Metamath Proof Explorer


Theorem sylow3lem6

Description: Lemma for sylow3 , second part. Using the lemma sylow2a , show that the number of sylow subgroups is equivalent mod P to the number of fixed points under the group action. But K is the unique element of the set of Sylow subgroups that is fixed under the group action, so there is exactly one fixed point and so ( ( #( P pSyl G ) ) mod P ) = 1 . (Contributed by Mario Carneiro, 19-Jan-2015)

Ref Expression
Hypotheses sylow3.x X = Base G
sylow3.g φ G Grp
sylow3.xf φ X Fin
sylow3.p φ P
sylow3lem5.a + ˙ = + G
sylow3lem5.d - ˙ = - G
sylow3lem5.k φ K P pSyl G
sylow3lem5.m ˙ = x K , y P pSyl G ran z y x + ˙ z - ˙ x
sylow3lem6.n N = x X | y X x + ˙ y s y + ˙ x s
Assertion sylow3lem6 φ P pSyl G mod P = 1

Proof

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
5 sylow3lem5.a + ˙ = + G
6 sylow3lem5.d - ˙ = - G
7 sylow3lem5.k φ K P pSyl G
8 sylow3lem5.m ˙ = x K , y P pSyl G ran z y x + ˙ z - ˙ x
9 sylow3lem6.n N = x X | y X x + ˙ y s y + ˙ x s
10 eqid Base G 𝑠 K = Base G 𝑠 K
11 1 2 3 4 5 6 7 8 sylow3lem5 φ ˙ G 𝑠 K GrpAct P pSyl G
12 eqid G 𝑠 K = G 𝑠 K
13 12 slwpgp K P pSyl G P pGrp G 𝑠 K
14 7 13 syl φ P pGrp G 𝑠 K
15 slwsubg K P pSyl G K SubGrp G
16 7 15 syl φ K SubGrp G
17 12 subgbas K SubGrp G K = Base G 𝑠 K
18 16 17 syl φ K = Base G 𝑠 K
19 1 subgss K SubGrp G K X
20 16 19 syl φ K X
21 3 20 ssfid φ K Fin
22 18 21 eqeltrrd φ Base G 𝑠 K Fin
23 pwfi X Fin 𝒫 X Fin
24 3 23 sylib φ 𝒫 X Fin
25 slwsubg x P pSyl G x SubGrp G
26 1 subgss x SubGrp G x X
27 25 26 syl x P pSyl G x X
28 25 27 elpwd x P pSyl G x 𝒫 X
29 28 ssriv P pSyl G 𝒫 X
30 ssfi 𝒫 X Fin P pSyl G 𝒫 X P pSyl G Fin
31 24 29 30 sylancl φ P pSyl G Fin
32 eqid s P pSyl G | g Base G 𝑠 K g ˙ s = s = s P pSyl G | g Base G 𝑠 K g ˙ s = s
33 eqid z w | z w P pSyl G h Base G 𝑠 K h ˙ z = w = z w | z w P pSyl G h Base G 𝑠 K h ˙ z = w
34 10 11 14 22 31 32 33 sylow2a φ P P pSyl G s P pSyl G | g Base G 𝑠 K g ˙ s = s
35 eqcom ran z s g + ˙ z - ˙ g = s s = ran z s g + ˙ z - ˙ g
36 20 adantr φ s P pSyl G K X
37 36 sselda φ s P pSyl G g K g X
38 37 biantrurd φ s P pSyl G g K s = ran z s g + ˙ z - ˙ g g X s = ran z s g + ˙ z - ˙ g
39 35 38 syl5bb φ s P pSyl G g K ran z s g + ˙ z - ˙ g = s g X s = ran z s g + ˙ z - ˙ g
40 simpr φ s P pSyl G g K g K
41 simplr φ s P pSyl G g K s P pSyl G
42 simpr x = g y = s y = s
43 simpl x = g y = s x = g
44 43 oveq1d x = g y = s x + ˙ z = g + ˙ z
45 44 43 oveq12d x = g y = s x + ˙ z - ˙ x = g + ˙ z - ˙ g
46 42 45 mpteq12dv x = g y = s z y x + ˙ z - ˙ x = z s g + ˙ z - ˙ g
47 46 rneqd x = g y = s ran z y x + ˙ z - ˙ x = ran z s g + ˙ z - ˙ g
48 vex s V
49 48 mptex z s g + ˙ z - ˙ g V
50 49 rnex ran z s g + ˙ z - ˙ g V
51 47 8 50 ovmpoa g K s P pSyl G g ˙ s = ran z s g + ˙ z - ˙ g
52 40 41 51 syl2anc φ s P pSyl G g K g ˙ s = ran z s g + ˙ z - ˙ g
53 52 eqeq1d φ s P pSyl G g K g ˙ s = s ran z s g + ˙ z - ˙ g = s
54 slwsubg s P pSyl G s SubGrp G
55 54 ad2antlr φ s P pSyl G g K s SubGrp G
56 eqid z s g + ˙ z - ˙ g = z s g + ˙ z - ˙ g
57 1 5 6 56 9 conjnmzb s SubGrp G g N g X s = ran z s g + ˙ z - ˙ g
58 55 57 syl φ s P pSyl G g K g N g X s = ran z s g + ˙ z - ˙ g
59 39 53 58 3bitr4d φ s P pSyl G g K g ˙ s = s g N
60 59 ralbidva φ s P pSyl G g K g ˙ s = s g K g N
