| Step | Hyp | Ref
| Expression |
| 1 | | sylow2b.h |
. . . 4
⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺)) |
| 2 | | eqid 2737 |
. . . . 5
⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻) |
| 3 | 2 | subggrp 19147 |
. . . 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 9357 |
. . . . 5
⊢ (𝑋 ∈ Fin ↔ 𝒫
𝑋 ∈
Fin) |
| 7 | 5, 6 | sylib 218 |
. . . 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 19196 |
. . . . . 6
⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋) |
| 12 | 8, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → ∼ Er 𝑋) |
| 13 | 12 | qsss 8818 |
. . . 4
⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫
𝑋) |
| 14 | 7, 13 | ssexd 5324 |
. . 3
⊢ (𝜑 → (𝑋 / ∼ ) ∈
V) |
| 15 | 4, 14 | jca 511 |
. 2
⊢ (𝜑 → ((𝐺 ↾s 𝐻) ∈ Grp ∧ (𝑋 / ∼ ) ∈
V)) |
| 16 | | sylow2b.m |
. . . . . . 7
⊢ · =
(𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran
(𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧))) |
| 17 | | vex 3484 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
| 18 | 17 | mptex 7243 |
. . . . . . . 8
⊢ (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V |
| 19 | 18 | rnex 7932 |
. . . . . . 7
⊢ ran
(𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)) ∈ V |
| 20 | 16, 19 | fnmpoi 8095 |
. . . . . 6
⊢ · Fn
(𝐻 × (𝑋 / ∼ )) |
| 21 | 20 | a1i 11 |
. . . . 5
⊢ (𝜑 → · Fn (𝐻 × (𝑋 / ∼
))) |
| 22 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑋 / ∼ ) = (𝑋 / ∼ ) |
| 23 | | oveq2 7439 |
. . . . . . . . 9
⊢ ([𝑠] ∼ = 𝑣 → (𝑢 · [𝑠] ∼ ) = (𝑢 · 𝑣)) |
| 24 | 23 | eleq1d 2826 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑣 → ((𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ ) ↔ (𝑢 · 𝑣) ∈ (𝑋 / ∼
))) |
| 25 | | sylow2b.a |
. . . . . . . . . . 11
⊢ + =
(+g‘𝐺) |
| 26 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 19638 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) = [(𝑢 + 𝑠)] ∼ ) |
| 27 | 10 | ovexi 7465 |
. . . . . . . . . . 11
⊢ ∼ ∈
V |
| 28 | | subgrcl 19149 |
. . . . . . . . . . . . . 14
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
| 29 | 1, 28 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 ∈ Grp) |
| 30 | 29 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝐺 ∈ Grp) |
| 31 | 9 | subgss 19145 |
. . . . . . . . . . . . . . 15
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋) |
| 32 | 1, 31 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐻 ⊆ 𝑋) |
| 33 | 32 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → 𝑢 ∈ 𝑋) |
| 34 | 33 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑢 ∈ 𝑋) |
| 35 | | simp3 1139 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) |
| 36 | 9, 25 | grpcl 18959 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋) |
| 37 | 30, 34, 35, 36 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 + 𝑠) ∈ 𝑋) |
| 38 | | ecelqsg 8812 |
. . . . . . . . . . 11
⊢ (( ∼ ∈
V ∧ (𝑢 + 𝑠) ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ )) |
| 39 | 27, 37, 38 | sylancr 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → [(𝑢 + 𝑠)] ∼ ∈ (𝑋 / ∼ )) |
| 40 | 26, 39 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ )) |
| 41 | 40 | 3expa 1119 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑠 ∈ 𝑋) → (𝑢 · [𝑠] ∼ ) ∈ (𝑋 / ∼ )) |
| 42 | 22, 24, 41 | ectocld 8824 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑢 ∈ 𝐻) ∧ 𝑣 ∈ (𝑋 / ∼ )) → (𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
| 43 | 42 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻) → ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
| 44 | 43 | ralrimiva 3146 |
. . . . 5
⊢ (𝜑 → ∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼ )) |
| 45 | | ffnov 7559 |
. . . . 5
⊢ ( ·
:(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ ( · Fn
(𝐻 × (𝑋 / ∼ )) ∧
∀𝑢 ∈ 𝐻 ∀𝑣 ∈ (𝑋 / ∼ )(𝑢 · 𝑣) ∈ (𝑋 / ∼
))) |
| 46 | 21, 44, 45 | sylanbrc 583 |
. . . 4
⊢ (𝜑 → · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ )) |
