![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > slwpss | Structured version Visualization version GIF version |
Description: A proper superset of a Sylow subgroup is not a 𝑃-group. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
slwispgp.1 | ⊢ 𝑆 = (𝐺 ↾s 𝐾) |
Ref | Expression |
---|---|
slwpss | ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1138 | . . 3 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ⊊ 𝐾) | |
2 | 1 | pssned 4118 | . 2 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ≠ 𝐾) |
3 | 1 | pssssd 4117 | . . . . 5 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → 𝐻 ⊆ 𝐾) |
4 | 3 | biantrurd 532 | . . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (𝑃 pGrp 𝑆 ↔ (𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆))) |
5 | slwispgp.1 | . . . . . 6 ⊢ 𝑆 = (𝐺 ↾s 𝐾) | |
6 | 5 | slwispgp 19648 | . . . . 5 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺)) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)) |
7 | 6 | 3adant3 1132 | . . . 4 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ((𝐻 ⊆ 𝐾 ∧ 𝑃 pGrp 𝑆) ↔ 𝐻 = 𝐾)) |
8 | 4, 7 | bitrd 279 | . . 3 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (𝑃 pGrp 𝑆 ↔ 𝐻 = 𝐾)) |
9 | 8 | necon3bbid 2980 | . 2 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → (¬ 𝑃 pGrp 𝑆 ↔ 𝐻 ≠ 𝐾)) |
10 | 2, 9 | mpbird 257 | 1 ⊢ ((𝐻 ∈ (𝑃 pSyl 𝐺) ∧ 𝐾 ∈ (SubGrp‘𝐺) ∧ 𝐻 ⊊ 𝐾) → ¬ 𝑃 pGrp 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 ⊆ wss 3970 ⊊ wpss 3971 class class class wbr 5169 ‘cfv 6572 (class class class)co 7445 ↾s cress 17282 SubGrpcsubg 19155 pGrp cpgp 19563 pSyl cslw 19564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-subg 19158 df-slw 19568 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |