Theorem sylow2blem3 19563

Description: Sylow's second theorem. Putting together the results of sylow2a 19560 and the orbit-stabilizer theorem to show that 𝑃 does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some 𝑔 ∈ 𝑋 with ℎ𝑔𝐾 = 𝑔𝐾 for all ℎ ∈ 𝐻. This implies that inv_g(𝑔)ℎ𝑔 ∈ 𝐾, so ℎ is in the conjugated subgroup 𝑔𝐾inv_g(𝑔). (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
sylow2b.x	⊢ 𝑋 = (Base‘𝐺)
sylow2b.xf	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2b.h	⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
sylow2b.k	⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
sylow2b.a	⊢ + = (+g‘𝐺)
sylow2b.r	⊢ ∼ = (𝐺 ~QG 𝐾)
sylow2b.m	⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
sylow2blem3.hp	⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))
sylow2blem3.kn	⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
sylow2blem3.d	⊢ − = (-g‘𝐺)

Assertion

Ref	Expression
sylow2blem3	⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

Distinct variable groups:   𝑥,𝑔,𝑦,𝑧,𝐺   𝑔,𝐾,𝑥,𝑦,𝑧   · ,𝑔,𝑥,𝑦,𝑧   + ,𝑔,𝑥,𝑦,𝑧   ∼ ,𝑔,𝑥,𝑦,𝑧   𝜑,𝑔,𝑧   𝑥, − ,𝑧   𝑔,𝐻,𝑥,𝑦,𝑧   𝑔,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑧,𝑔)   − (𝑦,𝑔)

