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Theorem sylow2a 19548

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 19547	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
9		inass 4180	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 4424	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 4169	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 4347	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2763	. . . . . 6 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅)
15		inundif 4431	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = (𝑌 / ∼ )
16	15	eqcomi 2745	. . . . . 6 ⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
18		pwfi 9219	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 19237	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 8713	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 9169	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3933	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9169	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 14279	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 12464	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 15664	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
32	21, 5	qshash 15750	. . . 4 ⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧))
33		inss1 4189	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 9097	. . . . . . . 8 ⊢ (((𝑌 / ∼ ) ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
35	23, 33, 34	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
36		ax-1cn 11084	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 15713	. . . . . . 7 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
38	35, 36, 37	sylancl 586	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
39		elin 3917	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3959	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		velpw 4559	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	bitr4di 289	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 5101	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈ 1_o ↔ 𝑧 ≈ 1_o))
45	43, 44	imbi12d 344	. . . . . . . . . . . . 13 ⊢ ([𝑤] ∼ = 𝑧 → (([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_o) ↔ (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_o)))
46	21	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
47		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
48	46, 47	erref 8655	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3444	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 8681	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3927	. . . . . . . . . . . . . . 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 19546	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 8958	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_o
56	54, 55	eqbrtrdi 5137	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈ 1_o)
57	56	ex 412	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_o))
58	57	adantr 480	. . . . . . . . . . . . . 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 8719	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_o))
61	60	impr 454	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1_o)
62	39, 61	sylan2b 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ≈ 1_o)
63		en1b 8962	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_o ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6838	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = (♯‘{∪ 𝑧}))
66		vuniex 7684	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 14292	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (♯‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∪ 𝑧}) = 1
69	65, 68	eqtrdi 2787	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = 1)
70	69	sumeq2dv 15625	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71	6	ssrab3 4034	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
72		ssfi 9097	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
73	5, 71, 72	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
74		hashcl 14279	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (♯‘𝑍) ∈ ℕ₀)
75	73, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑍) ∈ ℕ₀)
76	75	nn0cnd 12464	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) ∈ ℂ)
77	76	mulridd 11149	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = (♯‘𝑍))
78	6, 5	rabexd 5285	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
79		eqid 2736	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↦ {𝑤}) = (𝑤 ∈ 𝑍 ↦ {𝑤})
80	7	relopabiv 5769	. . . . . . . . . . . . . . 15 ⊢ Rel ∼
81		relssdmrn 6227	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
82	80, 81	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
83		erdm 8645	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
84	21, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ∼ = 𝑌)
85	84, 5	eqeltrd 2836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ∼ ∈ Fin)
86		errn 8657	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
87	21, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran ∼ = 𝑌)
88	87, 5	eqeltrd 2836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran ∼ ∈ Fin)
89	85, 88	xpexd 7696	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
90		ssexg 5268	. . . . . . . . . . . . . 14 ⊢ (( ∼ ⊆ (dom ∼ × ran ∼ ) ∧ (dom ∼ × ran ∼ ) ∈ V) → ∼ ∈ V)
91	82, 89, 90	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∼ ∈ V)
92		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
93	71, 92	sselid 3931	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
94		ecelqsw 8706	. . . . . . . . . . . . 13 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
95	91, 93, 94	syl2an2r 685	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
96	54, 95	eqeltrrd 2837	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
97		snelpwi 5392	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
98	97	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
99	96, 98	elind 4152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
100		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
101	100	elin2d 4157	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ 𝒫 𝑍)
102	101	elpwid 4563	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ⊆ 𝑍)
103	64, 102	eqsstrrd 3969	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
104	66	snss 4741	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
105	103, 104	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
106		sneq 4590	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
107	106	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∪ 𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧}))
108	64, 107	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
109	108	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
110		unieq 4874	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
111		unisnv 4883	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
112	110, 111	eqtr2di 2788	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧)
113	109, 112	impbid1 225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤}))
114	79, 99, 105, 113	f1o2d 7612	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ 𝑍 ↦ {𝑤}):𝑍–1-1-onto→((𝑌 / ∼ ) ∩ 𝒫 𝑍))
115	78, 114	hasheqf1od 14276	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
116	115	oveq1d 7373	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
117	77, 116	eqtr3d 2773	. . . . . 6 ⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
118	38, 70, 117	3eqtr4rd 2782	. . . . 5 ⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧))
119	118	oveq1d 7373	. . . 4 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
120	31, 32, 119	3eqtr4rd 2782	. . 3 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌))
121		hashcl 14279	. . . . . 6 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ₀)
122	5, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ₀)
123	122	nn0cnd 12464	. . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ)
124		diffi 9099	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
125	23, 124	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
126		eldifi 4083	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
127	126, 30	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℂ)
128	125, 127	fsumcl 15656	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ∈ ℂ)
129	123, 76, 128	subaddd 11510	. . 3 ⊢ (𝜑 → (((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ↔ ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌)))
130	120, 129	mpbird 257	. 2 ⊢ (𝜑 → ((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
131	8, 130	breqtrrd 5126	1 ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 {crab 3399 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 𝒫 cpw 4554 {csn 4580 {cpr 4582 ∪ cuni 4863 class class class wbr 5098 {copab 5160 ↦ cmpt 5179 × cxp 5622 dom cdm 5624 ran crn 5625 Rel wrel 5629 ‘cfv 6492 (class class class)co 7358 1_oc1o 8390 Er wer 8632 [cec 8633 / cqs 8634 ≈ cen 8880 Fincfn 8883 ℂcc 11024 1c1 11027 + caddc 11029 · cmul 11031 − cmin 11364 ℕ₀cn0 12401 ♯chash 14253 Σcsu 15609 ∥ cdvds 16179 Basecbs 17136 GrpAct cga 19218 pGrp cpgp 19455

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-xnn0 12475 df-z 12489 df-uz 12752 df-q 12862 df-rp 12906 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-sum 15610 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-0g 17361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-eqg 19055 df-ga 19219 df-od 19457 df-pgp 19459

This theorem is referenced by: sylow2blem3 19551 sylow3lem6 19561

W3C validator