Step |
Hyp |
Ref |
Expression |
1 |
|
pgpfac1.k |
⊢ 𝐾 = ( mrCls ‘ ( SubGrp ‘ 𝐺 ) ) |
2 |
|
pgpfac1.s |
⊢ 𝑆 = ( 𝐾 ‘ { 𝐴 } ) |
3 |
|
pgpfac1.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
4 |
|
pgpfac1.o |
⊢ 𝑂 = ( od ‘ 𝐺 ) |
5 |
|
pgpfac1.e |
⊢ 𝐸 = ( gEx ‘ 𝐺 ) |
6 |
|
pgpfac1.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
7 |
|
pgpfac1.l |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
8 |
|
pgpfac1.p |
⊢ ( 𝜑 → 𝑃 pGrp 𝐺 ) |
9 |
|
pgpfac1.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
10 |
|
pgpfac1.n |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
11 |
|
pgpfac1.oe |
⊢ ( 𝜑 → ( 𝑂 ‘ 𝐴 ) = 𝐸 ) |
12 |
|
pgpfac1.u |
⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
13 |
|
pgpfac1.au |
⊢ ( 𝜑 → 𝐴 ∈ 𝑈 ) |
14 |
|
pgpfac1.w |
⊢ ( 𝜑 → 𝑊 ∈ ( SubGrp ‘ 𝐺 ) ) |
15 |
|
pgpfac1.i |
⊢ ( 𝜑 → ( 𝑆 ∩ 𝑊 ) = { 0 } ) |
16 |
|
pgpfac1.ss |
⊢ ( 𝜑 → ( 𝑆 ⊕ 𝑊 ) ⊆ 𝑈 ) |
17 |
|
pgpfac1.2 |
⊢ ( 𝜑 → ∀ 𝑤 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤 ) → ¬ ( 𝑆 ⊕ 𝑊 ) ⊊ 𝑤 ) ) |
18 |
|
pgpfac1.c |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝑈 ∖ ( 𝑆 ⊕ 𝑊 ) ) ) |
19 |
|
pgpfac1.mg |
⊢ · = ( .g ‘ 𝐺 ) |
20 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
pgpfac1lem2 |
⊢ ( 𝜑 → ( 𝑃 · 𝐶 ) ∈ ( 𝑆 ⊕ 𝑊 ) ) |
21 |
|
ablgrp |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ Grp ) |
22 |
9 21
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
23 |
3
|
subgacs |
⊢ ( 𝐺 ∈ Grp → ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) ) |
24 |
|
acsmre |
⊢ ( ( SubGrp ‘ 𝐺 ) ∈ ( ACS ‘ 𝐵 ) → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
25 |
22 23 24
|
3syl |
⊢ ( 𝜑 → ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ) |
26 |
3
|
subgss |
⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ 𝐵 ) |
27 |
12 26
|
syl |
⊢ ( 𝜑 → 𝑈 ⊆ 𝐵 ) |
28 |
27 13
|
sseldd |
⊢ ( 𝜑 → 𝐴 ∈ 𝐵 ) |
29 |
1
|
mrcsncl |
⊢ ( ( ( SubGrp ‘ 𝐺 ) ∈ ( Moore ‘ 𝐵 ) ∧ 𝐴 ∈ 𝐵 ) → ( 𝐾 ‘ { 𝐴 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
30 |
25 28 29
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝐴 } ) ∈ ( SubGrp ‘ 𝐺 ) ) |
31 |
2 30
|
eqeltrid |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
32 |
7
|
lsmcom |
⊢ ( ( 𝐺 ∈ Abel ∧ 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑊 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑆 ⊕ 𝑊 ) = ( 𝑊 ⊕ 𝑆 ) ) |
33 |
9 31 14 32
|
syl3anc |
⊢ ( 𝜑 → ( 𝑆 ⊕ 𝑊 ) = ( 𝑊 ⊕ 𝑆 ) ) |
34 |
20 33
|
eleqtrd |
⊢ ( 𝜑 → ( 𝑃 · 𝐶 ) ∈ ( 𝑊 ⊕ 𝑆 ) ) |
35 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
36 |
35 7 14 31
|
lsmelvalm |
⊢ ( 𝜑 → ( ( 𝑃 · 𝐶 ) ∈ ( 𝑊 ⊕ 𝑆 ) ↔ ∃ 𝑤 ∈ 𝑊 ∃ 𝑠 ∈ 𝑆 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ) ) |
37 |
34 36
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑊 ∃ 𝑠 ∈ 𝑆 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ) |
38 |
|
eqid |
⊢ ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) = ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) |
39 |
3 19 38 1
|
cycsubg2 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝐴 ∈ 𝐵 ) → ( 𝐾 ‘ { 𝐴 } ) = ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ) |
