
⊢ ( 𝑔 = 𝑖 → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
128 |
127
|
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 ) ) ) |
129 |
128
|
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 ) ) |
130 |
124 129
|
sylib |
⊢ ( 𝜑 → ∀ 𝑖 ∈ 𝐹 ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
131 |
130
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
132 |
131
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
133 |
1 11 105 106 107 108 120 121 122 123 132
|
gsummptfsadd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
134 |
28
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → 𝑔 Fn Word 𝐴 ) |
135 |
134
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 Fn Word 𝐴 ) |
136 |
116
|
ffnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐹 ) → 𝑖 Fn Word 𝐴 ) |
137 |
136
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 Fn Word 𝐴 ) |
138 |
|
inidm |
⊢ ( Word 𝐴 ∩ Word 𝐴 ) = Word 𝐴 |
139 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) = ( 𝑔 ‘ 𝑤 ) ) |
140 |
|
eqidd |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑤 ) = ( 𝑖 ‘ 𝑤 ) ) |
141 |
135 137 107 107 138 139 140
|
ofval |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) = ( ( 𝑔 ‘ 𝑤 ) + ( 𝑖 ‘ 𝑤 ) ) ) |
142 |
141
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ‘ 𝑤 ) + ( 𝑖 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) ) |
143 |
20
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑅 ∈ Grp ) |
144 |
29
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
145 |
117
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑖 ‘ 𝑤 ) ∈ ℤ ) |
146 |
38
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) |
147 |
1 3 105
|
mulgdir |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑔 ‘ 𝑤 ) ∈ ℤ ∧ ( 𝑖 ‘ 𝑤 ) ∈ ℤ ∧ ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) + ( 𝑖 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
148 |
143 144 145 146 147
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) + ( 𝑖 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
149 |
142 148
|
eqtr2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
150 |
149
|
mpteq2dva |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
151 |
150
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
152 |
133 151
|
eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
153 |
|
fveq1 |
⊢ ( 𝑔 = ℎ → ( 𝑔 ‘ 𝑤 ) = ( ℎ ‘ 𝑤 ) ) |
154 |
153
|
oveq1d |
⊢ ( 𝑔 = ℎ → ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
155 |
154
|
mpteq2dv |
⊢ ( 𝑔 = ℎ → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
156 |
155
|
oveq2d |
⊢ ( 𝑔 = ℎ → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
157 |
156
|
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 ) ) ) ) ) |
158 |
|
fveq1 |
⊢ ( ℎ = ( 𝑔 ∘f + 𝑖 ) → ( ℎ ‘ 𝑤 ) = ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) ) |
159 |
158
|
oveq1d |
⊢ ( ℎ = ( 𝑔 ∘f + 𝑖 ) → ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
160 |
159
|
mpteq2dv |
⊢ ( ℎ = ( 𝑔 ∘f + 𝑖 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
161 |
160
|
oveq2d |
⊢ ( ℎ = ( 𝑔 ∘f + 𝑖 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
162 |
161
|
eqeq2d |
⊢ ( ℎ = ( 𝑔 ∘f + 𝑖 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
163 |
|
breq1 |
⊢ ( 𝑓 = ( 𝑔 ∘f + 𝑖 ) → ( 𝑓 finSupp 0 ↔ ( 𝑔 ∘f + 𝑖 ) finSupp 0 ) ) |
164 |
24
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ℤ ∈ V ) |
165 |
|
zaddcl |
⊢ ( ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) → ( 𝑥 + 𝑦 ) ∈ ℤ ) |
166 |
165
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) ∧ ( 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ) ) → ( 𝑥 + 𝑦 ) ∈ ℤ ) |
167 |
28
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
168 |
116
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 : Word 𝐴 ⟶ ℤ ) |
169 |
166 167 168 107 107 138
|
off |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) : Word 𝐴 ⟶ ℤ ) |
170 |
164 107 169
|
elmapdd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
171 |
|
zringring |
⊢ ℤring ∈ Ring |
172 |
|
ringmnd |
⊢ ( ℤring ∈ Ring → ℤring ∈ Mnd ) |
173 |
171 172
|
ax-mp |
⊢ ℤring ∈ Mnd |
174 |
173
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ℤring ∈ Mnd ) |
175 |
23
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ) |
176 |
112
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) |
177 |
47
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 finSupp 0 ) |
178 |
|
zring0 |
⊢ 0 = ( 0g ‘ ℤring ) |
179 |
177 178
|
breqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑔 finSupp ( 0g ‘ ℤring ) ) |