61 dfss3 K N g K g N
62 60 61 bitr4di φ s P pSyl G g K g ˙ s = s K N
63 18 adantr φ s P pSyl G K = Base G 𝑠 K
64 63 raleqdv φ s P pSyl G g K g ˙ s = s g Base G 𝑠 K g ˙ s = s
65 eqid Base G 𝑠 N = Base G 𝑠 N
66 2 ad2antrr φ s P pSyl G K N G Grp
67 9 1 5 nmzsubg G Grp N SubGrp G
68 66 67 syl φ s P pSyl G K N N SubGrp G
69 eqid G 𝑠 N = G 𝑠 N
70 69 subgbas N SubGrp G N = Base G 𝑠 N
71 68 70 syl φ s P pSyl G K N N = Base G 𝑠 N
72 3 ad2antrr φ s P pSyl G K N X Fin
73 1 subgss N SubGrp G N X
74 68 73 syl φ s P pSyl G K N N X
75 72 74 ssfid φ s P pSyl G K N N Fin
76 71 75 eqeltrrd φ s P pSyl G K N Base G 𝑠 N Fin
77 7 ad2antrr φ s P pSyl G K N K P pSyl G
78 simpr φ s P pSyl G K N K N
79 69 subgslw N SubGrp G K P pSyl G K N K P pSyl G 𝑠 N
80 68 77 78 79 syl3anc φ s P pSyl G K N K P pSyl G 𝑠 N
81 simplr φ s P pSyl G K N s P pSyl G
82 54 ad2antlr φ s P pSyl G K N s SubGrp G
83 9 1 5 ssnmz s SubGrp G s N
84 82 83 syl φ s P pSyl G K N s N
85 69 subgslw N SubGrp G s P pSyl G s N s P pSyl G 𝑠 N
86 68 81 84 85 syl3anc φ s P pSyl G K N s P pSyl G 𝑠 N
87 1 fvexi X V
88 9 87 rabex2 N V
89 69 5 ressplusg N V + ˙ = + G 𝑠 N
90 88 89 ax-mp + ˙ = + G 𝑠 N
91 eqid - G 𝑠 N = - G 𝑠 N
92 65 76 80 86 90 91 sylow2 φ s P pSyl G K N g Base G 𝑠 N K = ran z s g + ˙ z - G 𝑠 N g
93 9 1 5 69 nmznsg s SubGrp G s NrmSGrp G 𝑠 N
94 82 93 syl φ s P pSyl G K N s NrmSGrp G 𝑠 N
95 eqid z s g + ˙ z - G 𝑠 N g = z s g + ˙ z - G 𝑠 N g
96 65 90 91 95 conjnsg s NrmSGrp G 𝑠 N g Base G 𝑠 N s = ran z s g + ˙ z - G 𝑠 N g
97 94 96 sylan φ s P pSyl G K N g Base G 𝑠 N s = ran z s g + ˙ z - G 𝑠 N g
98 eqeq2 K = ran z s g + ˙ z - G 𝑠 N g s = K s = ran z s g + ˙ z - G 𝑠 N g
99 97 98 syl5ibrcom φ s P pSyl G K N g Base G 𝑠 N K = ran z s g + ˙ z - G 𝑠 N g s = K
100 99 rexlimdva φ s P pSyl G K N g Base G 𝑠 N K = ran z s g + ˙ z - G 𝑠 N g s = K
101 92 100 mpd φ s P pSyl G K N s = K
102 simpr φ s P pSyl G s = K s = K
103 16 ad2antrr φ s P pSyl G s = K K SubGrp G
104 102 103 eqeltrd φ s P pSyl G s = K s SubGrp G
105 104 83 syl φ s P pSyl G s = K s N
106 102 105 eqsstrrd φ s P pSyl G s = K K N
107 101 106 impbida φ s P pSyl G K N s = K
108 62 64 107 3bitr3d φ s P pSyl G g Base G 𝑠 K g ˙ s = s s = K
109 108 rabbidva φ s P pSyl G | g Base G 𝑠 K g ˙ s = s = s P pSyl G | s = K
110 rabsn K P pSyl G s P pSyl G | s = K = K
111 7 110 syl φ s P pSyl G | s = K = K
112 109 111 eqtrd φ s P pSyl G | g Base G 𝑠 K g ˙ s = s = K
113 112 fveq2d φ s P pSyl G | g Base G 𝑠 K g ˙ s = s = K
114 hashsng K P pSyl G K = 1
115 7 114 syl φ K = 1
116 113 115 eqtrd φ s P pSyl G | g Base G 𝑠 K g ˙ s = s = 1
117 116 oveq2d φ P pSyl G s P pSyl G | g Base G 𝑠 K g ˙ s = s = P pSyl G 1
118 34 117 breqtrd φ P P pSyl G 1
119 prmnn P P
120 4 119 syl φ P
121 hashcl P pSyl G Fin P pSyl G 0
122 31 121 syl φ P pSyl G 0
123 122 nn0zd φ P pSyl G
124 1zzd φ 1
125 moddvds P P pSyl G 1 P pSyl G mod P = 1 mod P P P pSyl G 1
126 120 123 124 125 syl3anc φ P pSyl G mod P = 1 mod P P P pSyl G 1
127 118 126 mpbird φ P pSyl G mod P = 1 mod P
128 prmuz2 P P 2
129 eluz2b2 P 2 P 1 < P
130 nnre P P
131 1mod P 1 < P 1 mod P = 1
132 130 131 sylan P 1 < P 1 mod P = 1
133 129 132 sylbi P 2 1 mod P = 1
134 4 128 133 3syl φ 1 mod P = 1
135 127 134 eqtrd φ P pSyl G mod P = 1