Step | Hyp | Ref
| Expression |
1 | | sylow2b.h |
. . . 4
⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
2 | | eqid 2622 |
. . . . 5
⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) |
3 | 2 | subggrp 17597 |
. . . 4
⊢ (𝐻 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝐻) ∈ Grp) |
4 | 1, 3 | syl 17 |
. . 3
⊢ (𝜑 → (𝐺 ↾s 𝐻) ∈ Grp) |
5 | | sylow2b.xf |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ Fin) |
6 | | pwfi 8261 |
. . . . 5
⊢ (𝑋 ∈ Fin ↔ 𝒫
𝑋 ∈
Fin) |
7 | 5, 6 | sylib 208 |
. . . 4
⊢ (𝜑 → 𝒫 𝑋 ∈ Fin) |
8 | | sylow2b.k |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺)) |
9 | | sylow2b.x |
. . . . . . 7
⊢ 𝑋 = (Base‘𝐺) |
10 | | sylow2b.r |
. . . . . . 7
⊢ ∼ =
(𝐺 ~QG
𝐾) |
11 | 9, 10 | eqger 17644 |
. . . . . 6
⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) |
12 | 8, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → ∼ Er 𝑋) |
13 | 12 | qsss 7808 |
. . . 4
⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫
𝑋) |
14 | 7, 13 | ssexd 4805 |
. . 3
⊢ (𝜑 → (𝑋 / ∼ ) ∈
V) |
15 | 4, 14 | jca 554 |
. 2
⊢ (𝜑 → ((𝐺 ↾s 𝐻) ∈ Grp ∧ (𝑋 / ∼ ) ∈
V)) |
16 | | sylow2b.m |
. . . . . . 7
⊢ · =
(𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran
(𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
17 | | vex 3203 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
18 | 17 | mptex 6486 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V |
19 | 18 | rnex 7100 |
. . . . . . 7
⊢ ran
(𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V |
20 | 16, 19 | fnmpt2i 7239 |
. . . . . 6
⊢ · Fn
(𝐻 × (𝑋 / ∼ )) |
21 | 20 | a1i 11 |
. . . . 5
⊢ (𝜑 → · Fn (𝐻 × (𝑋 / ∼
))) |
22 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑋 / ∼ ) = (𝑋 / ∼ ) |
23 | | oveq2 6658 |
. . . . . . . . 9
⊢ ([𝑠] ∼ = 𝑣 → (𝑢 · [𝑠] ∼ ) = (𝑢 · 𝑣)) |
24 | 23 | eleq1d 2686 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑣 → ((𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ ) ↔ (𝑢 · 𝑣) ∈ (𝑋 / ∼
))) |
25 | | sylow2b.a |
. . . . . . . . . . 11
⊢ + =
(+g‘𝐺) |
26 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 18035 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) = [(𝑢 + 𝑠)] ∼ ) |
27 | | ovex 6678 |
. . . . . . . . . . . 12
⊢ (𝐺 ~QG 𝐾) ∈ V |
28 | 10, 27 | eqeltri 2697 |
. . . . . . . . . . 11
⊢ ∼ ∈
V |
29 | | subgrcl 17599 |
. . . . . . . . . . . . . 14
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
30 | 1, 29 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 ∈ Grp) |
31 | 30 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝐺 ∈ Grp) |
32 | 9 | subgss 17595 |
. . . . . . . . . . . . . . 15
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
33 | 1, 32 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐻 ⊆ 𝑋) |
34 | 33 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → 𝑢 ∈ 𝑋) |
35 | 34 | 3adant3 1081 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑢 ∈ 𝑋) |
36 | | simp3 1063 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) |
37 | 9, 25 | grpcl 17430 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋) |
38 | 31, 35, 36, 37 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋) |
39 | | ecelqsg 7802 |
. . . . . . . . . . 11
⊢ (( ∼ ∈
V ∧ (𝑢 + 𝑠) ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ )) |
40 | 28, 38, 39 | sylancr 695 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ )) |
41 | 26, 40 | eqeltrd 2701 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ )) |
42 | 41 | 3expa 1265 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ )) |
43 | 22, 24, 42 | ectocld 7814 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑣 ∈ (𝑋 / ∼ )) → (𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
44 | 43 | ralrimiva 2966 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
45 | 44 | ralrimiva 2966 |
. . . . 5
⊢ (𝜑 → ∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
46 | | ffnov 6764 |
. . . . 5
⊢ ( ·
:(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ ( · Fn
(𝐻 × (𝑋 / ∼ )) ∧
∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼
