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Theorem sylow2a 19401

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 19400	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
9		inass 4179	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 4431	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 4169	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 4351	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2768	. . . . . 6 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅)
15		inundif 4438	. . . . . . 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 9122	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 19088	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 8717	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 9211	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3944	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9211	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 14256	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 12475	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 15626	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
32	21, 5	qshash 15712	. . . 4 ⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧))
33		inss1 4188	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 9117	. . . . . . . 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 11109	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 15675	. . . . . . 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 3926	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2736	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3969	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		velpw 4565	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	bitr4di 288	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 5108	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
47		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
48	46, 47	erref 8668	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3449	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 8692	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3937	. . . . . . . . . . . . . . 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 19399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 8961	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_o
56	54, 55	eqbrtrdi 5144	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈ 1_o)
57	56	ex 413	. . . . . . . . . . . . . . 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 8723	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_o))
61	60	impr 455	. . . . . . . . . . 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 8967	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_o ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = (♯‘{∪ 𝑧}))
66		vuniex 7676	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 14269	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (♯‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∪ 𝑧}) = 1
69	65, 68	eqtrdi 2792	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = 1)
70	69	sumeq2dv 15588	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71	6	ssrab3 4040	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
72		ssfi 9117	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
73	5, 71, 72	sylancl 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
74		hashcl 14256	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (♯‘𝑍) ∈ ℕ₀)
75	73, 74	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑍) ∈ ℕ₀)
76	75	nn0cnd 12475	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) ∈ ℂ)
77	76	mulid1d 11172	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = (♯‘𝑍))
78	6, 5	rabexd 5290	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
79		eqid 2736	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↦ {𝑤}) = (𝑤 ∈ 𝑍 ↦ {𝑤})
80	7	relopabiv 5776	. . . . . . . . . . . . . . 15 ⊢ Rel ∼
81		relssdmrn 6220	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
82	80, 81	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
83		erdm 8658	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
84	21, 83	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ∼ = 𝑌)
85	84, 5	eqeltrd 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ∼ ∈ Fin)
86		errn 8670	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
87	21, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran ∼ = 𝑌)
88	87, 5	eqeltrd 2838	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran ∼ ∈ Fin)
89	85, 88	xpexd 7685	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
90		ssexg 5280	. . . . . . . . . . . . . 14 ⊢ (( ∼ ⊆ (dom ∼ × ran ∼ ) ∧ (dom ∼ × ran ∼ ) ∈ V) → ∼ ∈ V)
91	82, 89, 90	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∼ ∈ V)
92		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
93	71, 92	sselid 3942	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
94		ecelqsg 8711	. . . . . . . . . . . . 13 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
95	91, 93, 94	syl2an2r 683	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
96	54, 95	eqeltrrd 2839	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
97		snelpwi 5400	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
98	97	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
99	96, 98	elind 4154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
100		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
101	100	elin2d 4159	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ 𝒫 𝑍)
102	101	elpwid 4569	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ⊆ 𝑍)
103	64, 102	eqsstrrd 3983	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
104	66	snss 4746	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
105	103, 104	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
106		sneq 4596	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
107	106	eqeq2d 2747	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∪ 𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧}))
108	64, 107	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
109	108	adantrl 714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
110		unieq 4876	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
111		unisnv 4888	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
112	110, 111	eqtr2di 2793	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧)
113	109, 112	impbid1 224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤}))
114	79, 99, 105, 113	f1o2d 7607	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ 𝑍 ↦ {𝑤}):𝑍–1-1-onto→((𝑌 / ∼ ) ∩ 𝒫 𝑍))
115	78, 114	hasheqf1od 14253	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
116	115	oveq1d 7372	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
117	77, 116	eqtr3d 2778	. . . . . 6 ⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
118	38, 70, 117	3eqtr4rd 2787	. . . . 5 ⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧))
119	118	oveq1d 7372	. . . 4 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
120	31, 32, 119	3eqtr4rd 2787	. . 3 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌))
121		hashcl 14256	. . . . . 6 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ₀)
122	5, 121	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ₀)
123	122	nn0cnd 12475	. . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ)
124		diffi 9123	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
125	23, 124	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
126		eldifi 4086	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
127	126, 30	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℂ)
128	125, 127	fsumcl 15618	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ∈ ℂ)
129	123, 76, 128	subaddd 11530	. . 3 ⊢ (𝜑 → (((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ↔ ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌)))
130	120, 129	mpbird 256	. 2 ⊢ (𝜑 → ((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
131	8, 130	breqtrrd 5133	1 ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ∖ cdif 3907 ∪ cun 3908 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 𝒫 cpw 4560 {csn 4586 {cpr 4588 ∪ cuni 4865 class class class wbr 5105 {copab 5167 ↦ cmpt 5188 × cxp 5631 dom cdm 5633 ran crn 5634 Rel wrel 5638 ‘cfv 6496 (class class class)co 7357 1_oc1o 8405 Er wer 8645 [cec 8646 / cqs 8647 ≈ cen 8880 Fincfn 8883 ℂcc 11049 1c1 11052 + caddc 11054 · cmul 11056 − cmin 11385 ℕ₀cn0 12413 ♯chash 14230 Σcsu 15570 ∥ cdvds 16136 Basecbs 17083 GrpAct cga 19069 pGrp cpgp 19308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-ec 8650 df-qs 8654 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-xnn0 12486 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-sum 15571 df-dvds 16137 df-gcd 16375 df-prm 16548 df-pc 16709 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-eqg 18927 df-ga 19070 df-od 19310 df-pgp 19312

This theorem is referenced by: sylow2blem3 19404 sylow3lem6 19414

W3C validator