Step |
Hyp |
Ref |
Expression |
1 |
|
archiabllem.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
archiabllem.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
archiabllem.e |
⊢ ≤ = ( le ‘ 𝑊 ) |
4 |
|
archiabllem.t |
⊢ < = ( lt ‘ 𝑊 ) |
5 |
|
archiabllem.m |
⊢ · = ( .g ‘ 𝑊 ) |
6 |
|
archiabllem.g |
⊢ ( 𝜑 → 𝑊 ∈ oGrp ) |
7 |
|
archiabllem.a |
⊢ ( 𝜑 → 𝑊 ∈ Archi ) |
8 |
|
archiabllem1.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝐵 ) |
9 |
|
archiabllem1.p |
⊢ ( 𝜑 → 0 < 𝑈 ) |
10 |
|
archiabllem1.s |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥 ) → 𝑈 ≤ 𝑥 ) |
11 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 0 ) → 0 ∈ ℤ ) |
12 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 0 ) → 𝑦 = 0 ) |
13 |
1 2 5
|
mulg0 |
⊢ ( 𝑈 ∈ 𝐵 → ( 0 · 𝑈 ) = 0 ) |
14 |
8 13
|
syl |
⊢ ( 𝜑 → ( 0 · 𝑈 ) = 0 ) |
15 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 0 ) → ( 0 · 𝑈 ) = 0 ) |
16 |
12 15
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 0 ) → 𝑦 = ( 0 · 𝑈 ) ) |
17 |
|
oveq1 |
⊢ ( 𝑛 = 0 → ( 𝑛 · 𝑈 ) = ( 0 · 𝑈 ) ) |
18 |
17
|
rspceeqv |
⊢ ( ( 0 ∈ ℤ ∧ 𝑦 = ( 0 · 𝑈 ) ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
19 |
11 16 18
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 = 0 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
20 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → 𝑚 ∈ ℕ ) |
21 |
20
|
nnzd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → 𝑚 ∈ ℤ ) |
22 |
21
|
znegcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → - 𝑚 ∈ ℤ ) |
23 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑈 ∈ 𝐵 ) |
24 |
23
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → 𝑈 ∈ 𝐵 ) |
25 |
|
eqid |
⊢ ( invg ‘ 𝑊 ) = ( invg ‘ 𝑊 ) |
26 |
1 5 25
|
mulgnegnn |
⊢ ( ( 𝑚 ∈ ℕ ∧ 𝑈 ∈ 𝐵 ) → ( - 𝑚 · 𝑈 ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑚 · 𝑈 ) ) ) |
27 |
20 24 26
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → ( - 𝑚 · 𝑈 ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑚 · 𝑈 ) ) ) |
28 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) |
29 |
28
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → ( ( invg ‘ 𝑊 ) ‘ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑚 · 𝑈 ) ) ) |
30 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑊 ∈ oGrp ) |
31 |
|
ogrpgrp |
⊢ ( 𝑊 ∈ oGrp → 𝑊 ∈ Grp ) |
32 |
30 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑊 ∈ Grp ) |
33 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑦 ∈ 𝐵 ) |
34 |
1 25
|
grpinvinv |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( ( invg ‘ 𝑊 ) ‘ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = 𝑦 ) |
35 |
32 33 34
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ( ( invg ‘ 𝑊 ) ‘ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = 𝑦 ) |
36 |
35
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → ( ( invg ‘ 𝑊 ) ‘ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = 𝑦 ) |
37 |
27 29 36
|
3eqtr2rd |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → 𝑦 = ( - 𝑚 · 𝑈 ) ) |
38 |
|
oveq1 |
⊢ ( 𝑛 = - 𝑚 → ( 𝑛 · 𝑈 ) = ( - 𝑚 · 𝑈 ) ) |
39 |
38
|
rspceeqv |
⊢ ( ( - 𝑚 ∈ ℤ ∧ 𝑦 = ( - 𝑚 · 𝑈 ) ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
40 |
22 37 39
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑚 ∈ ℕ ) ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
41 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑊 ∈ Archi ) |
42 |
9
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 0 < 𝑈 ) |
43 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝜑 ) |
44 |
43 10
|
syl3an1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥 ) → 𝑈 ≤ 𝑥 ) |
45 |
1 25
|
grpinvcl |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ∈ 𝐵 ) |
46 |
32 33 45
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ∈ 𝐵 ) |
47 |
1 2
|
grpidcl |
⊢ ( 𝑊 ∈ Grp → 0 ∈ 𝐵 ) |
48 |
32 47
|
syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 0 ∈ 𝐵 ) |
49 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 𝑦 < 0 ) |
50 |
|
eqid |
⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 ) |
51 |
1 4 50
|
ogrpaddlt |
⊢ ( ( 𝑊 ∈ oGrp ∧ ( 𝑦 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ∈ 𝐵 ) ∧ 𝑦 < 0 ) → ( 𝑦 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) < ( 0 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) ) |
52 |
30 33 48 46 49 51
|
syl131anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ( 𝑦 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) < ( 0 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) ) |
53 |
1 50 2 25
|
grprinv |
⊢ ( ( 𝑊 ∈ Grp ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = 0 ) |
54 |
32 33 53
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ( 𝑦 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = 0 ) |
55 |
1 50 2
|
grplid |
⊢ ( ( 𝑊 ∈ Grp ∧ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ∈ 𝐵 ) → ( 0 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) |
56 |
32 46 55
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ( 0 ( +g ‘ 𝑊 ) ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) = ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) |
57 |
52 54 56
|
3brtr3d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → 0 < ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) ) |
58 |
1 2 3 4 5 30 41 23 42 44 46 57
|
archiabllem1a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ∃ 𝑚 ∈ ℕ ( ( invg ‘ 𝑊 ) ‘ 𝑦 ) = ( 𝑚 · 𝑈 ) ) |
59 |
40 58
|
r19.29a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 < 0 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
60 |
59
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑦 < 0 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
61 |
|
nnssz |
⊢ ℕ ⊆ ℤ |
62 |
6
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 𝑊 ∈ oGrp ) |
63 |
7
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 𝑊 ∈ Archi ) |
64 |
8
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 𝑈 ∈ 𝐵 ) |
65 |
9
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 0 < 𝑈 ) |
66 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 𝜑 ) |
67 |
66 10
|
syl3an1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) ∧ 𝑥 ∈ 𝐵 ∧ 0 < 𝑥 ) → 𝑈 ≤ 𝑥 ) |
68 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 𝑦 ∈ 𝐵 ) |
69 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → 0 < 𝑦 ) |
70 |
1 2 3 4 5 62 63 64 65 67 68 69
|
archiabllem1a |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 0 < 𝑦 ) → ∃ 𝑛 ∈ ℕ 𝑦 = ( 𝑛 · 𝑈 ) ) |
71 |
70
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 0 < 𝑦 ) → ∃ 𝑛 ∈ ℕ 𝑦 = ( 𝑛 · 𝑈 ) ) |
72 |
|
ssrexv |
⊢ ( ℕ ⊆ ℤ → ( ∃ 𝑛 ∈ ℕ 𝑦 = ( 𝑛 · 𝑈 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) ) |
73 |
61 71 72
|
mpsyl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 0 < 𝑦 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |
74 |
|
isogrp |
⊢ ( 𝑊 ∈ oGrp ↔ ( 𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd ) ) |
75 |
74
|
simprbi |
⊢ ( 𝑊 ∈ oGrp → 𝑊 ∈ oMnd ) |
76 |
|
omndtos |
⊢ ( 𝑊 ∈ oMnd → 𝑊 ∈ Toset ) |
77 |
6 75 76
|
3syl |
⊢ ( 𝜑 → 𝑊 ∈ Toset ) |
78 |
77
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑊 ∈ Toset ) |
79 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
80 |
6 31 47
|
3syl |
⊢ ( 𝜑 → 0 ∈ 𝐵 ) |
81 |
80
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → 0 ∈ 𝐵 ) |
82 |
1 4
|
tlt3 |
⊢ ( ( 𝑊 ∈ Toset ∧ 𝑦 ∈ 𝐵 ∧ 0 ∈ 𝐵 ) → ( 𝑦 = 0 ∨ 𝑦 < 0 ∨ 0 < 𝑦 ) ) |
83 |
78 79 81 82
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 = 0 ∨ 𝑦 < 0 ∨ 0 < 𝑦 ) ) |
84 |
19 60 73 83
|
mpjao3dan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑛 ∈ ℤ 𝑦 = ( 𝑛 · 𝑈 ) ) |