Step |
Hyp |
Ref |
Expression |
1 |
|
lgsqr.y |
⊢ 𝑌 = ( ℤ/nℤ ‘ 𝑃 ) |
2 |
|
lgsqr.s |
⊢ 𝑆 = ( Poly1 ‘ 𝑌 ) |
3 |
|
lgsqr.b |
⊢ 𝐵 = ( Base ‘ 𝑆 ) |
4 |
|
lgsqr.d |
⊢ 𝐷 = ( deg1 ‘ 𝑌 ) |
5 |
|
lgsqr.o |
⊢ 𝑂 = ( eval1 ‘ 𝑌 ) |
6 |
|
lgsqr.e |
⊢ ↑ = ( .g ‘ ( mulGrp ‘ 𝑆 ) ) |
7 |
|
lgsqr.x |
⊢ 𝑋 = ( var1 ‘ 𝑌 ) |
8 |
|
lgsqr.m |
⊢ − = ( -g ‘ 𝑆 ) |
9 |
|
lgsqr.u |
⊢ 1 = ( 1r ‘ 𝑆 ) |
10 |
|
lgsqr.t |
⊢ 𝑇 = ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) |
11 |
|
lgsqr.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑌 ) |
12 |
|
lgsqr.1 |
⊢ ( 𝜑 → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
13 |
|
lgsqrlem1.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℤ ) |
14 |
|
lgsqrlem1.4 |
⊢ ( 𝜑 → ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) mod 𝑃 ) = ( 1 mod 𝑃 ) ) |
15 |
10
|
fveq2i |
⊢ ( 𝑂 ‘ 𝑇 ) = ( 𝑂 ‘ ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) ) |
16 |
15
|
fveq1i |
⊢ ( ( 𝑂 ‘ 𝑇 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( 𝑂 ‘ ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) |
17 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
18 |
12
|
eldifad |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
19 |
1
|
znfld |
⊢ ( 𝑃 ∈ ℙ → 𝑌 ∈ Field ) |
20 |
18 19
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ Field ) |
21 |
|
fldidom |
⊢ ( 𝑌 ∈ Field → 𝑌 ∈ IDomn ) |
22 |
20 21
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ IDomn ) |
23 |
|
isidom |
⊢ ( 𝑌 ∈ IDomn ↔ ( 𝑌 ∈ CRing ∧ 𝑌 ∈ Domn ) ) |
24 |
23
|
simplbi |
⊢ ( 𝑌 ∈ IDomn → 𝑌 ∈ CRing ) |
25 |
22 24
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ CRing ) |
26 |
|
crngring |
⊢ ( 𝑌 ∈ CRing → 𝑌 ∈ Ring ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ Ring ) |
28 |
11
|
zrhrhm |
⊢ ( 𝑌 ∈ Ring → 𝐿 ∈ ( ℤring RingHom 𝑌 ) ) |
29 |
27 28
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ( ℤring RingHom 𝑌 ) ) |
30 |
|
zringbas |
⊢ ℤ = ( Base ‘ ℤring ) |
31 |
30 17
|
rhmf |
⊢ ( 𝐿 ∈ ( ℤring RingHom 𝑌 ) → 𝐿 : ℤ ⟶ ( Base ‘ 𝑌 ) ) |
32 |
29 31
|
syl |
⊢ ( 𝜑 → 𝐿 : ℤ ⟶ ( Base ‘ 𝑌 ) ) |
33 |
32 13
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐿 ‘ 𝐴 ) ∈ ( Base ‘ 𝑌 ) ) |
34 |
5 7 17 2 3 25 33
|
evl1vard |
⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ∧ ( ( 𝑂 ‘ 𝑋 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 𝐿 ‘ 𝐴 ) ) ) |
35 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ 𝑌 ) ) = ( .g ‘ ( mulGrp ‘ 𝑌 ) ) |
36 |
|
oddprm |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) → ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ ) |
37 |
12 36
|
syl |
⊢ ( 𝜑 → ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ ) |
38 |
37
|
nnnn0d |
⊢ ( 𝜑 → ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ0 ) |
39 |
5 2 17 3 25 33 34 6 35 38
|
evl1expd |
⊢ ( 𝜑 → ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) ) ) |
40 |
|
zringmpg |
⊢ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) = ( mulGrp ‘ ℤring ) |
41 |
|
eqid |
⊢ ( mulGrp ‘ 𝑌 ) = ( mulGrp ‘ 𝑌 ) |
42 |
40 41
|
rhmmhm |
⊢ ( 𝐿 ∈ ( ℤring RingHom 𝑌 ) → 𝐿 ∈ ( ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) MndHom ( mulGrp ‘ 𝑌 ) ) ) |
43 |
29 42
|
syl |
⊢ ( 𝜑 → 𝐿 ∈ ( ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) MndHom ( mulGrp ‘ 𝑌 ) ) ) |
44 |
40 30
|
mgpbas |
⊢ ℤ = ( Base ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) |
45 |
|
eqid |
⊢ ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) = ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) |