| 47 | 2 | subgbas 19148 |
. . . . . . 7
⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
| 48 | 1, 47 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
| 49 | 48 | xpeq1d 5714 |
. . . . 5
⊢ (𝜑 → (𝐻 × (𝑋 / ∼ )) =
((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))) |
| 50 | 49 | feq2d 6722 |
. . . 4
⊢ (𝜑 → ( · :(𝐻 × (𝑋 / ∼ ))⟶(𝑋 / ∼ ) ↔ ·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼
))) |
| 51 | 46, 50 | mpbid 232 |
. . 3
⊢ (𝜑 → ·
:((Base‘(𝐺
↾s 𝐻))
× (𝑋 / ∼
))⟶(𝑋 / ∼
)) |
| 52 | | oveq2 7439 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 →
((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ ) =
((0g‘(𝐺
↾s 𝐻))
·
𝑢)) |
| 53 | | id 22 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 → [𝑠] ∼ = 𝑢) |
| 54 | 52, 53 | eqeq12d 2753 |
. . . . . 6
⊢ ([𝑠] ∼ = 𝑢 →
(((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ ) = [𝑠] ∼ ↔
((0g‘(𝐺
↾s 𝐻))
·
𝑢) = 𝑢)) |
| 55 | | oveq2 7439 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑢 → ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢)) |
| 56 | | oveq2 7439 |
. . . . . . . . 9
⊢ ([𝑠] ∼ = 𝑢 → (𝑏 · [𝑠] ∼ ) = (𝑏 · 𝑢)) |
| 57 | 56 | oveq2d 7447 |
. . . . . . . 8
⊢ ([𝑠] ∼ = 𝑢 → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · (𝑏 · 𝑢))) |
| 58 | 55, 57 | eqeq12d 2753 |
. . . . . . 7
⊢ ([𝑠] ∼ = 𝑢 → (((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · 𝑢) = (𝑎 · (𝑏 · 𝑢)))) |
| 59 | 58 | 2ralbidv 3221 |
. . . . . 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)))) |
| 60 | 54, 59 | anbi12d 632 |
. . . . 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))))) |
| 61 | | simpl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝜑) |
| 62 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 ∈ (SubGrp‘𝐺)) |
| 63 | | eqid 2737 |
. . . . . . . . . 10
⊢
(0g‘𝐺) = (0g‘𝐺) |
| 64 | 63 | subg0cl 19152 |
. . . . . . . . 9
⊢ (𝐻 ∈ (SubGrp‘𝐺) →
(0g‘𝐺)
∈ 𝐻) |
| 65 | 62, 64 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) ∈ 𝐻) |
| 66 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋) |
| 67 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 19638 |
. . . . . . . 8
⊢ ((𝜑 ∧ (0g‘𝐺) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
[((0g‘𝐺)
+ 𝑠)] ∼ ) |
| 68 | 61, 65, 66, 67 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
[((0g‘𝐺)
+ 𝑠)] ∼ ) |
| 69 | 2, 63 | subg0 19150 |
. . . . . . . . 9
⊢ (𝐻 ∈ (SubGrp‘𝐺) →
(0g‘𝐺) =
(0g‘(𝐺
↾s 𝐻))) |
| 70 | 62, 69 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (0g‘𝐺) = (0g‘(𝐺 ↾s 𝐻))) |
| 71 | 70 | oveq1d 7446 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) · [𝑠] ∼ ) =
((0g‘(𝐺
↾s 𝐻))
·
[𝑠] ∼ )) |
| 72 | 9, 25, 63 | grplid 18985 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Grp ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠) |
| 73 | 29, 72 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘𝐺) + 𝑠) = 𝑠) |
| 74 | 73 | eceq1d 8785 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → [((0g‘𝐺) + 𝑠)] ∼ = [𝑠] ∼ ) |
| 75 | 68, 71, 74 | 3eqtr3d 2785 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((0g‘(𝐺 ↾s 𝐻)) · [𝑠] ∼ ) = [𝑠] ∼ ) |
| 76 | 62 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ∈ (SubGrp‘𝐺)) |
| 77 | 76, 28 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐺 ∈ Grp) |
| 78 | 76, 31 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝐻 ⊆ 𝑋) |
| 79 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝐻) |
| 80 | 78, 79 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑎 ∈ 𝑋) |
| 81 | | simprr 773 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝐻) |
| 82 | 78, 81 | sseldd 3984 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑏 ∈ 𝑋) |
| 83 | 66 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝑠 ∈ 𝑋) |
| 84 | 9, 25 | grpass 18960 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠))) |
| 85 | 77, 80, 82, 83, 84 | syl13anc 1374 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) + 𝑠) = (𝑎 + (𝑏 + 𝑠))) |