))) |
47 | 21, 45, 46 | sylanbrc 698 |
. . . 4
⊢ (𝜑 → · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ )) |
48 | 2 | subgbas 17598 |
. . . . . . 7
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
49 | 1, 48 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
50 | 49 | xpeq1d 5138 |
. . . . 5
⊢ (𝜑 → (𝐻 × (𝑋 / ∼ )) =
((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))) |
51 | 50 | feq2d 6031 |
. . . 4
⊢ (𝜑 → ( · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ ·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼
))) |
52 | 47, 51 | mpbid 222 |
. . 3
⊢ (𝜑 → ·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼
)) |
53 | | oveq2 6658 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 →
((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ ) =
((0g‘(𝐺
↾s 𝐻))
·
𝑢)) |
54 | | id 22 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 → [𝑠] ∼ = 𝑢) |
55 | 53, 54 | eqeq12d 2637 |
. . . . . 6
⊢ ([𝑠] ∼ = 𝑢 →
(((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ ) = [𝑠] ∼ ↔
((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢)) |
56 | | oveq2 6658 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑢 → ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢)) |
57 | | oveq2 6658 |
. . . . . . . . 9
⊢ ([𝑠] ∼ = 𝑢 → (𝑏 · [𝑠] ∼ ) = (𝑏 · 𝑢)) |
58 | 57 | oveq2d 6666 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑢 → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · (𝑏 · 𝑢))) |
59 | 56, 58 | eqeq12d 2637 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 → (((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))) |
60 | 59 | 2ralbidv 2989 |
. . . . . 6
⊢ ([𝑠] ∼ = 𝑢 → (∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔
∀𝑎 ∈
(Base‘(𝐺
↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))) |
61 | 55, 60 | anbi12d 747 |
. . . . 5
⊢ ([𝑠] ∼ = 𝑢 →
((((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ ) = [𝑠] ∼ ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ ))) ↔
(((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))) |
62 | | simpl 473 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝜑) |
63 | 1 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 ∈ (SubGrp‘𝐺)) |
64 | | eqid 2622 |
. . . . . . . . . 10
⊢
(0g‘𝐺) = (0g‘𝐺) |
65 | 64 | subg0cl 17602 |
. . . . . . . . 9
⊢ (𝐻 ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ 𝐻) |
66 | 63, 65 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) ∈ 𝐻) |
67 | | simpr 477 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) |
68 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 18035 |
. . . . . . . 8
⊢ ((𝜑 ∧ (0g‘𝐺) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
[((0g‘𝐺)
+ 𝑠)] ∼ ) |
69 | 62, 66, 67, 68 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
[((0g‘𝐺)
+ 𝑠)] ∼ ) |
70 | 2, 64 | subg0 17600 |
. . . . . . . . 9
⊢ (𝐻 ∈ (SubGrp‘𝐺) →
(0g‘𝐺) =
(0g‘(𝐺
↾s 𝐻))) |
71 | 63, 70 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) = (0g‘(𝐺 ↾s 𝐻))) |
72 | 71 | oveq1d 6665 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ )) |
73 | 9, 25, 64 | grplid 17452 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠) |
74 | 30, 73 | sylan 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠) |
75 | 74 | eceq1d 7783 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → [((0g‘𝐺) + 𝑠)] ∼ = [𝑠] ∼ ) |
76 | 69, 72, 75 | 3eqtr3d 2664 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ) |
77 | 63 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ∈ (SubGrp‘𝐺)) |
78 | 77, 29 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐺 ∈ Grp) |
79 | 77, 32 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ⊆ 𝑋) |
80 | | simprl 794 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝐻) |
81 | 79, 80 | sseldd 3604 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝑋) |
82 | | simprr 796 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻) |
83 | 79, 82 | sseldd 3604 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝑋) |
84 | 67 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑠 ∈ 𝑋) |
