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Theorem sylow2a 18034

Description: A named lemma of Sylow's second and third theorems. If 𝐺 is a finite 𝑃-group that acts on the finite set 𝑌, then the set 𝑍 of all points of 𝑌 fixed by every element of 𝐺 has cardinality equivalent to the cardinality of 𝑌, mod 𝑃. (Contributed by Mario Carneiro, 17-Jan-2015.)

**Hypotheses**
Ref	Expression
sylow2a.x	⊢ 𝑋 = (Base‘𝐺)
sylow2a.m	⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))
sylow2a.p	⊢ (𝜑 → 𝑃 pGrp 𝐺)
sylow2a.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow2a.y	⊢ (𝜑 → 𝑌 ∈ Fin)
sylow2a.z	⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
sylow2a.r	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

**Assertion**
Ref	Expression
sylow2a	⊢ (𝜑 → 𝑃 ∥ ((#‘𝑌) − (#‘𝑍)))

Distinct variable groups: ∼ ,ℎ 𝑔,ℎ,𝑢,𝑥,𝑦 𝑔,𝐺,𝑥,𝑦 ⊕ ,𝑔,ℎ,𝑢,𝑥,𝑦 𝑔,𝑋,ℎ,𝑢,𝑥,𝑦 𝜑,ℎ 𝑔,𝑌,ℎ,𝑢,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑢,𝑔) 𝑃(𝑥,𝑦,𝑢,𝑔,ℎ) ∼ (𝑥,𝑦,𝑢,𝑔) 𝐺(𝑢,ℎ) 𝑍(𝑥,𝑦,𝑢,𝑔,ℎ)

