Theorem sylow2blem3 19715

Description: Sylow's second theorem. Putting together the results of sylow2a 19712 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 19686	. . . . . . . . 9 ⊢ (𝑃 pGrp (𝐺 ↾s 𝐻) → 𝑃 ∈ ℙ)
3	1, 2	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
4		sylow2b.h	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝐺))
5		subgrcl 19220	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
6	4, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ Grp)
7		sylow2b.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
8	7	grpbn0 19056	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
9	6, 8	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
10		sylow2b.xf	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ Fin)
11		hashnncl 14415	. . . . . . . . . 10 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
12	10, 11	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
13	9, 12	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
14		pcndvds2 16945	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑋) ∈ ℕ) → ¬ 𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
15	3, 13, 14	syl2anc 596	. . . . . . 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 19288	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) = ((♯‘(𝑋 / ∼ )) · (♯‘𝐾)))
19	18	oveq1d 7431	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)))
20		sylow2blem3.kn	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) = (𝑃↑(𝑃 pCnt (♯‘𝑋))))
21	20	oveq2d 7432	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) / (♯‘𝐾)) = ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))))
22		pwfi 9281	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
23	10, 22	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
24	7, 16	eqger 19269	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → ∼ Er 𝑋)
25	17, 24	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∼ Er 𝑋)
26	25	qsss 8775	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋 / ∼ ) ⊆ 𝒫 𝑋)
27	23, 26	ssfid 9232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑋 / ∼ ) ∈ Fin)
28		hashcl 14405	. . . . . . . . . . . 12 ⊢ ((𝑋 / ∼ ) ∈ Fin → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
29	27, 28	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℕ0)
30	29	nn0cnd 12578	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℂ)
31		eqid 2765	. . . . . . . . . . . . . . 15 ⊢ (0g‘𝐺) = (0g‘𝐺)
32	31	subg0cl 19223	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐾)
33	17, 32	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐾)
34	33	ne0d 4295	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ≠ ∅)
35	7	subgss 19216	. . . . . . . . . . . . . . 15 ⊢ (𝐾 ∈ (SubGrp‘𝐺) → 𝐾 ⊆ 𝑋)
36	17, 35	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ⊆ 𝑋)
37	10, 36	ssfid 9232	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ Fin)
38		hashnncl 14415	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ Fin → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
39	37, 38	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝐾) ∈ ℕ ↔ 𝐾 ≠ ∅))
40	34, 39	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐾) ∈ ℕ)
41	40	nncnd 12260	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ∈ ℂ)
42	40	nnne0d 12297	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐾) ≠ 0)
43	30, 41, 42	divcan4d 12008	. . . . . . . . 9 ⊢ (𝜑 → (((♯‘(𝑋 / ∼ )) · (♯‘𝐾)) / (♯‘𝐾)) = (♯‘(𝑋 / ∼ )))
44	19, 21, 43	3eqtr3d 2808	. . . . . . . 8 ⊢ (𝜑 → ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) = (♯‘(𝑋 / ∼ )))
45	44	breq2d 5123	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∥ ((♯‘𝑋) / (𝑃↑(𝑃 pCnt (♯‘𝑋)))) ↔ 𝑃 ∥ (♯‘(𝑋 / ∼ ))))
46	15, 45	mtbid 327	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘(𝑋 / ∼ )))
47		prmz 16750	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
48	3, 47	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℤ)
49	29	nn0zd 12627	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝑋 / ∼ )) ∈ ℤ)
50		ssrab2 4035	. . . . . . . . . 10 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ⊆ (𝑋 / ∼ )
51		ssfi 9160	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (𝜑 → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ∈ Fin)
53		hashcl 14405	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (𝜑 → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℕ0)
55	54	nn0zd 12627	. . . . . . 7 ⊢ (𝜑 → (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}) ∈ ℤ)
56		eqid 2765	. . . . . . . 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 19714	. . . . . . . 8 ⊢ (𝜑 → · ∈ ((𝐺 ↾s 𝐻) GrpAct (𝑋 / ∼ )))
60		eqid 2765	. . . . . . . . . . 11 ⊢ (𝐺 ↾s 𝐻) = (𝐺 ↾s 𝐻)
61	60	subgbas 19219	. . . . . . . . . 10 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
62	4, 61	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘(𝐺 ↾s 𝐻)))
63	7	subgss 19216	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐺) → 𝐻 ⊆ 𝑋)
64	4, 63	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ 𝑋)