46 |
44 45 35
|
mhmmulg |
⊢ ( ( 𝐿 ∈ ( ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) MndHom ( mulGrp ‘ 𝑌 ) ) ∧ ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ0 ∧ 𝐴 ∈ ℤ ) → ( 𝐿 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) ) |
47 |
43 38 13 46
|
syl3anc |
⊢ ( 𝜑 → ( 𝐿 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) ) |
48 |
|
zsubrg |
⊢ ℤ ∈ ( SubRing ‘ ℂfld ) |
49 |
|
eqid |
⊢ ( mulGrp ‘ ℂfld ) = ( mulGrp ‘ ℂfld ) |
50 |
49
|
subrgsubm |
⊢ ( ℤ ∈ ( SubRing ‘ ℂfld ) → ℤ ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ) |
51 |
48 50
|
mp1i |
⊢ ( 𝜑 → ℤ ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ) |
52 |
|
eqid |
⊢ ( .g ‘ ( mulGrp ‘ ℂfld ) ) = ( .g ‘ ( mulGrp ‘ ℂfld ) ) |
53 |
|
eqid |
⊢ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) = ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) |
54 |
52 53 45
|
submmulg |
⊢ ( ( ℤ ∈ ( SubMnd ‘ ( mulGrp ‘ ℂfld ) ) ∧ ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ0 ∧ 𝐴 ∈ ℤ ) → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ ℂfld ) ) 𝐴 ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) |
55 |
51 38 13 54
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ ℂfld ) ) 𝐴 ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) |
56 |
13
|
zcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
57 |
|
cnfldexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ0 ) → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ ℂfld ) ) 𝐴 ) = ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) |
58 |
56 38 57
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ ℂfld ) ) 𝐴 ) = ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) |
59 |
55 58
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) = ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) |
60 |
59
|
fveq2d |
⊢ ( 𝜑 → ( 𝐿 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) = ( 𝐿 ‘ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) ) |
61 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
62 |
18 61
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
63 |
|
zexpcl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( ( 𝑃 − 1 ) / 2 ) ∈ ℕ0 ) → ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ∈ ℤ ) |
64 |
13 38 63
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ∈ ℤ ) |
65 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
66 |
|
moddvds |
⊢ ( ( 𝑃 ∈ ℕ ∧ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ∈ ℤ ∧ 1 ∈ ℤ ) → ( ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) mod 𝑃 ) = ( 1 mod 𝑃 ) ↔ 𝑃 ∥ ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) − 1 ) ) ) |
67 |
62 64 65 66
|
syl3anc |
⊢ ( 𝜑 → ( ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) mod 𝑃 ) = ( 1 mod 𝑃 ) ↔ 𝑃 ∥ ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) − 1 ) ) ) |
68 |
14 67
|
mpbid |
⊢ ( 𝜑 → 𝑃 ∥ ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) − 1 ) ) |
69 |
62
|
nnnn0d |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
70 |
1 11
|
zndvds |
⊢ ( ( 𝑃 ∈ ℕ0 ∧ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ∈ ℤ ∧ 1 ∈ ℤ ) → ( ( 𝐿 ‘ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) = ( 𝐿 ‘ 1 ) ↔ 𝑃 ∥ ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) − 1 ) ) ) |
71 |
69 64 65 70
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐿 ‘ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) = ( 𝐿 ‘ 1 ) ↔ 𝑃 ∥ ( ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) − 1 ) ) ) |