Proof of Theorem sylow2blem3

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow2blem3.hp	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 pGrp (𝐺 ↾s 𝐻))
2		pgpprm 19534	. . . . . . . . 9 ⊢ (𝑃 pGrp (𝐺 ↾s 𝐻) → 𝑃 ∈ ℙ)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
4		sylow2b.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 19073	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		sylow2b.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
8	7	grpbn0 18908	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
9	6, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
10		sylow2b.xf	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
11		hashnncl 14301	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	9, 12	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
14		pcndvds2 16808	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
15	3, 13, 14	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ¬ 𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
16		sylow2b.r	. . . . . . . . . . 11 ⊢ ∼ = (𝐺 ~QG 𝐾)
17		sylow2b.k	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾 ∈ (SubGrp‘𝐺))
18	7, 16, 17, 10	lagsubg2 19135	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐾)))
19	18	oveq1d 7383	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)))
20		sylow2blem3.kn	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
21	20	oveq2d 7384	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
22		pwfi 9231	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	10, 22	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24	7, 16	eqger 19119	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
25	17, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∼ Er 𝑋)
26	25	qsss 8724	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
27	23, 26	ssfid 9181	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin)
28		hashcl 14291	. . . . . . . . . . . 12 ⊢ ((𝑋 / ∼ ) ∈ Fin → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
29	27, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
30	29	nn0cnd 12476	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℂ)
31		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐺) = (0g‘𝐺)
32	31	subg0cl 19076	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐾)
33	17, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐾)
34	33	ne0d 4296	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≠ ∅)
35	7	subgss 19069	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
36	17, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
37	10, 36	ssfid 9181	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Fin)
38		hashnncl 14301	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Fin → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
40	34, 39	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐾) ∈ ℕ)
41	40	nncnd 12173	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
42	40	nnne0d 12207	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ≠ 0)
43	30, 41, 42	divcan4d 11935	. . . . . . . . 9 ⊢ (𝜑 → (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)) = (♯‘(𝑋 / ∼ )))
44	19, 21, 43	3eqtr3d 2780	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) = (♯‘(𝑋 / ∼ )))
45	44	breq2d 5112	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ↔ 𝑃 ∥ (♯‘(𝑋 / ∼ ))))
46	15, 45	mtbid 324	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘(𝑋 / ∼ )))
47		prmz 16614	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
48	3, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℤ)
49	29	nn0zd 12525	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℤ)
50		ssrab2 4034	. . . . . . . . . 10 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ⊆ (𝑋 / ∼ )
51		ssfi 9109	. . . . . . . . . 10 ⊢ (((𝑋 / ∼ ) ∈ Fin ∧ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ⊆ (𝑋 / ∼ )) → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin)
52	27, 50, 51	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin)
53		hashcl 14291	. . . . . . . . 9 ⊢ ({𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℕ0)
54	52, 53	syl 17	. . . . . . . 8 ⊢ (𝜑 → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℕ0)
55	54	nn0zd 12525	. . . . . . 7 ⊢ (𝜑 → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℤ)
56		eqid 2737	. . . . . . . 8 ⊢ (Base‘(𝐺 ↾s 𝐻)) = (Base‘(𝐺 ↾s 𝐻))
57		sylow2b.a	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
58		sylow2b.m	. . . . . . . . 9 ⊢ · = (𝑥 ∈ 𝐻, 𝑦 ∈ (𝑋 / ∼ ) ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
59	7, 10, 4, 17, 57, 16, 58	sylow2blem2 19562	. . . . . . . 8 ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))
60		eqid 2737	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
61	60	subgbas 19072	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
62	4, 61	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
63	7	subgss 19069	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
64	4, 63	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
65	10, 64	ssfid 9181	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Fin)
66	62, 65	eqeltrrd 2838	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝐻)) ∈ Fin)
67		eqid 2737	. . . . . . . 8 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}
68		eqid 2737	. . . . . . . 8 ⊢ {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑋 / ∼ ) ∧ ∃𝑔 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑔 · 𝑥) = 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ (𝑋 / ∼ ) ∧ ∃𝑔 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑔 · 𝑥) = 𝑦)}
69	56, 59, 1, 66, 27, 67, 68	sylow2a 19560	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑋 / ∼ )) − (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
70		dvdssub2 16240	. . . . . . 7 ⊢ (((𝑃 ∈ ℤ ∧ (♯‘(𝑋 / ∼ )) ∈ ℤ ∧ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℤ) ∧ 𝑃 ∥ ((♯‘(𝑋 / ∼ )) − (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}))) → (𝑃 ∥ (♯‘(𝑋 / ∼ )) ↔ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
71	48, 49, 55, 69, 70	syl31anc 1376	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (♯‘(𝑋 / ∼ )) ↔ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
72	46, 71	mtbid 324	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}))
73		hasheq0 14298	. . . . . . . 8 ⊢ ({𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin → ((♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) = 0 ↔ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = ∅))
74	52, 73	syl 17	. . . . . . 7 ⊢ (𝜑 → ((♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) = 0 ↔ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = ∅))
75		dvds0 16210	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 0)
76	48, 75	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∥ 0)
77		breq2 5104	. . . . . . . 8 ⊢ ((♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) = 0 → (𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ↔ 𝑃 ∥ 0))
78	76, 77	syl5ibrcom 247	. . . . . . 7 ⊢ (𝜑 → ((♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) = 0 → 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
79	74, 78	sylbird 260	. . . . . 6 ⊢ (𝜑 → ({𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = ∅ → 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
80	79	necon3bd 2947	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ≠ ∅))
81	72, 80	mpd 15	. . . 4 ⊢ (𝜑 → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ≠ ∅)
82		rabn0 4343	. . . 4 ⊢ ({𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ≠ ∅ ↔ ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧)
83	81, 82	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧)
