Theorem sylow2blem3 18749

Description: Sylow's second theorem. Putting together the results of sylow2a 18746 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 18720	. . . . . . . . 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 18286	. . . . . . . . . . 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 18134	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
9	6, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
10		sylow2b.xf	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
11		hashnncl 13730	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	9, 12	mpbird 259	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
14		pcndvds2 16206	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
15	3, 13, 14	syl2anc 586	. . . . . . 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 18343	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐾)))
19	18	oveq1d 7173	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)))
20		sylow2blem3.kn	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
21	20	oveq2d 7174	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
22		pwfi 8821	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	10, 22	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24	7, 16	eqger 18332	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
25	17, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∼ Er 𝑋)
26	25	qsss 8360	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
27	23, 26	ssfid 8743	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin)
28		hashcl 13720	. . . . . . . . . . . 12 ⊢ ((𝑋 / ∼ ) ∈ Fin → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
29	27, 28	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
30	29	nn0cnd 11960	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℂ)
31		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐺) = (0g‘𝐺)
32	31	subg0cl 18289	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐾)
33	17, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐾)
34	33	ne0d 4303	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≠ ∅)
35	7	subgss 18282	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
36	17, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
37	10, 36	ssfid 8743	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Fin)
38		hashnncl 13730	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Fin → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
40	34, 39	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐾) ∈ ℕ)
41	40	nncnd 11656	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
42	40	nnne0d 11690	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ≠ 0)
43	30, 41, 42	divcan4d 11424	. . . . . . . . 9 ⊢ (𝜑 → (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)) = (♯‘(𝑋 / ∼ )))
44	19, 21, 43	3eqtr3d 2866	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) = (♯‘(𝑋 / ∼ )))
45	44	breq2d 5080	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ↔ 𝑃 ∥ (♯‘(𝑋 / ∼ ))))
46	15, 45	mtbid 326	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘(𝑋 / ∼ )))
47		prmz 16021	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
48	3, 47	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℤ)
49	29	nn0zd 12088	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℤ)
50		ssrab2 4058	. . . . . . . . . 10 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ⊆ (𝑋 / ∼ )
51		ssfi 8740	. . . . . . . . . 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 588	. . . . . . . . 9 ⊢ (𝜑 → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin)
53		hashcl 13720	. . . . . . . . 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 12088	. . . . . . 7 ⊢ (𝜑 → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℤ)
56		eqid 2823	. . . . . . . 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 18748	. . . . . . . 8 ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))
60		eqid 2823	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
61	60	subgbas 18285	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
62	4, 61	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
63	7	subgss 18282	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
64	4, 63	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
65	10, 64	ssfid 8743	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Fin)
66	62, 65	eqeltrrd 2916	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝐻)) ∈ Fin)
67		eqid 2823	. . . . . . . 8 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}
68		eqid 2823	. . . . . . . 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 18746	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑋 / ∼ )) − (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
70		dvdssub2 15653	. . . . . . 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 1369	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (♯‘(𝑋 / ∼ )) ↔ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
72	46, 71	mtbid 326	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}))
73		hasheq0 13727	. . . . . . . 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 15627	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 0)
76	48, 75	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∥ 0)
77		breq2 5072	. . . . . . . 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 249	. . . . . . 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 262	. . . . . 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 3032	. . . . 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 4341	. . . 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 220	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧)
84	62	raleqdv 3417	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
85	84	rexbidv 3299	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
86	83, 85	mpbird 259	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧)
87		vex 3499	. . . . 5 ⊢ 𝑧 ∈ V
88	87	elqs 8351	. . . 4 ⊢ (𝑧 ∈ (𝑋 / ∼ ) ↔ ∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ )
89		simplrr 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [𝑔] ∼ )
90	89	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = (𝑢 · [𝑔] ∼ ))