180 |
111
|
simprbi |
⊢ ( 𝑖 ∈ 𝐹 → 𝑖 finSupp 0 ) |
181 |
180
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 finSupp 0 ) |
182 |
181 178
|
breqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 𝑖 finSupp ( 0g ‘ ℤring ) ) |
183 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
184 |
183
|
mndpfsupp |
⊢ ( ( ( ℤring ∈ Mnd ∧ Word 𝐴 ∈ V ) ∧ ( 𝑔 ∈ ( ℤ ↑m Word 𝐴 ) ∧ 𝑖 ∈ ( ℤ ↑m Word 𝐴 ) ) ∧ ( 𝑔 finSupp ( 0g ‘ ℤring ) ∧ 𝑖 finSupp ( 0g ‘ ℤring ) ) ) → ( 𝑔 ∘f ( +g ‘ ℤring ) 𝑖 ) finSupp ( 0g ‘ ℤring ) ) |
185 |
174 107 175 176 179 182 184
|
syl222anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f ( +g ‘ ℤring ) 𝑖 ) finSupp ( 0g ‘ ℤring ) ) |
186 |
|
zringplusg |
⊢ + = ( +g ‘ ℤring ) |
187 |
186
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → + = ( +g ‘ ℤring ) ) |
188 |
187
|
ofeqd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∘f + = ∘f ( +g ‘ ℤring ) ) |
189 |
188
|
oveqd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) = ( 𝑔 ∘f ( +g ‘ ℤring ) 𝑖 ) ) |
190 |
178
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → 0 = ( 0g ‘ ℤring ) ) |
191 |
185 189 190
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) finSupp 0 ) |
192 |
163 170 191
|
elrabd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
193 |
192 5
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑔 ∘f + 𝑖 ) ∈ 𝐹 ) |
194 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
195 |
162 193 194
|
rspcedvdw |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ∃ ℎ ∈ 𝐹 ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
196 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ V ) |
197 |
157 195 196
|
elrnmptd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
198 |
197 8
|
eleqtrrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ∘f + 𝑖 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑆 ) |
199 |
152 198
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
200 |
199
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
201 |
200
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑖 ∈ 𝐹 ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
202 |
201
|
ad4ant13 |
⊢ ( ( ( ( ( ( ( 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 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ∈ 𝑆 ) |
203 |
104 202
|
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 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
204 |
8
|
eleq2i |
⊢ ( 𝑦 ∈ 𝑆 ↔ 𝑦 ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
205 |
127
|
oveq2d |
⊢ ( 𝑔 = 𝑖 → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
206 |
205
|
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 ) ) ) ) ) |
207 |
206
|
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 ) ) ) ) ) ) |
208 |
207
|
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 ) ) ) ) ) |
209 |
204 208
|
sylbb |
⊢ ( 𝑦 ∈ 𝑆 → ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
210 |
209
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑖 ∈ 𝐹 𝑦 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑖 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
211 |
210
|
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 ) ) ) ) ) |
212 |
203 211
|
r19.29a |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
213 |
59
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ∃ 𝑔 ∈ 𝐹 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
214 |
212 213
|
r19.29a |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑦 ∈ 𝑆 ) → ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
215 |
214
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ) |
216 |
9
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → 𝑅 ∈ Grp ) |
217 |
29
|
znegcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → - ( 𝑔 ‘ 𝑤 ) ∈ ℤ ) |
218 |
1 3 20 217 38
|
mulgcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ∈ 𝐵 ) |
219 |
218
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) : Word 𝐴 ⟶ 𝐵 ) |
220 |
28
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑔 : Word 𝐴 ⟶ ℤ ) |
221 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → 𝑤 ∈ Word 𝐴 ) |
222 |
220 221
|
fvco3d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) ‘ 𝑤 ) = ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ‘ ( 𝑔 ‘ 𝑤 ) ) ) |
223 |
|
eqid |
⊢ ( 𝑧 ∈ ℤ ↦ - 𝑧 ) = ( 𝑧 ∈ ℤ ↦ - 𝑧 ) |
224 |
|
negeq |
⊢ ( 𝑧 = ( 𝑔 ‘ 𝑤 ) → - 𝑧 = - ( 𝑔 ‘ 𝑤 ) ) |
225 |
223 224 29 217
|
fvmptd3 |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ‘ ( 𝑔 ‘ 𝑤 ) ) = - ( 𝑔 ‘ 𝑤 ) ) |
