Step |
Hyp |
Ref |
Expression |
1 |
|
sylow3.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
sylow3.g |
⊢ ( 𝜑 → 𝐺 ∈ Grp ) |
3 |
|
sylow3.xf |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
4 |
|
sylow3.p |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
5 |
|
sylow3.n |
⊢ 𝑁 = ( ♯ ‘ ( 𝑃 pSyl 𝐺 ) ) |
6 |
1
|
slwn0 |
⊢ ( ( 𝐺 ∈ Grp ∧ 𝑋 ∈ Fin ∧ 𝑃 ∈ ℙ ) → ( 𝑃 pSyl 𝐺 ) ≠ ∅ ) |
7 |
2 3 4 6
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 pSyl 𝐺 ) ≠ ∅ ) |
8 |
|
n0 |
⊢ ( ( 𝑃 pSyl 𝐺 ) ≠ ∅ ↔ ∃ 𝑘 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) |
9 |
7 8
|
sylib |
⊢ ( 𝜑 → ∃ 𝑘 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) |
10 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → 𝐺 ∈ Grp ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → 𝑋 ∈ Fin ) |
12 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → 𝑃 ∈ ℙ ) |
13 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
14 |
|
eqid |
⊢ ( -g ‘ 𝐺 ) = ( -g ‘ 𝐺 ) |
15 |
|
oveq2 |
⊢ ( 𝑐 = 𝑧 → ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) = ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ) |
16 |
15
|
oveq1d |
⊢ ( 𝑐 = 𝑧 → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) = ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑎 ) ) |
17 |
16
|
cbvmptv |
⊢ ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) = ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑎 ) ) |
18 |
|
oveq1 |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ) |
19 |
|
id |
⊢ ( 𝑎 = 𝑥 → 𝑎 = 𝑥 ) |
20 |
18 19
|
oveq12d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑎 ) = ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) |
21 |
20
|
mpteq2dv |
⊢ ( 𝑎 = 𝑥 → ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑎 ) ) = ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
22 |
17 21
|
syl5eq |
⊢ ( 𝑎 = 𝑥 → ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) = ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
23 |
22
|
rneqd |
⊢ ( 𝑎 = 𝑥 → ran ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) = ran ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
24 |
|
mpteq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) = ( 𝑧 ∈ 𝑦 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
25 |
24
|
rneqd |
⊢ ( 𝑏 = 𝑦 → ran ( 𝑧 ∈ 𝑏 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) = ran ( 𝑧 ∈ 𝑦 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
26 |
23 25
|
cbvmpov |
⊢ ( 𝑎 ∈ 𝑋 , 𝑏 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) ) = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑧 ∈ 𝑦 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
27 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) |
28 |
|
eqid |
⊢ { 𝑢 ∈ 𝑋 ∣ ( 𝑢 ( 𝑎 ∈ 𝑋 , 𝑏 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) ) 𝑘 ) = 𝑘 } = { 𝑢 ∈ 𝑋 ∣ ( 𝑢 ( 𝑎 ∈ 𝑋 , 𝑏 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) ) 𝑘 ) = 𝑘 } |
29 |
|
eqid |
⊢ { 𝑥 ∈ 𝑋 ∣ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑘 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝑘 ) } = { 𝑥 ∈ 𝑋 ∣ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑘 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝑘 ) } |
30 |
1 10 11 12 13 14 26 27 28 29
|
sylow3lem4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → ( ♯ ‘ ( 𝑃 pSyl 𝐺 ) ) ∥ ( ( ♯ ‘ 𝑋 ) / ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ) |
31 |
5 30
|
eqbrtrid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → 𝑁 ∥ ( ( ♯ ‘ 𝑋 ) / ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ) |
32 |
5
|
oveq1i |
⊢ ( 𝑁 mod 𝑃 ) = ( ( ♯ ‘ ( 𝑃 pSyl 𝐺 ) ) mod 𝑃 ) |
33 |
23 25
|
cbvmpov |
⊢ ( 𝑎 ∈ 𝑘 , 𝑏 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑐 ∈ 𝑏 ↦ ( ( 𝑎 ( +g ‘ 𝐺 ) 𝑐 ) ( -g ‘ 𝐺 ) 𝑎 ) ) ) = ( 𝑥 ∈ 𝑘 , 𝑦 ∈ ( 𝑃 pSyl 𝐺 ) ↦ ran ( 𝑧 ∈ 𝑦 ↦ ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑧 ) ( -g ‘ 𝐺 ) 𝑥 ) ) ) |
34 |
|
eqid |
⊢ { 𝑥 ∈ 𝑋 ∣ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑠 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝑠 ) } = { 𝑥 ∈ 𝑋 ∣ ∀ 𝑦 ∈ 𝑋 ( ( 𝑥 ( +g ‘ 𝐺 ) 𝑦 ) ∈ 𝑠 ↔ ( 𝑦 ( +g ‘ 𝐺 ) 𝑥 ) ∈ 𝑠 ) } |
35 |
1 10 11 12 13 14 27 33 34
|
sylow3lem6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → ( ( ♯ ‘ ( 𝑃 pSyl 𝐺 ) ) mod 𝑃 ) = 1 ) |
36 |
32 35
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → ( 𝑁 mod 𝑃 ) = 1 ) |
37 |
31 36
|
jca |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑃 pSyl 𝐺 ) ) → ( 𝑁 ∥ ( ( ♯ ‘ 𝑋 ) / ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ∧ ( 𝑁 mod 𝑃 ) = 1 ) ) |
38 |
9 37
|
exlimddv |
⊢ ( 𝜑 → ( 𝑁 ∥ ( ( ♯ ‘ 𝑋 ) / ( 𝑃 ↑ ( 𝑃 pCnt ( ♯ ‘ 𝑋 ) ) ) ) ∧ ( 𝑁 mod 𝑃 ) = 1 ) ) |