Step |
Hyp |
Ref |
Expression |
1 |
|
frgpcyg.g |
⊢ 𝐺 = ( freeGrp ‘ 𝐼 ) |
2 |
|
brdom2 |
⊢ ( 𝐼 ≼ 1o ↔ ( 𝐼 ≺ 1o ∨ 𝐼 ≈ 1o ) ) |
3 |
|
sdom1 |
⊢ ( 𝐼 ≺ 1o ↔ 𝐼 = ∅ ) |
4 |
|
fveq2 |
⊢ ( 𝐼 = ∅ → ( freeGrp ‘ 𝐼 ) = ( freeGrp ‘ ∅ ) ) |
5 |
1 4
|
syl5eq |
⊢ ( 𝐼 = ∅ → 𝐺 = ( freeGrp ‘ ∅ ) ) |
6 |
|
0ex |
⊢ ∅ ∈ V |
7 |
|
eqid |
⊢ ( freeGrp ‘ ∅ ) = ( freeGrp ‘ ∅ ) |
8 |
7
|
frgpgrp |
⊢ ( ∅ ∈ V → ( freeGrp ‘ ∅ ) ∈ Grp ) |
9 |
6 8
|
ax-mp |
⊢ ( freeGrp ‘ ∅ ) ∈ Grp |
10 |
|
eqid |
⊢ ( Base ‘ ( freeGrp ‘ ∅ ) ) = ( Base ‘ ( freeGrp ‘ ∅ ) ) |
11 |
7 10
|
0frgp |
⊢ ( Base ‘ ( freeGrp ‘ ∅ ) ) ≈ 1o |
12 |
10
|
0cyg |
⊢ ( ( ( freeGrp ‘ ∅ ) ∈ Grp ∧ ( Base ‘ ( freeGrp ‘ ∅ ) ) ≈ 1o ) → ( freeGrp ‘ ∅ ) ∈ CycGrp ) |
13 |
9 11 12
|
mp2an |
⊢ ( freeGrp ‘ ∅ ) ∈ CycGrp |
14 |
5 13
|
eqeltrdi |
⊢ ( 𝐼 = ∅ → 𝐺 ∈ CycGrp ) |
15 |
3 14
|
sylbi |
⊢ ( 𝐼 ≺ 1o → 𝐺 ∈ CycGrp ) |
16 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
17 |
|
eqid |
⊢ ( .g ‘ 𝐺 ) = ( .g ‘ 𝐺 ) |
18 |
|
relen |
⊢ Rel ≈ |
19 |
18
|
brrelex1i |
⊢ ( 𝐼 ≈ 1o → 𝐼 ∈ V ) |
20 |
1
|
frgpgrp |
⊢ ( 𝐼 ∈ V → 𝐺 ∈ Grp ) |
21 |
19 20
|
syl |
⊢ ( 𝐼 ≈ 1o → 𝐺 ∈ Grp ) |
22 |
|
eqid |
⊢ ( ~FG ‘ 𝐼 ) = ( ~FG ‘ 𝐼 ) |
23 |
|
eqid |
⊢ ( varFGrp ‘ 𝐼 ) = ( varFGrp ‘ 𝐼 ) |
24 |
22 23 1 16
|
vrgpf |
⊢ ( 𝐼 ∈ V → ( varFGrp ‘ 𝐼 ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
25 |
19 24
|
syl |
⊢ ( 𝐼 ≈ 1o → ( varFGrp ‘ 𝐼 ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
26 |
|
en1uniel |
⊢ ( 𝐼 ≈ 1o → ∪ 𝐼 ∈ 𝐼 ) |
27 |
25 26
|
ffvelrnd |
⊢ ( 𝐼 ≈ 1o → ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ∈ ( Base ‘ 𝐺 ) ) |
28 |
|
zringgrp |
⊢ ℤring ∈ Grp |
29 |
19
|
uniexd |
⊢ ( 𝐼 ≈ 1o → ∪ 𝐼 ∈ V ) |
30 |
|
1zzd |
⊢ ( 𝐼 ≈ 1o → 1 ∈ ℤ ) |
31 |
29 30
|
fsnd |
⊢ ( 𝐼 ≈ 1o → { 〈 ∪ 𝐼 , 1 〉 } : { ∪ 𝐼 } ⟶ ℤ ) |
32 |
|
en1b |
⊢ ( 𝐼 ≈ 1o ↔ 𝐼 = { ∪ 𝐼 } ) |
33 |
32
|
biimpi |
⊢ ( 𝐼 ≈ 1o → 𝐼 = { ∪ 𝐼 } ) |
34 |
33
|
feq2d |
⊢ ( 𝐼 ≈ 1o → ( { 〈 ∪ 𝐼 , 1 〉 } : 𝐼 ⟶ ℤ ↔ { 〈 ∪ 𝐼 , 1 〉 } : { ∪ 𝐼 } ⟶ ℤ ) ) |
35 |
31 34
|
mpbird |
⊢ ( 𝐼 ≈ 1o → { 〈 ∪ 𝐼 , 1 〉 } : 𝐼 ⟶ ℤ ) |
36 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
37 |
1 36 23
|
frgpup3 |
⊢ ( ( ℤring ∈ Grp ∧ 𝐼 ∈ V ∧ { 〈 ∪ 𝐼 , 1 〉 } : 𝐼 ⟶ ℤ ) → ∃! 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } ) |
38 |
28 19 35 37
|
mp3an2i |
⊢ ( 𝐼 ≈ 1o → ∃! 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } ) |
39 |
38
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ∃! 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } ) |
40 |
|
reurex |
⊢ ( ∃! 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ∃ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } ) |
41 |
39 40
|
syl |
⊢ ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ∃ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } ) |
42 |
|
fveq1 |
⊢ ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) ‘ ∪ 𝐼 ) = ( { 〈 ∪ 𝐼 , 1 〉 } ‘ ∪ 𝐼 ) ) |
43 |
25 26
|
fvco3d |
⊢ ( 𝐼 ≈ 1o → ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) ‘ ∪ 𝐼 ) = ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
44 |
|
1z |
⊢ 1 ∈ ℤ |
45 |
|
fvsng |
⊢ ( ( ∪ 𝐼 ∈ V ∧ 1 ∈ ℤ ) → ( { 〈 ∪ 𝐼 , 1 〉 } ‘ ∪ 𝐼 ) = 1 ) |
46 |
29 44 45
|
sylancl |
⊢ ( 𝐼 ≈ 1o → ( { 〈 ∪ 𝐼 , 1 〉 } ‘ ∪ 𝐼 ) = 1 ) |
47 |
43 46
|
eqeq12d |
⊢ ( 𝐼 ≈ 1o → ( ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) ‘ ∪ 𝐼 ) = ( { 〈 ∪ 𝐼 , 1 〉 } ‘ ∪ 𝐼 ) ↔ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) |
48 |
42 47
|
syl5ib |
⊢ ( 𝐼 ≈ 1o → ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) |
49 |
48
|
ad2antrr |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ) → ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) |
50 |
16 36