40 |
22 28 39
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ‘ { 𝐴 } ) = ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ) |
41 |
2 40
|
syl5eq |
⊢ ( 𝜑 → 𝑆 = ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ) |
42 |
41
|
rexeqdv |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝑆 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ∃ 𝑠 ∈ ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ) ) |
43 |
|
ovex |
⊢ ( 𝑘 · 𝐴 ) ∈ V |
44 |
43
|
rgenw |
⊢ ∀ 𝑘 ∈ ℤ ( 𝑘 · 𝐴 ) ∈ V |
45 |
|
oveq2 |
⊢ ( 𝑠 = ( 𝑘 · 𝐴 ) → ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
46 |
45
|
eqeq2d |
⊢ ( 𝑠 = ( 𝑘 · 𝐴 ) → ( ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
47 |
38 46
|
rexrnmptw |
⊢ ( ∀ 𝑘 ∈ ℤ ( 𝑘 · 𝐴 ) ∈ V → ( ∃ 𝑠 ∈ ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
48 |
44 47
|
ax-mp |
⊢ ( ∃ 𝑠 ∈ ran ( 𝑘 ∈ ℤ ↦ ( 𝑘 · 𝐴 ) ) ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
49 |
42 48
|
bitrdi |
⊢ ( 𝜑 → ( ∃ 𝑠 ∈ 𝑆 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
50 |
49
|
rexbidv |
⊢ ( 𝜑 → ( ∃ 𝑤 ∈ 𝑊 ∃ 𝑠 ∈ 𝑆 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) 𝑠 ) ↔ ∃ 𝑤 ∈ 𝑊 ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
51 |
37 50
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑊 ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
52 |
|
rexcom |
⊢ ( ∃ 𝑤 ∈ 𝑊 ∃ 𝑘 ∈ ℤ ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ∃ 𝑘 ∈ ℤ ∃ 𝑤 ∈ 𝑊 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
53 |
51 52
|
sylib |
⊢ ( 𝜑 → ∃ 𝑘 ∈ ℤ ∃ 𝑤 ∈ 𝑊 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
54 |
22
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → 𝐺 ∈ Grp ) |
55 |
3
|
subgss |
⊢ ( 𝑊 ∈ ( SubGrp ‘ 𝐺 ) → 𝑊 ⊆ 𝐵 ) |
56 |
14 55
|
syl |
⊢ ( 𝜑 → 𝑊 ⊆ 𝐵 ) |
57 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → 𝑊 ⊆ 𝐵 ) |
58 |
57
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → 𝑤 ∈ 𝐵 ) |
59 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → 𝑘 ∈ ℤ ) |
60 |
28
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → 𝐴 ∈ 𝐵 ) |
61 |
3 19
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑘 ∈ ℤ ∧ 𝐴 ∈ 𝐵 ) → ( 𝑘 · 𝐴 ) ∈ 𝐵 ) |
62 |
54 59 60 61
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → ( 𝑘 · 𝐴 ) ∈ 𝐵 ) |
63 |
|
pgpprm |
⊢ ( 𝑃 pGrp 𝐺 → 𝑃 ∈ ℙ ) |
64 |
|
prmz |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℤ ) |
65 |
8 63 64
|
3syl |
⊢ ( 𝜑 → 𝑃 ∈ ℤ ) |
66 |
18
|
eldifad |
⊢ ( 𝜑 → 𝐶 ∈ 𝑈 ) |
67 |
27 66
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
68 |
3 19
|
mulgcl |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑃 ∈ ℤ ∧ 𝐶 ∈ 𝐵 ) → ( 𝑃 · 𝐶 ) ∈ 𝐵 ) |
69 |
22 65 67 68
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 · 𝐶 ) ∈ 𝐵 ) |
70 |
69
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → ( 𝑃 · 𝐶 ) ∈ 𝐵 ) |
71 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
72 |
3 71 35
|
grpsubadd |
⊢ ( ( 𝐺 ∈ Grp ∧ ( 𝑤 ∈ 𝐵 ∧ ( 𝑘 · 𝐴 ) ∈ 𝐵 ∧ ( 𝑃 · 𝐶 ) ∈ 𝐵 ) ) → ( ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = ( 𝑃 · 𝐶 ) ↔ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = 𝑤 ) ) |
73 |
54 58 62 70 72
|
syl13anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → ( ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = ( 𝑃 · 𝐶 ) ↔ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = 𝑤 ) ) |
74 |
|
eqcom |
⊢ ( ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = ( 𝑃 · 𝐶 ) ) |
75 |
|
eqcom |
⊢ ( 𝑤 = ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) = 𝑤 ) |
76 |
73 74 75
|
3bitr4g |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) ∧ 𝑤 ∈ 𝑊 ) → ( ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ 𝑤 = ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
77 |
76
|
rexbidva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → ( ∃ 𝑤 ∈ 𝑊 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ∃ 𝑤 ∈ 𝑊 𝑤 = ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) ) |
78 |
|
risset |
⊢ ( ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ↔ ∃ 𝑤 ∈ 𝑊 𝑤 = ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ) |
79 |
77 78
|
bitr4di |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℤ ) → ( ∃ 𝑤 ∈ 𝑊 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) |
80 |
79
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑘 ∈ ℤ ∃ 𝑤 ∈ 𝑊 ( 𝑃 · 𝐶 ) = ( 𝑤 ( -g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ↔ ∃ 𝑘 ∈ ℤ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) |
81 |
53 80
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑘 ∈ ℤ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) |
82 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝑃 pGrp 𝐺 ) |
83 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝐺 ∈ Abel ) |
84 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝐵 ∈ Fin ) |
85 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ( 𝑂 ‘ 𝐴 ) = 𝐸 ) |
86 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) |
87 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝐴 ∈ 𝑈 ) |
88 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝑊 ∈ ( SubGrp ‘ 𝐺 ) ) |
89 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ( 𝑆 ∩ 𝑊 ) = { 0 } ) |
90 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ( 𝑆 ⊕ 𝑊 ) ⊆ 𝑈 ) |
91 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ∀ 𝑤 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑤 ⊊ 𝑈 ∧ 𝐴 ∈ 𝑤 ) → ¬ ( 𝑆 ⊕ 𝑊 ) ⊊ 𝑤 ) ) |
92 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝐶 ∈ ( 𝑈 ∖ ( 𝑆 ⊕ 𝑊 ) ) ) |
93 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → 𝑘 ∈ ℤ ) |
94 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) |
95 |
|
eqid |
⊢ ( 𝐶 ( +g ‘ 𝐺 ) ( ( 𝑘 / 𝑃 ) · 𝐴 ) ) = ( 𝐶 ( +g ‘ 𝐺 ) ( ( 𝑘 / 𝑃 ) · 𝐴 ) ) |
96 |
1 2 3 4 5 6 7 82 83 84 85 86 87 88 89 90 91 92 19 93 94 95
|
pgpfac1lem3 |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℤ ∧ ( ( 𝑃 · 𝐶 ) ( +g ‘ 𝐺 ) ( 𝑘 · 𝐴 ) ) ∈ 𝑊 ) ) → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑈 ) ) |
97 |
81 96
|
rexlimddv |
⊢ ( 𝜑 → ∃ 𝑡 ∈ ( SubGrp ‘ 𝐺 ) ( ( 𝑆 ∩ 𝑡 ) = { 0 } ∧ ( 𝑆 ⊕ 𝑡 ) = 𝑈 ) ) |