| 86 | 85 | eceq1d 8785 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
| 87 | 61 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → 𝜑) |
| 88 | 9, 25 | grpcl 18959 |
. . . . . . . . . . . 12
⊢ ((𝐺 ∈ Grp ∧ 𝑏 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑏 + 𝑠) ∈ 𝑋) |
| 89 | 77, 82, 83, 88 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 + 𝑠) ∈ 𝑋) |
| 90 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 19638 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐻 ∧ (𝑏 + 𝑠) ∈ 𝑋) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
| 91 | 87, 79, 89, 90 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · [(𝑏 + 𝑠)] ∼ ) = [(𝑎 + (𝑏 + 𝑠))] ∼ ) |
| 92 | 86, 91 | eqtr4d 2780 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → [((𝑎 + 𝑏) + 𝑠)] ∼ = (𝑎 · [(𝑏 + 𝑠)] ∼ )) |
| 93 | 25 | subgcl 19154 |
. . . . . . . . . . 11
⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻) → (𝑎 + 𝑏) ∈ 𝐻) |
| 94 | 76, 79, 81, 93 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 + 𝑏) ∈ 𝐻) |
| 95 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 19638 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 + 𝑏) ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ ) |
| 96 | 87, 94, 83, 95 | syl3anc 1373 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = [((𝑎 + 𝑏) + 𝑠)] ∼ ) |
| 97 | 9, 5, 1, 8, 25, 10, 16 | sylow2blem1 19638 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐻 ∧ 𝑠 ∈ 𝑋) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ ) |
| 98 | 87, 81, 83, 97 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑏 · [𝑠] ∼ ) = [(𝑏 + 𝑠)] ∼ ) |
| 99 | 98 | oveq2d 7447 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → (𝑎 · (𝑏 · [𝑠] ∼ )) = (𝑎 · [(𝑏 + 𝑠)] ∼ )) |
| 100 | 92, 96, 99 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ 𝑋) ∧ (𝑎 ∈ 𝐻 ∧ 𝑏 ∈ 𝐻)) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
| 101 | 100 | ralrimivva 3202 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ 𝐻 ∀𝑏 ∈ 𝐻 ((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
| 102 | 62, 47 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝐻 = (Base‘(𝐺 ↾s 𝐻))) |
| 103 | 2, 25 | ressplusg 17334 |
. . . . . . . . . . . . 13
⊢ (𝐻 ∈ (SubGrp‘𝐺) → + =
(+g‘(𝐺
↾s 𝐻))) |
| 104 | 1, 103 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → + =
(+g‘(𝐺
↾s 𝐻))) |
| 105 | 104 | oveqdr 7459 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑎(+g‘(𝐺 ↾s 𝐻))𝑏)) |
| 106 | 105 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑎 + 𝑏) · [𝑠] ∼ ) = ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ )) |
| 107 | 106 | eqeq1d 2739 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑎 + 𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼ )) ↔ ((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
)))) |
| 108 | 102, 107 | raleqbidv 3346 |
. . . . . . . 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] ∼
)))) |
| 109 | 102, 108 | raleqbidv 3346 |
. . . . . . 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] ∼
)))) |
| 110 | 101, 109 | mpbid 232 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ∀𝑎 ∈ (Base‘(𝐺 ↾s 𝐻))∀𝑏 ∈ (Base‘(𝐺 ↾s 𝐻))((𝑎(+g‘(𝐺 ↾s 𝐻))𝑏) · [𝑠] ∼ ) = (𝑎 · (𝑏 · [𝑠] ∼
))) |
| 111 | 75, 110 | jca 511 |
. . . . 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] ∼
)))) |
| 112 | 22, 60, 111 | ectocld 8824 |
. . . 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)))) |
| 113 | 112 | ralrimiva 3146 |
. . 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)))) |
| 114 | 51, 113 | jca 511 |
. 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))))) |
| 115 | | eqid 2737 |
. . 3
⊢
(Base‘(𝐺
↾s 𝐻)) =
(Base‘(𝐺
↾s 𝐻)) |
| 116 | | eqid 2737 |
. . 3
⊢
(+g‘(𝐺 ↾s 𝐻)) = (+g‘(𝐺 ↾s 𝐻)) |
| 117 | | eqid 2737 |
. . 3
⊢
(0g‘(𝐺 ↾s 𝐻)) = (0g‘(𝐺 ↾s 𝐻)) |
| 118 | 115, 116,
117 | isga 19309 |
. 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)))))) |
| 119 | 15, 114, 118 | sylanbrc 583 |
1
⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼
))) |