226 |
222 225
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) ‘ 𝑤 ) = - ( 𝑔 ‘ 𝑤 ) ) |
227 |
226
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
228 |
227
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
229 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℤ ) → 𝑧 ∈ ℤ ) |
230 |
229
|
znegcld |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℤ ) → - 𝑧 ∈ ℤ ) |
231 |
230
|
fmpttd |
⊢ ( 𝜑 → ( 𝑧 ∈ ℤ ↦ - 𝑧 ) : ℤ ⟶ ℤ ) |
232 |
231
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑧 ∈ ℤ ↦ - 𝑧 ) : ℤ ⟶ ℤ ) |
233 |
232 28
|
fcod |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) : Word 𝐴 ⟶ ℤ ) |
234 |
24
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ℤ ∈ V ) |
235 |
|
negeq |
⊢ ( 𝑧 = 0 → - 𝑧 = - 0 ) |
236 |
|
neg0 |
⊢ - 0 = 0 |
237 |
235 236
|
eqtrdi |
⊢ ( 𝑧 = 0 → - 𝑧 = 0 ) |
238 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
239 |
223 237 238 238
|
fvmptd3 |
⊢ ( 𝜑 → ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ‘ 0 ) = 0 ) |
240 |
239
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ‘ 0 ) = 0 ) |
241 |
42 28 232 19 234 47 240
|
fsuppco2 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) finSupp 0 ) |
242 |
41 42 19 43 38 233 241 49
|
fisuppov1 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( ( 𝑧 ∈ ℤ ↦ - 𝑧 ) ∘ 𝑔 ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
243 |
228 242
|
eqbrtrrd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) finSupp ( 0g ‘ 𝑅 ) ) |
244 |
1 11 13 19 219 243
|
gsumcl |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
245 |
244
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) |
246 |
10
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
247 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
248 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
249 |
1 11 105 13 19 39 218 247 248 50 243
|
gsummptfsadd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
250 |
249
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
251 |
29
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑤 ) ∈ ℂ ) |
252 |
251
|
negidd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑔 ‘ 𝑤 ) + - ( 𝑔 ‘ 𝑤 ) ) = 0 ) |
253 |
252
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) + - ( 𝑔 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) = ( 0 · ( 𝑀 Σg 𝑤 ) ) ) |
254 |
1 3 105
|
mulgdir |
⊢ ( ( 𝑅 ∈ Grp ∧ ( ( 𝑔 ‘ 𝑤 ) ∈ ℤ ∧ - ( 𝑔 ‘ 𝑤 ) ∈ ℤ ∧ ( 𝑀 Σg 𝑤 ) ∈ 𝐵 ) ) → ( ( ( 𝑔 ‘ 𝑤 ) + - ( 𝑔 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
255 |
20 29 217 38 254
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) + - ( 𝑔 ‘ 𝑤 ) ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
256 |
38 85
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( 0 · ( 𝑀 Σg 𝑤 ) ) = ( 0g ‘ 𝑅 ) ) |
257 |
253 255 256
|
3eqtr3d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 0g ‘ 𝑅 ) ) |
258 |
257
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) |
259 |
258
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) ) |
260 |
92
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( 0g ‘ 𝑅 ) ) ) = ( 0g ‘ 𝑅 ) ) |
261 |
259 260
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
262 |
261
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ( +g ‘ 𝑅 ) ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
263 |
246 250 262
|
3eqtr2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑥 ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) |
264 |
|
eqid |
⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 ) |
265 |
1 105 11 264
|
grpinvid1 |
⊢ ( ( 𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) → ( ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 𝑥 ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) ) |
266 |
265
|
biimpar |
⊢ ( ( ( 𝑅 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝐵 ) ∧ ( 𝑥 ( +g ‘ 𝑅 ) ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) = ( 0g ‘ 𝑅 ) ) → ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
267 |
216 53 245 263 266
|
syl31anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
268 |
|
fveq1 |
⊢ ( ℎ = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( ℎ ‘ 𝑤 ) = ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) ) |
269 |
268
|
oveq1d |
⊢ ( ℎ = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
270 |
269
|
mpteq2dv |
⊢ ( ℎ = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
271 |
270
|
oveq2d |
⊢ ( ℎ = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
272 |
271
|
eqeq2d |