65	10, 64	ssfid 9232	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 ∈ Fin)
66	62, 65	eqeltrrd 2866	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐺 ↾s 𝐻)) ∈ Fin)
67		eqid 2765	. . . . . . . 8 ⊢ {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} = {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}
68		eqid 2765	. . . . . . . 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 19712	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∥ ((♯‘(𝑋 / ∼ )) − (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
70		dvdssub2 16376	. . . . . . 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 1400	. . . . . 6 ⊢ (𝜑 → (𝑃 ∥ (♯‘(𝑋 / ∼ )) ↔ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧})))
72	46, 71	mtbid 327	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ∥ (♯‘{𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧}))
73		hasheq0 14412	. . . . . . . 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 18	. . . . . . 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 16346	. . . . . . . . 9 ⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 0)
76	48, 75	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∥ 0)
77		breq2 5115	. . . . . . . 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 250	. . . . . . 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 263	. . . . . 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 2974	. . . . 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 16	. . . 4 ⊢ (𝜑 → {𝑧 ∈ (𝑋 / ∼ ) ∣ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧} ≠ ∅)
82		rabn0 4346	. . . 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 221	. . 3 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧)
84	62	raleqdv 3325	. . . 4 ⊢ (𝜑 → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
85	84	rexbidv 3191	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 ↔ ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ (Base‘(𝐺 ↾s 𝐻))(𝑢 · 𝑧) = 𝑧))
86	83, 85	mpbird 260	. 2 ⊢ (𝜑 → ∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧)
87		vex 3461	. . . . 5 ⊢ 𝑧 ∈ V
88	87	elqs 8764	. . . 4 ⊢ (𝑧 ∈ (𝑋 / ∼ ) ↔ ∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ )
89		simplrr 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [𝑔] ∼ )
90	89	oveq2d 7432	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = (𝑢 · [𝑔] ∼ ))
91		simprr 785	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · 𝑧) = 𝑧)
92		simpll 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝜑)
93		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝐻)
94		simplrl 789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∈ 𝑋)
95	7, 10, 4, 17, 57, 16, 58	sylow2blem1 19713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝐻 ∧ 𝑔 ∈ 𝑋) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
96	92, 93, 94, 95	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 · [𝑔] ∼ ) = [(𝑢 + 𝑔)] ∼ )
97	90, 91, 96	3eqtr3d 2808	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑧 = [(𝑢 + 𝑔)] ∼ )
98	89, 97	eqtr3d 2802	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ )
99	25	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ∼ Er 𝑋)
100	99, 94	erth 8751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 ∼ (𝑢 + 𝑔) ↔ [𝑔] ∼ = [(𝑢 + 𝑔)] ∼ ))
101	98, 100	mpbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑔 ∼ (𝑢 + 𝑔))
102	6	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐺 ∈ Grp)
103	36	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐾 ⊆ 𝑋)
104		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (invg‘𝐺) = (invg‘𝐺)
105	7, 104, 57, 16	eqgval 19268	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝐾 ⊆ 𝑋) → (𝑔 ∼ (𝑢 + 𝑔) ↔ (𝑔 ∈ 𝑋 ∧ (𝑢 + 𝑔) ∈ 𝑋 ∧ (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)))
106	102, 103, 105	syl2anc 596	. . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . 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 1162	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) ∈ 𝐾)
109		oveq2 7424	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → (𝑔 + 𝑥) = (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))))
110	109	oveq1d 7431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔)) → ((𝑔 + 𝑥) − 𝑔) = ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔))
111		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) = (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))
112		ovex 7449	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) − 𝑔) ∈ V
113	110, 111, 112	fvmpt 6993	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 19080	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
116	102, 94, 115	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + ((invg‘𝐺)‘𝑔)) = (0g‘𝐺))
117	116	oveq1d 7431	. . . . . . . . . . . . . . . . 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 19077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑔 ∈ 𝑋) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
119	102, 94, 118	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((invg‘𝐺)‘𝑔) ∈ 𝑋)
