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Theorem sylow2a 17857

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 17856	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧))
9		inass 3785	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 3992	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 3773	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 3920	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2636	. . . . . 6 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
14	13	a1i 11	. . . . 5 ⊢ (𝜑 → (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅)
15		inundif 3998	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = (𝑌 / ∼ )
16	15	eqcomi 2619	. . . . . 6 ⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
18		pwfi 8144	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 17564	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 7695	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 8068	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3568	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 8068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 13009	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (#‘𝑧) ∈ ℕ₀)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (#‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 11230	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (#‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 14318	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(#‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)))
32	21, 5	qshash 14398	. . . 4 ⊢ (𝜑 → (#‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(#‘𝑧))
33		inss1 3795	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 8065	. . . . . . . 8 ⊢ (((𝑌 / ∼ ) ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
35	23, 33, 34	sylancl 693	. . . . . . 7 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
36		ax-1cn 9873	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 14364	. . . . . . 7 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
38	35, 36, 37	sylancl 693	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
39		elin 3758	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3589	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		selpw 4115	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	syl6bbr 277	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 4586	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈ 1_𝑜 ↔ 𝑧 ≈ 1_𝑜))
45	43, 44	imbi12d 333	. . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
48	46, 47	erref 7649	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3176	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 7673	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 223	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3562	. . . . . . . . . . . . . . 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 17855	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 7906	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_𝑜
56	54, 55	syl6eqbr 4622	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈ 1_𝑜)
57	56	ex 449	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
58	57	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
59	53, 58	syld 46	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_𝑜))
60	40, 45, 59	ectocld 7701	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_𝑜))
61	60	impr 647	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1_𝑜)
62	39, 61	sylan2b 491	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ≈ 1_𝑜)
63		en1b 7910	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_𝑜 ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 207	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (#‘𝑧) = (#‘{∪ 𝑧}))
66		vuniex 6852	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 13020	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (#‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (#‘{∪ 𝑧}) = 1
69	65, 68	syl6eq 2660	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (#‘𝑧) = 1)
70	69	sumeq2dv 14281	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71		ssrab2 3650	. . . . . . . . . . . 12 ⊢ {𝑢 ∈ 𝑌 ∣ ∀ℎ ∈ 𝑋 (ℎ ⊕ 𝑢) = 𝑢} ⊆ 𝑌
72	6, 71	eqsstri 3598	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
73		ssfi 8065	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
74	5, 72, 73	sylancl 693	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
75		hashcl 13009	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (#‘𝑍) ∈ ℕ₀)
76	74, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (#‘𝑍) ∈ ℕ₀)
77	76	nn0cnd 11230	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑍) ∈ ℂ)
78	77	mulid1d 9936	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑍) · 1) = (#‘𝑍))
79	6, 5	rabexd 4741	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ V)
80		inss2 3796	. . . . . . . . . . 11 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ 𝒫 𝑍
81		pwexg 4776	. . . . . . . . . . . 12 ⊢ (𝑍 ∈ Fin → 𝒫 𝑍 ∈ V)
82	74, 81	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝒫 𝑍 ∈ V)
83		ssexg 4732	. . . . . . . . . . 11 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ 𝒫 𝑍 ∧ 𝒫 𝑍 ∈ V) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ V)
84	80, 82, 83	sylancr 694	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ V)
85	7	relopabi 5167	. . . . . . . . . . . . . . . . 17 ⊢ Rel ∼
86		relssdmrn 5573	. . . . . . . . . . . . . . . . 17 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
87	85, 86	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
88		erdm 7639	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
89	21, 88	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ∼ = 𝑌)
90	89, 5	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom ∼ ∈ Fin)
91		errn 7651	. . . . . . . . . . . . . . . . . . 19 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
92	21, 91	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ran ∼ = 𝑌)
93	92, 5	eqeltrd 2688	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran ∼ ∈ Fin)
94		xpexg 6858	. . . . . . . . . . . . . . . . 17 ⊢ ((dom ∼ ∈ Fin ∧ ran ∼ ∈ Fin) → (dom ∼ × ran ∼ ) ∈ V)
95	90, 93, 94	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
96		ssexg 4732	. . . . . . . . . . . . . . . 16 ⊢ (( ∼ ⊆ (dom ∼ × ran ∼ ) ∧ (dom ∼ × ran ∼ ) ∈ V) → ∼ ∈ V)
97	87, 95, 96	sylancr 694	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∼ ∈ V)
98	97	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → ∼ ∈ V)
99		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
100	72, 99	sseldi 3566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
101		ecelqsg 7689	. . . . . . . . . . . . . 14 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
102	98, 100, 101	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
103	54, 102	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
104		snelpwi 4839	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
105	104	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
106	103, 105	elind 3760	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
107	106	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
108		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
109	80, 108	sseldi 3566	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ 𝒫 𝑍)
110	109	elpwid 4118	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ⊆ 𝑍)
111	64, 110	eqsstr3d 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
112	66	snss 4259	. . . . . . . . . . . 12 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
113	111, 112	sylibr 223	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
114	113	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) → ∪ 𝑧 ∈ 𝑍))
115		sneq 4135	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
116	115	eqeq2d 2620	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧}))
117	64, 116	syl5ibrcom 236	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
118	117	adantrl 748	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
119		unieq 4380	. . . . . . . . . . . . 13 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
120	49	unisn 4387	. . . . . . . . . . . . 13 ⊢ ∪ {𝑤} = 𝑤
121	119, 120	syl6req 2661	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧)
122	118, 121	impbid1 214	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤}))
123	122	ex 449	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤})))
124	79, 84, 107, 114, 123	en3d 7878	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
125		hashen 12997	. . . . . . . . . 10 ⊢ ((𝑍 ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin) → ((#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) ↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
126	74, 35, 125	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) ↔ 𝑍 ≈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
127	124, 126	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → (#‘𝑍) = (#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
128	127	oveq1d 6564	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑍) · 1) = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
129	78, 128	eqtr3d 2646	. . . . . 6 ⊢ (𝜑 → (#‘𝑍) = ((#‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
130	38, 70, 129	3eqtr4rd 2655	. . . . 5 ⊢ (𝜑 → (#‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧))
131	130	oveq1d 6564	. . . 4 ⊢ (𝜑 → ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(#‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)))
132	31, 32, 131	3eqtr4rd 2655	. . 3 ⊢ (𝜑 → ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (#‘𝑌))
133		hashcl 13009	. . . . . 6 ⊢ (𝑌 ∈ Fin → (#‘𝑌) ∈ ℕ₀)
134	5, 133	syl 17	. . . . 5 ⊢ (𝜑 → (#‘𝑌) ∈ ℕ₀)
135	134	nn0cnd 11230	. . . 4 ⊢ (𝜑 → (#‘𝑌) ∈ ℂ)
136		diffi 8077	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
137	23, 136	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
138		eldifi 3694	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
139	138, 30	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (#‘𝑧) ∈ ℂ)
140	137, 139	fsumcl 14311	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧) ∈ ℂ)
141	135, 77, 140	subaddd 10289	. . 3 ⊢ (𝜑 → (((#‘𝑌) − (#‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧) ↔ ((#‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧)) = (#‘𝑌)))
142	132, 141	mpbird 246	. 2 ⊢ (𝜑 → ((#‘𝑌) − (#‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(#‘𝑧))
143	8, 142	breqtrrd 4611	1 ⊢ (𝜑 → 𝑃 ∥ ((#‘𝑌) − (#‘𝑍)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 {cpr 4127 ∪ cuni 4372 class class class wbr 4583 {copab 4642 × cxp 5036 dom cdm 5038 ran crn 5039 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 1_𝑜c1o 7440 Er wer 7626 [cec 7627 / cqs 7628 ≈ cen 7838 Fincfn 7841 ℂcc 9813 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 ℕ₀cn0 11169 #chash 12979 Σcsu 14264 ∥ cdvds 14821 Basecbs 15695 GrpAct cga 17545 pGrp cpgp 17769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-eqg 17416 df-ga 17546 df-od 17771 df-pgp 17773

This theorem is referenced by: sylow2blem3 17860 sylow3lem6 17870

W3C validator