⊢ ( ℎ = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ↔ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
273 |
|
breq1 |
⊢ ( 𝑓 = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) → ( 𝑓 finSupp 0 ↔ ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) finSupp 0 ) ) |
274 |
28
|
ffvelcdmda |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → ( 𝑔 ‘ 𝑣 ) ∈ ℤ ) |
275 |
274
|
znegcld |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ Word 𝐴 ) → - ( 𝑔 ‘ 𝑣 ) ∈ ℤ ) |
276 |
275
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) : Word 𝐴 ⟶ ℤ ) |
277 |
234 19 276
|
elmapdd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ∈ ( ℤ ↑m Word 𝐴 ) ) |
278 |
276
|
ffund |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → Fun ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ) |
279 |
134
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → 𝑔 Fn Word 𝐴 ) |
280 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → Word 𝐴 ∈ V ) |
281 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → 0 ∈ ℤ ) |
282 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) |
283 |
279 280 281 282
|
fvdifsupp |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → ( 𝑔 ‘ 𝑣 ) = 0 ) |
284 |
283
|
negeqd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → - ( 𝑔 ‘ 𝑣 ) = - 0 ) |
285 |
284 236
|
eqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑣 ∈ ( Word 𝐴 ∖ ( 𝑔 supp 0 ) ) ) → - ( 𝑔 ‘ 𝑣 ) = 0 ) |
286 |
285 19
|
suppss2 |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) supp 0 ) ⊆ ( 𝑔 supp 0 ) ) |
287 |
277 42 278 47 286
|
fsuppsssuppgd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) finSupp 0 ) |
288 |
273 277 287
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ∈ { 𝑓 ∈ ( ℤ ↑m Word 𝐴 ) ∣ 𝑓 finSupp 0 } ) |
289 |
288 5
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ∈ 𝐹 ) |
290 |
|
eqid |
⊢ ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) = ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) |
291 |
|
fveq2 |
⊢ ( 𝑣 = 𝑤 → ( 𝑔 ‘ 𝑣 ) = ( 𝑔 ‘ 𝑤 ) ) |
292 |
291
|
negeqd |
⊢ ( 𝑣 = 𝑤 → - ( 𝑔 ‘ 𝑣 ) = - ( 𝑔 ‘ 𝑤 ) ) |
293 |
290 292 221 217
|
fvmptd3 |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) = - ( 𝑔 ‘ 𝑤 ) ) |
294 |
293
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → - ( 𝑔 ‘ 𝑤 ) = ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) ) |
295 |
294
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑤 ∈ Word 𝐴 ) → ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) = ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) |
296 |
295
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) = ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) |
297 |
296
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ( 𝑣 ∈ Word 𝐴 ↦ - ( 𝑔 ‘ 𝑣 ) ) ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
298 |
272 289 297
|
rspcedvdw |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ∃ ℎ ∈ 𝐹 ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( ℎ ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) |
299 |
157 298 244
|
elrnmptd |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ ran ( 𝑔 ∈ 𝐹 ↦ ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) ) |
300 |
299 8
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑔 ∈ 𝐹 ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑆 ) |
301 |
300
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( - ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ∈ 𝑆 ) |
302 |
267 301
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) ∧ 𝑔 ∈ 𝐹 ) ∧ 𝑥 = ( 𝑅 Σg ( 𝑤 ∈ Word 𝐴 ↦ ( ( 𝑔 ‘ 𝑤 ) · ( 𝑀 Σg 𝑤 ) ) ) ) ) → ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) |
303 |
302 59
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) |
304 |
215 303
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑆 ) → ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ∧ ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) ) |
305 |
304
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ∧ ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) ) |
306 |
1 105 264
|
issubg2 |
⊢ ( 𝑅 ∈ Grp → ( 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ∧ ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) ) ) ) |
307 |
306
|
biimpar |
⊢ ( ( 𝑅 ∈ Grp ∧ ( 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝑆 ( ∀ 𝑦 ∈ 𝑆 ( 𝑥 ( +g ‘ 𝑅 ) 𝑦 ) ∈ 𝑆 ∧ ( ( invg ‘ 𝑅 ) ‘ 𝑥 ) ∈ 𝑆 ) ) ) → 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ) |
308 |
9 64 101 305 307
|
syl13anc |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝑅 ) ) |