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Theorem sylow2a 19549

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 19548	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
9		inass 4191	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 4435	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 4180	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 4358	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2756	. . . . . 6 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅)
15		inundif 4442	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = (𝑌 / ∼ )
16	15	eqcomi 2738	. . . . . 6 ⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
18		pwfi 9268	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 19240	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 8749	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 9212	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3946	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4572	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 14321	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 12505	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 15707	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
32	21, 5	qshash 15793	. . . 4 ⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧))
33		inss1 4200	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 9137	. . . . . . . 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 11126	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 15756	. . . . . . 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 3930	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3972	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		velpw 4568	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	bitr4di 289	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 5110	. . . . . . . . . . . . . 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 8691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 8717	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3940	. . . . . . . . . . . . . . 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 19547	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 8992	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_o
56	54, 55	eqbrtrdi 5146	. . . . . . . . . . . . . . . 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 8755	. . . . . . . . . . . 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 8996	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_o ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = (♯‘{∪ 𝑧}))
66		vuniex 7715	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 14334	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (♯‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∪ 𝑧}) = 1
69	65, 68	eqtrdi 2780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = 1)
70	69	sumeq2dv 15668	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71	6	ssrab3 4045	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
72		ssfi 9137	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
73	5, 71, 72	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
74		hashcl 14321	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (♯‘𝑍) ∈ ℕ₀)
75	73, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑍) ∈ ℕ₀)
76	75	nn0cnd 12505	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) ∈ ℂ)
77	76	mulridd 11191	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = (♯‘𝑍))
78	6, 5	rabexd 5295	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
79		eqid 2729	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↦ {𝑤}) = (𝑤 ∈ 𝑍 ↦ {𝑤})
80	7	relopabiv 5783	. . . . . . . . . . . . . . 15 ⊢ Rel ∼
81		relssdmrn 6241	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
82	80, 81	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
83		erdm 8681	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
84	21, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ∼ = 𝑌)
85	84, 5	eqeltrd 2828	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ∼ ∈ Fin)
86		errn 8693	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
87	21, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran ∼ = 𝑌)
88	87, 5	eqeltrd 2828	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran ∼ ∈ Fin)
89	85, 88	xpexd 7727	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
90		ssexg 5278	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
94		ecelqsw 8742	. . . . . . . . . . . . 13 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
95	91, 93, 94	syl2an2r 685	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
96	54, 95	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
97		snelpwi 5403	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
98	97	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
99	96, 98	elind 4163	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
100		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
101	100	elin2d 4168	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ 𝒫 𝑍)
102	101	elpwid 4572	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ⊆ 𝑍)
103	64, 102	eqsstrrd 3982	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
104	66	snss 4749	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
105	103, 104	sylibr 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
106		sneq 4599	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
107	106	eqeq2d 2740	. . . . . . . . . . . . 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 4882	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
111		unisnv 4891	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
112	110, 111	eqtr2di 2781	. . . . . . . . . . 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 7643	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ 𝑍 ↦ {𝑤}):𝑍–1-1-onto→((𝑌 / ∼ ) ∩ 𝒫 𝑍))
115	78, 114	hasheqf1od 14318	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
116	115	oveq1d 7402	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
117	77, 116	eqtr3d 2766	. . . . . 6 ⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
118	38, 70, 117	3eqtr4rd 2775	. . . . 5 ⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧))
119	118	oveq1d 7402	. . . 4 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
120	31, 32, 119	3eqtr4rd 2775	. . 3 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌))
121		hashcl 14321	. . . . . 6 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ₀)
122	5, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ₀)
123	122	nn0cnd 12505	. . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ)
124		diffi 9139	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
125	23, 124	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
126		eldifi 4094	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
127	126, 30	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℂ)
128	125, 127	fsumcl 15699	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ∈ ℂ)
129	123, 76, 128	subaddd 11551	. . 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 5135	1 ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3405 Vcvv 3447 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 𝒫 cpw 4563 {csn 4589 {cpr 4591 ∪ cuni 4871 class class class wbr 5107 {copab 5169 ↦ cmpt 5188 × cxp 5636 dom cdm 5638 ran crn 5639 Rel wrel 5643 ‘cfv 6511 (class class class)co 7387 1_oc1o 8427 Er wer 8668 [cec 8669 / cqs 8670 ≈ cen 8915 Fincfn 8918 ℂcc 11066 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 ℕ₀cn0 12442 ♯chash 14295 Σcsu 15652 ∥ cdvds 16222 Basecbs 17179 GrpAct cga 19221 pGrp cpgp 19456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-xnn0 12516 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-eqg 19057 df-ga 19222 df-od 19458 df-pgp 19460

This theorem is referenced by: sylow2blem3 19552 sylow3lem6 19562

W3C validator