Proof of Theorem sylow2a Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow2a.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		sylow2a.m	. . 3 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑌))
3		sylow2a.p	. . 3 ⊢ (𝜑 → 𝑃 pGrp 𝐺)
4		sylow2a.f	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
5		sylow2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ Fin)
6		sylow2a.z	. . 3 ⊢ 𝑍 = {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢}
7		sylow2a.r	. . 3 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
8	1, 2, 3, 4, 5, 6, 7	sylow2alem2 18033	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧))
9		inass 3823	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 4040	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 3811	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 3968	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2648	. . . . . 6 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅)
15		inundif 4046	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = (𝑌 / ∼ )
16	15	eqcomi 2631	. . . . . 6 ⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
18		pwfi 8261	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 208	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 17741	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 7808	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 8183	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4170	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 8183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 13147	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (#‘𝑧) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (#‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 11353	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (#‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 14471	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(#‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)))
32	21, 5	qshash 14559	. . . 4 ⊢ (𝜑 → (#‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(#‘𝑧))
33		inss1 3833	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 8180	. . . . . . . 8 ⊢ (((𝑌 / ∼ ) ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
35	23, 33, 34	sylancl 694	. . . . . . 7 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
36		ax-1cn 9994	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 14522	. . . . . . 7 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
38	35, 36, 37	sylancl 694	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
39		elin 3796	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2622	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3626	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		selpw 4165	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	syl6bbr 278	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 4656	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈ 1_𝑜 ↔ 𝑧 ≈ 1_𝑜))
45	43, 44	imbi12d 334	. . . . . . . . . . . . 13 ⊢ ([𝑤] ∼ = 𝑧 → (([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_𝑜) ↔ (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_𝑜)))
46	21	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
47		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
48	46, 47	erref 7762	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3203	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 7786	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3597	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ ⊆ 𝑍 → (𝑤 ∈ [𝑤] ∼ → 𝑤 ∈ 𝑍))
53	51, 52	syl5com 31	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → 𝑤 ∈ 𝑍))
54	1, 2, 3, 4, 5, 6, 7	sylow2alem1 18032	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 8020	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_𝑜
56	54, 55	syl6eqbr 4692	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈ 1_𝑜)
57	56	ex 450	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
58	57	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
59	53, 58	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
60	40, 45, 59	ectocld 7814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_𝑜))
61	60	impr 649	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1_𝑜)
62	39, 61	sylan2b 492	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ≈ 1_𝑜)
63		en1b 8024	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_𝑜 ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (#‘𝑧) = (#‘{∪ 𝑧}))
66		vuniex 6954	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 13159	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (#‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (#‘{∪ 𝑧}) = 1
69	65, 68	syl6eq 2672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (#‘𝑧) = 1)
70	69	sumeq2dv 14433	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71		ssrab2 3687	. . . . . . . . . . . 12 ⊢ {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} ⊆ 𝑌
72	6, 71	eqsstri 3635	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
73		ssfi 8180	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
74	5, 72, 73	sylancl 694	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
75		hashcl 13147	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (#‘𝑍) ∈ ℕ₀)
76	74, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑍) ∈ ℕ₀)
77	76	nn0cnd 11353	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑍) ∈ ℂ)
78	77	mulid1d 10057	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑍) · 1) = (#‘𝑍))
79	6, 5	rabexd 4814	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ V)
80		inss2 3834	. . . . . . . . . . 11 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ 𝒫 𝑍
81		pwexg 4850	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ Fin → 𝒫 𝑍 ∈ V)
82	74, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑍 ∈ V)
83		ssexg 4804	. . . . . . . . . . 11 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ 𝒫 𝑍 ∧ 𝒫 𝑍 ∈ V) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ V)
84	80, 82, 83	sylancr 695	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ V)
85	7	relopabi 5245	. . . . . . . . . . . . . . . . 17 ⊢ Rel ∼
86		relssdmrn 5656	. . . . . . . . . . . . . . . . 17 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
87	85, 86	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
88		erdm 7752	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
89	21, 88	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ∼ = 𝑌)
90	89, 5	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom ∼ ∈ Fin)
91		errn 7764	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
92	21, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran ∼ = 𝑌)
93	92, 5	eqeltrd 2701	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran ∼ ∈ Fin)
94		xpexg 6960	. . . . . . . . . . . . . . . . 17 ⊢ ((dom ∼ ∈ Fin ∧ ran ∼ ∈ Fin) → (dom ∼ × ran ∼ ) ∈ V)
95	90, 93, 94	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
96		ssexg 4804	. . . . . . . . . . . . . . . 16 ⊢ (( ∼ ⊆ (dom ∼ × ran ∼ ) ∧ (dom ∼ × ran ∼ ) ∈ V) → ∼ ∈ V)
97	87, 95, 96	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∼ ∈ V)
98	97	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∼ ∈ V)
99		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
100	72, 99	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
101		ecelqsg 7802	. . . . . . . . . . . . . 14 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
102	98, 100, 101	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
103	54, 102	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
104		snelpwi 4912	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
105	104	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
106	103, 105	elind 3798	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
107	106	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
108		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
109	80, 108	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ 𝒫 𝑍)
110	109	elpwid 4170	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ⊆ 𝑍)
111	64, 110	eqsstr3d 3640	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
112	66	snss 4316	. . . . . . . . . . . 12 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
113	111, 112	sylibr 224	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
114	113	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) → ∪ 𝑧 ∈ 𝑍))
115		sneq 4187	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
116	115	eqeq2d 2632	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧}))
117	64, 116	syl5ibrcom 237	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
118	117	adantrl 752	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
119		unieq 4444	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
120	49	unisn 4451	. . . . . . . . . . . . 13 ⊢ ∪ {𝑤} = 𝑤
121	119, 120	syl6req 2673	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧)
122	118, 121	impbid1 215	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤}))
123	122	ex 450	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤})))
124	79, 84, 107, 114, 123	en3d 7992	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
125		hashen 13135	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin) → ((#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) ↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
126	74, 35, 125	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) ↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
127	124, 126	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
128	127	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑍) · 1) = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
129	78, 128	eqtr3d 2658	. . . . . 6 ⊢ (𝜑 → (#‘𝑍) = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
130	38, 70, 129	3eqtr4rd 2667	. . . . 5 ⊢ (𝜑 → (#‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧))
131	130	oveq1d 6665	. . . 4 ⊢ (𝜑 → ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)))
132	31, 32, 131	3eqtr4rd 2667	. . 3 ⊢ (𝜑 → ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (#‘𝑌))
133		hashcl 13147	. . . . . 6 ⊢ (𝑌 ∈ Fin → (#‘𝑌) ∈ ℕ₀)
134	5, 133	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝑌) ∈ ℕ₀)
135	134	nn0cnd 11353	. . . 4 ⊢ (𝜑 → (#‘𝑌) ∈ ℂ)
136		diffi 8192	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
137	23, 136	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
138		eldifi 3732	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
139	138, 30	sylan2 491	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℂ)
140	137, 139	fsumcl 14464	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧) ∈ ℂ)
141	135, 77, 140	subaddd 10410	. . 3 ⊢ (𝜑 → (((#‘𝑌) − (#‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧) ↔ ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (#‘𝑌)))
142	132, 141	mpbird 247	. 2 ⊢ (𝜑 → ((#‘𝑌) − (#‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧))
143	8, 142	breqtrrd 4681	1 ⊢ (𝜑 → 𝑃 ∥ ((#‘𝑌) − (#‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 {cpr 4179 ∪ cuni 4436 class class class wbr 4653 {copab 4712 × cxp 5112 dom cdm 5114 ran crn 5115 Rel wrel 5119 ‘cfv 5888 (class class class)co 6650 1_𝑜c1o 7553 Er wer 7739 [cec 7740 / cqs 7741 ≈ cen 7952 Fincfn 7955 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕ₀cn0 11292 #chash 13117 Σcsu 14416 ∥ cdvds 14983 Basecbs 15857 GrpAct cga 17722 pGrp cpgp 17946

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-eqg 17593 df-ga 17723 df-od 17948 df-pgp 17950

This theorem is referenced by: sylow2blem3 18037 sylow3lem6 18047

W3C validator