|
ghmf |
⊢ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) → 𝑓 : ( Base ‘ 𝐺 ) ⟶ ℤ ) |
51 |
50
|
ad2antrl |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝑓 : ( Base ‘ 𝐺 ) ⟶ ℤ ) |
52 |
51
|
ffvelrnda |
⊢ ( ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( 𝑓 ‘ 𝑥 ) ∈ ℤ ) |
53 |
52
|
an32s |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑓 ‘ 𝑥 ) ∈ ℤ ) |
54 |
|
mptresid |
⊢ ( I ↾ ( Base ‘ 𝐺 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ 𝑥 ) |
55 |
1 16 23
|
frgpup3 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐼 ∈ V ∧ ( varFGrp ‘ 𝐼 ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) → ∃! 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
56 |
21 19 25 55
|
syl3anc |
⊢ ( 𝐼 ≈ 1o → ∃! 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
57 |
|
reurmo |
⊢ ( ∃! 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) → ∃* 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
58 |
56 57
|
syl |
⊢ ( 𝐼 ≈ 1o → ∃* 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
59 |
58
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ∃* 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
60 |
21
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝐺 ∈ Grp ) |
61 |
16
|
idghm |
⊢ ( 𝐺 ∈ Grp → ( I ↾ ( Base ‘ 𝐺 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) ) |
62 |
60 61
|
syl |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( I ↾ ( Base ‘ 𝐺 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) ) |
63 |
25
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( varFGrp ‘ 𝐼 ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) ) |
64 |
|
fcoi2 |
⊢ ( ( varFGrp ‘ 𝐼 ) : 𝐼 ⟶ ( Base ‘ 𝐺 ) → ( ( I ↾ ( Base ‘ 𝐺 ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
65 |
63 64
|
syl |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( I ↾ ( Base ‘ 𝐺 ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
66 |
51
|
feqmptd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝑓 = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( 𝑓 ‘ 𝑥 ) ) ) |
67 |
|
eqidd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
68 |
|
oveq1 |
⊢ ( 𝑛 = ( 𝑓 ‘ 𝑥 ) → ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
69 |
52 66 67 68
|
fmptco |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ 𝑓 ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
70 |
27
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ∈ ( Base ‘ 𝐺 ) ) |
71 |
|
eqid |
⊢ ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) = ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
72 |
17 71 16
|
mulgghm2 |
⊢ ( ( 𝐺 ∈ Grp ∧ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ∈ ( Base ‘ 𝐺 ) ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∈ ( ℤring GrpHom 𝐺 ) ) |
73 |
60 70 72
|
syl2anc |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∈ ( ℤring GrpHom 𝐺 ) ) |
74 |
|
simprl |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ) |
75 |
|
ghmco |
⊢ ( ( ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∈ ( ℤring GrpHom 𝐺 ) ∧ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ) → ( ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ 𝑓 ) ∈ ( 𝐺 GrpHom 𝐺 ) ) |
76 |
73 74 75
|
syl2anc |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑛 ∈ ℤ ↦ ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ 𝑓 ) ∈ ( 𝐺 GrpHom 𝐺 ) ) |
77 |
69 76
|
eqeltrrd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∈ ( 𝐺 GrpHom 𝐺 ) ) |
78 |
33
|
adantr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝐼 = { ∪ 𝐼 } ) |
79 |
78
|
eleq2d |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑦 ∈ 𝐼 ↔ 𝑦 ∈ { ∪ 𝐼 } ) ) |