91		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = 𝑧)
92		simpll 765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝜑)
93		simprl 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝐻)
94		simplrl 775	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∈ 𝑋)
95	7, 10, 4, 17, 57, 16, 58	sylow2blem1 18747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑔 ∈ 𝑋) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
96	92, 93, 94, 95	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
97	90, 91, 96	3eqtr3d 2866	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [(𝑢 + 𝑔)] ∼ )
98	89, 97	eqtr3d 2860	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ )
99	25	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ∼ Er 𝑋)
100	99, 94	erth 8340	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 ∼ (𝑢 + 𝑔) ↔ [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ ))
101	98, 100	mpbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∼ (𝑢 + 𝑔))
102	6	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐺 ∈ Grp)
103	36	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐾 ⊆ 𝑋)
104		eqid 2823	. . . . . . . . . . . . . . . . . . . 20 ⊢ (invg‘𝐺) = (invg‘𝐺)
105	7, 104, 57, 16	eqgval 18331	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝑔 ∼ (𝑢 + 𝑔) ↔ (𝑔 ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)))
106	102, 103, 105	syl2anc 586	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . 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 1140	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)
109		oveq2 7166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → (𝑔 + 𝑥) = (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))))
110	109	oveq1d 7173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → ((𝑔 + 𝑥) − 𝑔) = ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔))
111		eqid 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))
112		ovex 7191	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔) ∈ V
113	110, 111, 112	fvmpt 6770	. . . . . . . . . . . . . . . 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 18155	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
116	102, 94, 115	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
117	116	oveq1d 7173	. . . . . . . . . . . . . . . . 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 18153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
119	102, 94, 118	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
120	64	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐻 ⊆ 𝑋)
121	120, 93	sseldd 3970	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝑋)
122	7, 57	grpcl 18113	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑢 + 𝑔) ∈ 𝑋)
123	102, 121, 94, 122	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 + 𝑔) ∈ 𝑋)
124	7, 57	grpass 18114	. . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . 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 18135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 + 𝑔) ∈ 𝑋) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
127	102, 123, 126	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
128	117, 125, 127	3eqtr3d 2866	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = (𝑢 + 𝑔))
129	128	oveq1d 7173	. . . . . . . . . . . . . . 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 18192	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
132	102, 121, 94, 131	syl3anc 1367	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
133	114, 129, 132	3eqtrd 2862	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = 𝑢)
134		ovex 7191	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 + 𝑥) − 𝑔) ∈ V
135	134, 111	fnmpti 6493	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) Fn 𝐾
136		fnfvelrn 6850	. . . . . . . . . . . . . . 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 589	. . . . . . . . . . . . . 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 2916	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
139	138	expr 459	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ 𝑢 ∈ 𝐻) → ((𝑢 · 𝑧) = 𝑧 → 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
140	139	ralimdva 3179	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
141	140	imp 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
142	141	an32s 650	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
143		dfss3 3958	. . . . . . . . 9 ⊢ (𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ↔ ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
144	142, 143	sylibr 236	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
145	144	expr 459	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ 𝑔 ∈ 𝑋) → (𝑧 = [𝑔] ∼ → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
146	145	reximdva 3276	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
147	146	ex 415	. . . . 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	syl5bi 244	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋 / ∼ ) → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
150	149	rexlimdv 3285	. 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {cpr 4571   class class class wbr 5068  {copab 5130   ↦ cmpt 5148  ran crn 5558   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160   Er wer 8288  [cec 8289   / cqs 8290  Fincfn 8511  0cc0 10539   · cmul 10544   − cmin 10872   / cdiv 11299  ℕcn 11640  ℕ₀cn0 11900  ℤcz 11984  ↑cexp 13432  ♯chash 13693   ∥ cdvds 15609  ℙcprime 16017   pCnt cpc 16175  Basecbs 16485   ↾_s cress 16486  +_gcplusg 16567  0_gc0g 16715  Grpcgrp 18105  inv_gcminusg 18106  -_gcsg 18107  SubGrpcsubg 18275   ~_QG cqg 18277   pGrp cpgp 18656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-omul 8109  df-er 8291  df-ec 8293  df-qs 8297  df-map 8410  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-acn 9373  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-fac 13637  df-bc 13666  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-dvds 15610  df-gcd 15846  df-prm 16018  df-pc 16176  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-eqg 18280  df-ga 18422  df-od 18658  df-pgp 18660

This theorem is referenced by:  sylow2b  18750