85 | 9, 25 | grpass 17431 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠))) |
86 | 78, 81, 83, 84, 85 | syl13anc 1328 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠))) |
87 | 86 | eceq1d 7783 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
88 | 62 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝜑) |
89 | 9, 25 | grpcl 17430 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑏 + 𝑠) ∈ 𝑋) |
90 | 78, 83, 84, 89 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 + 𝑠) ∈ 𝑋) |
91 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 18035 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻 ∧ (𝑏 + 𝑠) ∈ 𝑋) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
92 | 88, 80, 90, 91 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
93 | 87, 92 | eqtr4d 2659 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = (𝑎 · [(𝑏 + 𝑠)] ∼ )) |
94 | 25 | subgcl 17604 |
. . . . . . . . . . 11
⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻) → (𝑎 + 𝑏) ∈ 𝐻) |
95 | 77, 80, 82, 94 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 + 𝑏) ∈ 𝐻) |
96 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 18035 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 + 𝑏) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ ) |
97 | 88, 95, 84, 96 | syl3anc 1326 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ ) |
98 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 18035 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ ) |
99 | 88, 82, 84, 98 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ ) |
100 | 99 | oveq2d 6666 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · [(𝑏 + 𝑠)] ∼ )) |
101 | 93, 97, 100 | 3eqtr4d 2666 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
102 | 101 | ralrimivva 2971 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ 𝐻 ∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
103 | 63, 48 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
104 | 2, 25 | ressplusg 15993 |
. . . . . . . . . . . . 13
⊢ (𝐻 ∈ (SubGrp‘𝐺) → + =
(+g‘(𝐺
↾s 𝐻))) |
105 | 1, 104 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → + =
(+g‘(𝐺
↾s 𝐻))) |
106 | 105 | oveqdr 6674 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑎(+g‘(𝐺 ↾s 𝐻))𝑏)) |
107 | 106 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ )) |
108 | 107 | eqeq1d 2624 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
)))) |
109 | 103, 108 | raleqbidv 3152 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔
∀𝑏 ∈
(Base‘(𝐺
↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
)))) |
110 | 103, 109 | raleqbidv 3152 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (∀𝑎 ∈ 𝐻 ∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔
∀𝑎 ∈
(Base‘(𝐺
↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
)))) |
111 | 102, 110 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
112 | 76, 111 | jca 554 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
)))) |
113 | 22, 61, 112 | ectocld 7814 |
. . . 4
⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 / ∼ )) →
(((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))) |
114 | 113 | ralrimiva 2966 |
. . 3
⊢ (𝜑 → ∀𝑢 ∈ (𝑋 / ∼
)(((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))) |
115 | 52, 114 | jca 554 |
. 2
⊢ (𝜑 → ( ·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼ )
∧ ∀𝑢 ∈
(𝑋 / ∼
)(((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢))))) |
116 | | eqid 2622 |
. . 3
⊢
(Base‘(𝐺
↾s 𝐻)) =
(Base‘(𝐺
↾s 𝐻)) |
117 | | eqid 2622 |
. . 3
⊢
(+g‘(𝐺 ↾s 𝐻)) = (+g‘(𝐺 ↾s 𝐻)) |
118 | | eqid 2622 |
. . 3
⊢
(0g‘(𝐺 ↾s 𝐻)) = (0g‘(𝐺 ↾s 𝐻)) |
119 | 116, 117,
118 | isga 17724 |
. 2
⊢ ( · ∈
((𝐺 ↾s
𝐻) GrpAct (𝑋 / ∼ )) ↔ (((𝐺 ↾s 𝐻) ∈ Grp ∧ (𝑋 / ∼ ) ∈ V) ∧ (
·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼ )
∧ ∀𝑢 ∈
(𝑋 / ∼
)(((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢 ∧ ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))))) |
120 | 15, 115, 119 | sylanbrc 698 |
1
⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼
))) |