120	64	ad2antrr 739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝐻 ⊆ 𝑋)
121	120, 93	sseldd 3939	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ 𝑋)
122	7, 57	grpcl 19031	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → (𝑢 + 𝑔) ∈ 𝑋)
123	102, 121, 94, 122	syl3anc 1398	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑢 + 𝑔) ∈ 𝑋)
124	7, 57	grpass 19032	. . . . . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . . . . 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 19057	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ (𝑢 + 𝑔) ∈ 𝑋) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
127	102, 123, 126	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((0g‘𝐺) + (𝑢 + 𝑔)) = (𝑢 + 𝑔))
128	117, 125, 127	3eqtr3d 2808	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → (𝑔 + (((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = (𝑢 + 𝑔))
129	128	oveq1d 7431	. . . . . . . . . . . . . . 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 19120	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∈ Grp ∧ 𝑢 ∈ 𝑋 ∧ 𝑔 ∈ 𝑋) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
132	102, 121, 94, 131	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑢 + 𝑔) − 𝑔) = 𝑢)
133	114, 129, 132	3eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → ((𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))‘(((invg‘𝐺)‘𝑔) + (𝑢 + 𝑔))) = 𝑢)
134		ovex 7449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 + 𝑥) − 𝑔) ∈ V
135	134, 111	fnmpti 6682	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) Fn 𝐾
136		fnfvelrn 7079	. . . . . . . . . . . . . . 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 599	. . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ (𝑢 ∈ 𝐻 ∧ (𝑢 · 𝑧) = 𝑧)) → 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
139	138	expr 462	. . . . . . . . . . . 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 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
142	141	an32s 665	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
143		dfss3 3927	. . . . . . . . 9 ⊢ (𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)) ↔ ∀𝑢 ∈ 𝐻 𝑢 ∈ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
144	142, 143	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ (𝑔 ∈ 𝑋 ∧ 𝑧 = [𝑔] ∼ )) → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))
145	144	expr 462	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) ∧ 𝑔 ∈ 𝑋) → (𝑧 = [𝑔] ∼ → 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
146	145	reximdva 3180	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧) → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
147	146	ex 418	. . . . 5 ⊢ (𝜑 → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
148	147	com23 87	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝑋 𝑧 = [𝑔] ∼ → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
149	88, 148	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋 / ∼ ) → (∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))))
150	149	rexlimdv 3166	. 2 ⊢ (𝜑 → (∃𝑧 ∈ (𝑋 / ∼ )∀𝑢 ∈ 𝐻 (𝑢 · 𝑧) = 𝑧 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔))))
151	86, 150	mpd 16	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝑋 𝐻 ⊆ ran (𝑥 ∈ 𝐾 ↦ ((𝑔 + 𝑥) − 𝑔)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {cpr 4593   class class class wbr 5111  {copab 5175   ↦ cmpt 5194  ran crn 5664   Fn wfn 6535  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418   Er wer 8693  [cec 8694   / cqs 8695  Fincfn 8945  0cc0 11111   · cmul 11116   − cmin 11452   / cdiv 11882  ℕcn 12244  ℕ₀cn0 12515  ℤcz 12602  ↑cexp 14110  ♯chash 14379   ∥ cdvds 16327  ℙcprime 16746   pCnt cpc 16913  Basecbs 17286   ↾_s cress 17307  +_gcplusg 17327  0_gc0g 17509  Grpcgrp 19023  inv_gcminusg 19024  -_gcsg 19025  SubGrpcsubg 19209   ~_QG cqg 19211   pGrp cpgp 19619

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-inf2 9613  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188  ax-pre-sup 11189

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-disj 5079  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-omul 8460  df-er 8696  df-ec 8698  df-qs 8702  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9405  df-inf 9406  df-oi 9475  df-dju 9899  df-card 9937  df-acn 9940  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-div 11883  df-nn 12245  df-2 12314  df-3 12315  df-n0 12516  df-xnn0 12589  df-z 12603  df-uz 12874  df-q 12984  df-rp 13028  df-fz 13547  df-fzo 13695  df-fl 13838  df-mod 13916  df-seq 14051  df-exp 14111  df-fac 14323  df-bc 14352  df-hash 14380  df-cj 15169  df-re 15170  df-im 15171  df-sqrt 15305  df-abs 15306  df-clim 15558  df-sum 15757  df-dvds 16328  df-gcd 16570  df-prm 16747  df-pc 16914  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-0g 17511  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-submnd 18865  df-grp 19026  df-minusg 19027  df-sbg 19028  df-mulg 19157  df-subg 19212  df-eqg 19214  df-ga 19383  df-od 19621  df-pgp 19623

This theorem is used by:  sylow2b  19716