72 |
68 71
|
mpbird |
⊢ ( 𝜑 → ( 𝐿 ‘ ( 𝐴 ↑ ( ( 𝑃 − 1 ) / 2 ) ) ) = ( 𝐿 ‘ 1 ) ) |
73 |
|
zring1 |
⊢ 1 = ( 1r ‘ ℤring ) |
74 |
|
eqid |
⊢ ( 1r ‘ 𝑌 ) = ( 1r ‘ 𝑌 ) |
75 |
73 74
|
rhm1 |
⊢ ( 𝐿 ∈ ( ℤring RingHom 𝑌 ) → ( 𝐿 ‘ 1 ) = ( 1r ‘ 𝑌 ) ) |
76 |
29 75
|
syl |
⊢ ( 𝜑 → ( 𝐿 ‘ 1 ) = ( 1r ‘ 𝑌 ) ) |
77 |
60 72 76
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐿 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( ( mulGrp ‘ ℂfld ) ↾s ℤ ) ) 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) |
78 |
47 77
|
eqtr3d |
⊢ ( 𝜑 → ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) |
79 |
78
|
eqeq2d |
⊢ ( 𝜑 → ( ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) ↔ ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) |
80 |
79
|
anbi2d |
⊢ ( 𝜑 → ( ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( ( 𝑃 − 1 ) / 2 ) ( .g ‘ ( mulGrp ‘ 𝑌 ) ) ( 𝐿 ‘ 𝐴 ) ) ) ↔ ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) ) |
81 |
39 80
|
mpbid |
⊢ ( 𝜑 → ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) |
82 |
|
eqid |
⊢ ( algSc ‘ 𝑆 ) = ( algSc ‘ 𝑆 ) |
83 |
17 74
|
ringidcl |
⊢ ( 𝑌 ∈ Ring → ( 1r ‘ 𝑌 ) ∈ ( Base ‘ 𝑌 ) ) |
84 |
27 83
|
syl |
⊢ ( 𝜑 → ( 1r ‘ 𝑌 ) ∈ ( Base ‘ 𝑌 ) ) |
85 |
5 2 17 82 3 25 84 33
|
evl1scad |
⊢ ( 𝜑 → ( ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) |
86 |
2 82 74 9
|
ply1scl1 |
⊢ ( 𝑌 ∈ Ring → ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) = 1 ) |
87 |
27 86
|
syl |
⊢ ( 𝜑 → ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) = 1 ) |
88 |
87
|
eleq1d |
⊢ ( 𝜑 → ( ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ∈ 𝐵 ↔ 1 ∈ 𝐵 ) ) |
89 |
87
|
fveq2d |
⊢ ( 𝜑 → ( 𝑂 ‘ ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ) = ( 𝑂 ‘ 1 ) ) |
90 |
89
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( 𝑂 ‘ 1 ) ‘ ( 𝐿 ‘ 𝐴 ) ) ) |
91 |
90
|
eqeq1d |
⊢ ( 𝜑 → ( ( ( 𝑂 ‘ ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ↔ ( ( 𝑂 ‘ 1 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) |
92 |
88 91
|
anbi12d |
⊢ ( 𝜑 → ( ( ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( algSc ‘ 𝑆 ) ‘ ( 1r ‘ 𝑌 ) ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ↔ ( 1 ∈ 𝐵 ∧ ( ( 𝑂 ‘ 1 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) ) |
93 |
85 92
|
mpbid |
⊢ ( 𝜑 → ( 1 ∈ 𝐵 ∧ ( ( 𝑂 ‘ 1 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 1r ‘ 𝑌 ) ) ) |
94 |
|
eqid |
⊢ ( -g ‘ 𝑌 ) = ( -g ‘ 𝑌 ) |
95 |
5 2 17 3 25 33 81 93 8 94
|
evl1subd |
⊢ ( 𝜑 → ( ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) ∈ 𝐵 ∧ ( ( 𝑂 ‘ ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( 1r ‘ 𝑌 ) ( -g ‘ 𝑌 ) ( 1r ‘ 𝑌 ) ) ) ) |
96 |
95
|
simprd |
⊢ ( 𝜑 → ( ( 𝑂 ‘ ( ( ( ( 𝑃 − 1 ) / 2 ) ↑ 𝑋 ) − 1 ) ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( 1r ‘ 𝑌 ) ( -g ‘ 𝑌 ) ( 1r ‘ 𝑌 ) ) ) |
97 |
16 96
|
syl5eq |
⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑇 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( ( 1r ‘ 𝑌 ) ( -g ‘ 𝑌 ) ( 1r ‘ 𝑌 ) ) ) |
98 |
|
ringgrp |
⊢ ( 𝑌 ∈ Ring → 𝑌 ∈ Grp ) |
99 |
27 98
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ Grp ) |
100 |
|
eqid |
⊢ ( 0g ‘ 𝑌 ) = ( 0g ‘ 𝑌 ) |
101 |
17 100 94
|
grpsubid |
⊢ ( ( 𝑌 ∈ Grp ∧ ( 1r ‘ 𝑌 ) ∈ ( Base ‘ 𝑌 ) ) → ( ( 1r ‘ 𝑌 ) ( -g ‘ 𝑌 ) ( 1r ‘ 𝑌 ) ) = ( 0g ‘ 𝑌 ) ) |
102 |
99 84 101
|
syl2anc |
⊢ ( 𝜑 → ( ( 1r ‘ 𝑌 ) ( -g ‘ 𝑌 ) ( 1r ‘ 𝑌 ) ) = ( 0g ‘ 𝑌 ) ) |
103 |
97 102
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑇 ) ‘ ( 𝐿 ‘ 𝐴 ) ) = ( 0g ‘ 𝑌 ) ) |