80 |
|
simprr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) |
81 |
80
|
oveq1d |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( 1 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
82 |
16 17
|
mulg1 |
⊢ ( ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ∈ ( Base ‘ 𝐺 ) → ( 1 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) |
83 |
70 82
|
syl |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 1 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) |
84 |
81 83
|
eqtrd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) |
85 |
|
elsni |
⊢ ( 𝑦 ∈ { ∪ 𝐼 } → 𝑦 = ∪ 𝐼 ) |
86 |
85
|
fveq2d |
⊢ ( 𝑦 ∈ { ∪ 𝐼 } → ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) = ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) |
87 |
86
|
fveq2d |
⊢ ( 𝑦 ∈ { ∪ 𝐼 } → ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) = ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
88 |
87
|
oveq1d |
⊢ ( 𝑦 ∈ { ∪ 𝐼 } → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
89 |
88 86
|
eqeq12d |
⊢ ( 𝑦 ∈ { ∪ 𝐼 } → ( ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ↔ ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
90 |
84 89
|
syl5ibrcom |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑦 ∈ { ∪ 𝐼 } → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ) |
91 |
79 90
|
sylbid |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑦 ∈ 𝐼 → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ) |
92 |
91
|
imp |
⊢ ( ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) |
93 |
92
|
mpteq2dva |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑦 ∈ 𝐼 ↦ ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) = ( 𝑦 ∈ 𝐼 ↦ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ) |
94 |
63
|
ffvelrnda |
⊢ ( ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) ∧ 𝑦 ∈ 𝐼 ) → ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) ) |
95 |
63
|
feqmptd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( varFGrp ‘ 𝐼 ) = ( 𝑦 ∈ 𝐼 ↦ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ) |
96 |
|
eqidd |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
97 |
|
fveq2 |
⊢ ( 𝑥 = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) → ( 𝑓 ‘ 𝑥 ) = ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ) |
98 |
97
|
oveq1d |
⊢ ( 𝑥 = ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) → ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
99 |
94 95 96 98
|
fmptco |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( 𝑦 ∈ 𝐼 ↦ ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ 𝑦 ) ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
100 |
93 99 95
|
3eqtr4d |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) |
101 |
|
coeq1 |
⊢ ( 𝑔 = ( I ↾ ( Base ‘ 𝐺 ) ) → ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( ( I ↾ ( Base ‘ 𝐺 ) ) ∘ ( varFGrp ‘ 𝐼 ) ) ) |
102 |
101
|
eqeq1d |
⊢ ( 𝑔 = ( I ↾ ( Base ‘ 𝐺 ) ) → ( ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ↔ ( ( I ↾ ( Base ‘ 𝐺 ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) ) |
103 |
|
coeq1 |
⊢ ( 𝑔 = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) → ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ ( varFGrp ‘ 𝐼 ) ) ) |
104 |
103
|
eqeq1d |
⊢ ( 𝑔 = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) → ( ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ↔ ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) ) |
105 |
102 104
|
rmoi |
⊢ ( ( ∃* 𝑔 ∈ ( 𝐺 GrpHom 𝐺 ) ( 𝑔 ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ∧ ( ( I ↾ ( Base ‘ 𝐺 ) ) ∈ ( 𝐺 GrpHom 𝐺 ) ∧ ( ( I ↾ ( Base ‘ 𝐺 ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) ∧ ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∈ ( 𝐺 GrpHom 𝐺 ) ∧ ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ∘ ( varFGrp ‘ 𝐼 ) ) = ( varFGrp ‘ 𝐼 ) ) ) → ( I ↾ ( Base ‘ 𝐺 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
106 |
59 62 65 77 100 105
|
syl122anc |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( I ↾ ( Base ‘ 𝐺 ) ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
107 |
54 106
|
syl5eqr |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ 𝑥 ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
108 |
|
mpteqb |
⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) 𝑥 ∈ ( Base ‘ 𝐺 ) → ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ 𝑥 ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
109 |
|
id |
⊢ ( 𝑥 ∈ ( Base ‘ 𝐺 ) → 𝑥 ∈ ( Base ‘ 𝐺 ) ) |
110 |
108 109
|
mprg |
⊢ ( ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ 𝑥 ) = ( 𝑥 ∈ ( Base ‘ 𝐺 ) ↦ ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ↔ ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
111 |
107 110
|
sylib |
⊢ ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ∀ 𝑥 ∈ ( Base ‘ 𝐺 ) 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
112 |
111
|
r19.21bi |
⊢ ( ( ( 𝐼 ≈ 1o ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
113 |
112
|
an32s |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
114 |
68
|
rspceeqv |
⊢ ( ( ( 𝑓 ‘ 𝑥 ) ∈ ℤ ∧ 𝑥 = ( ( 𝑓 ‘ 𝑥 ) ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
115 |
53 113 114
|
syl2anc |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ ( 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ∧ ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 ) ) → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
116 |
115
|
expr |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ) → ( ( 𝑓 ‘ ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) = 1 → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
117 |
49 116
|
syld |
⊢ ( ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) ∧ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ) → ( ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
118 |
117
|
rexlimdva |
⊢ ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ( ∃ 𝑓 ∈ ( 𝐺 GrpHom ℤring ) ( 𝑓 ∘ ( varFGrp ‘ 𝐼 ) ) = { 〈 ∪ 𝐼 , 1 〉 } → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) ) |
119 |
41 118
|
mpd |
⊢ ( ( 𝐼 ≈ 1o ∧ 𝑥 ∈ ( Base ‘ 𝐺 ) ) → ∃ 𝑛 ∈ ℤ 𝑥 = ( 𝑛 ( .g ‘ 𝐺 ) ( ( varFGrp ‘ 𝐼 ) ‘ ∪ 𝐼 ) ) ) |
120 |
16 17 21 27 119
|
iscygd |
⊢ ( 𝐼 ≈ 1o → 𝐺 ∈ CycGrp ) |
121 |
15 120
|
jaoi |
⊢ ( ( 𝐼 ≺ 1o ∨ 𝐼 ≈ 1o ) → 𝐺 ∈ CycGrp ) |
122 |
2 121
|
sylbi |
⊢ ( 𝐼 ≼ 1o → 𝐺 ∈ CycGrp ) |
123 |
|
cygabl |
⊢ ( 𝐺 ∈ CycGrp → 𝐺 ∈ Abel ) |
124 |
1
|
frgpnabl |
⊢ ( 1o ≺ 𝐼 → ¬ 𝐺 ∈ Abel ) |
125 |
124
|
con2i |
⊢ ( 𝐺 ∈ Abel → ¬ 1o ≺ 𝐼 ) |
126 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
127 |
|
eqid |
⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 ) |
128 |
16 127
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) ) |
129 |
1 16
|
elbasfv |
⊢ ( ( 0g ‘ 𝐺 ) ∈ ( Base ‘ 𝐺 ) → 𝐼 ∈ V ) |
130 |
126 128 129
|
3syl |
⊢ ( 𝐺 ∈ Abel → 𝐼 ∈ V ) |
131 |
|
1onn |
⊢ 1o ∈ ω |
132 |
|
nnfi |
⊢ ( 1o ∈ ω → 1o ∈ Fin ) |
133 |
131 132
|
ax-mp |
⊢ 1o ∈ Fin |
134 |
|
fidomtri2 |
⊢ ( ( 𝐼 ∈ V ∧ 1o ∈ Fin ) → ( 𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼 ) ) |
135 |
130 133 134
|
sylancl |
⊢ ( 𝐺 ∈ Abel → ( 𝐼 ≼ 1o ↔ ¬ 1o ≺ 𝐼 ) ) |
136 |
125 135
|
mpbird |
⊢ ( 𝐺 ∈ Abel → 𝐼 ≼ 1o ) |
137 |
123 136
|
syl |
⊢ ( 𝐺 ∈ CycGrp → 𝐼 ≼ 1o ) |
138 |
122 137
|
impbii |
⊢ ( 𝐼 ≼ 1o ↔ 𝐺 ∈ CycGrp ) |