84	62	raleqdv 3298	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
85	84	rexbidv 3162	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
86	83, 85	mpbird 257	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧)
87		vex 3446	. . . . 5 ⊢ 𝑧 ∈ V
88	87	elqs 8713	. . . 4 ⊢ (𝑧 ∈ (𝑋 / ∼ ) ↔ ∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ )
89		simplrr 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [𝑔] ∼ )
90	89	oveq2d 7384	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = (𝑢 · [𝑔] ∼ ))
91		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = 𝑧)
92		simpll 767	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝜑)
93		simprl 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝐻)
94		simplrl 777	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∈ 𝑋)
95	7, 10, 4, 17, 57, 16, 58	sylow2blem1 19561	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑔 ∈ 𝑋) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
96	92, 93, 94, 95	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
97	90, 91, 96	3eqtr3d 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [(𝑢 + 𝑔)] ∼ )
98	89, 97	eqtr3d 2774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ )
99	25	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ∼ Er 𝑋)
100	99, 94	erth 8700	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 ∼ (𝑢 + 𝑔) ↔ [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ ))
101	98, 100	mpbird 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∼ (𝑢 + 𝑔))
102	6	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐺 ∈ Grp)
103	36	ad2antrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐾 ⊆ 𝑋)
104		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (invg‘𝐺) = (invg‘𝐺)
105	7, 104, 57, 16	eqgval 19118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝑔 ∼ (𝑢 + 𝑔) ↔ (𝑔 ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)))
106	102, 103, 105	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 ∼ (𝑢 + 𝑔) ↔ (𝑔 ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)))
107	101, 106	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾))
108	107	simp3d 1145	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)
109		oveq2 7376	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → (𝑔 + 𝑥) = (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))))
110	109	oveq1d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → ((𝑔 + 𝑥) − 𝑔) = ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔))
111		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))
112		ovex 7401	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔) ∈ V
113	110, 111, 112	fvmpt 6949	. . . . . . . . . . . . . . . 16 ⊢ ((((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾 → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔))
114	108, 113	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔))
115	7, 57, 31, 104	grprinv 18932	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
116	102, 94, 115	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
117	116	oveq1d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑔 + ((invg‘𝐺)‘𝑔)) + (𝑢 + 𝑔)) = ((0g‘𝐺) + (𝑢 + 𝑔)))
118	7, 104	grpinvcl 18929	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
119	102, 94, 118	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
120	64	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐻 ⊆ 𝑋)
121	120, 93	sseldd 3936	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝑋)
122	7, 57	grpcl 18883	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑢 + 𝑔) ∈ 𝑋)
123	102, 121, 94, 122	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 + 𝑔) ∈ 𝑋)
124	7, 57	grpass 18884	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (𝑔 ∈ 𝑋 ∧ ((invg‘𝐺)‘𝑔) ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋)) → ((𝑔 + ((invg‘𝐺)‘𝑔)) + (𝑢 + 𝑔)) = (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))))
125	102, 94, 119, 123, 124	syl13anc 1375	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑔 + ((invg‘𝐺)‘𝑔)) + (𝑢 + 𝑔)) = (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))))
126	7, 57, 31	grplid 18909	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 + 𝑔) ∈ 𝑋) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
127	102, 123, 126	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
128	117, 125, 127	3eqtr3d 2780	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = (𝑢 + 𝑔))
129	128	oveq1d 7383	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔) = ((𝑢 + 𝑔) − 𝑔))
130		sylow2blem3.d	. . . . . . . . . . . . . . . . 17 ⊢ − = (-g‘𝐺)
131	7, 57, 130	grppncan 18973	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
132	102, 121, 94, 131	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
133	114, 129, 132	3eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = 𝑢)
134		ovex 7401	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 + 𝑥) − 𝑔) ∈ V
135	134, 111	fnmpti 6643	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) Fn 𝐾
136		fnfvelrn 7034	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) Fn 𝐾 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
137	135, 108, 136	sylancr 588	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
138	133, 137	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
139	138	expr 456	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ 𝑢 ∈ 𝐻) → ((𝑢 · 𝑧) = 𝑧 → 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
140	139	ralimdva 3150	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
141	140	imp 406	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
142	141	an32s 653	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
143		dfss3 3924	. . . . . . . . 9 ⊢ (𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ↔ ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
144	142, 143	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
145	144	expr 456	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ 𝑔 ∈ 𝑋) → (𝑧 = [𝑔] ∼ → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
146	145	reximdva 3151	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
147	146	ex 412	. . . . 5 ⊢ (𝜑 → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
148	147	com23 86	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
149	88, 148	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋 / ∼ ) → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
150	149	rexlimdv 3137	. 2 ⊢ (𝜑 → (∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
151	86, 150	mpd 15	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {cpr 4584   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  ran crn 5633   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370   Er wer 8642  [cec 8643   / cqs 8644  Fincfn 8895  0cc0 11038   · cmul 11043   − cmin 11376   / cdiv 11806  ℕcn 12157  ℕ₀cn0 12413  ℤcz 12500  ↑cexp 13996  ♯chash 14265   ∥ cdvds 16191  ℙcprime 16610   pCnt cpc 16776  Basecbs 17148   ↾_s cress 17169  +_gcplusg 17189  0_gc0g 17371  Grpcgrp 18875  inv_gcminusg 18876  -_gcsg 18877  SubGrpcsubg 19062   ~_QG cqg 19064   pGrp cpgp 19467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-omul 8412  df-er 8645  df-ec 8647  df-qs 8651  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-acn 9866  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-fac 14209  df-bc 14238  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-dvds 16192  df-gcd 16434  df-prm 16611  df-pc 16777  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19010  df-subg 19065  df-eqg 19067  df-ga 19231  df-od 19469  df-pgp 19471

This theorem is referenced by:  sylow2b  19564