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Theorem sylow2a 19866

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 19865	. 2 ⊢ (𝜑 → 𝑃 ∥ Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
9		inass 4127	. . . . . . 7 ⊢ (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
10		disjdif 4429	. . . . . . . 8 ⊢ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) = ∅
11	10	ineq2i 4117	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ (𝒫 𝑍 ∩ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))) = ((𝑌 / ∼ ) ∩ ∅)
12		in0 4348	. . . . . . 7 ⊢ ((𝑌 / ∼ ) ∩ ∅) = ∅
13	9, 11, 12	3eqtri 2790	. . . . . 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 2772	. . . . . 6 ⊢ (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍))
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) = (((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∪ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)))
18		pwfi 9341	. . . . . . 7 ⊢ (𝑌 ∈ Fin ↔ 𝒫 𝑌 ∈ Fin)
19	5, 18	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑌 ∈ Fin)
20	7, 1	gaorber 19555	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
21	2, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑌)
22	21	qsss 8827	. . . . . 6 ⊢ (𝜑 → (𝑌 / ∼ ) ⊆ 𝒫 𝑌)
23	19, 22	ssfid 9291	. . . . 5 ⊢ (𝜑 → (𝑌 / ∼ ) ∈ Fin)
24	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑌 ∈ Fin)
25	22	sselda 3913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ 𝒫 𝑌)
26	25	elpwid 4569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ⊆ 𝑌)
27	24, 26	ssfid 9291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → 𝑧 ∈ Fin)
28		hashcl 14530	. . . . . . 7 ⊢ (𝑧 ∈ Fin → (♯‘𝑧) ∈ ℕ₀)
29	27, 28	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℕ₀)
30	29	nn0cnd 12699	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (♯‘𝑧) ∈ ℂ)
31	14, 17, 23, 30	fsumsplit 15939	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
32	21, 5	qshash 16026	. . . 4 ⊢ (𝜑 → (♯‘𝑌) = Σ𝑧 ∈ (𝑌 / ∼ )(♯‘𝑧))
33		inss1 4136	. . . . . . . 8 ⊢ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )
34		ssfi 9219	. . . . . . . 8 ⊢ (((𝑌 / ∼ ) ∈ Fin ∧ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ⊆ (𝑌 / ∼ )) → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
35	23, 33, 34	sylancl 598	. . . . . . 7 ⊢ (𝜑 → ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin)
36		ax-1cn 11288	. . . . . . 7 ⊢ 1 ∈ ℂ
37		fsumconst 15988	. . . . . . 7 ⊢ ((((𝑌 / ∼ ) ∩ 𝒫 𝑍) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
38	35, 36, 37	sylancl 598	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1 = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
39		elin 4096	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍) ↔ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍))
40		eqid 2763	. . . . . . . . . . . . 13 ⊢ (𝑌 / ∼ ) = (𝑌 / ∼ )
41		sseq1 3938	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ⊆ 𝑍))
42		velpw 4565	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝒫 𝑍 ↔ 𝑧 ⊆ 𝑍)
43	41, 42	bitr4di 292	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ⊆ 𝑍 ↔ 𝑧 ∈ 𝒫 𝑍))
44		breq1 5109	. . . . . . . . . . . . . 14 ⊢ ([𝑤] ∼ = 𝑧 → ([𝑤] ∼ ≈ 1_o ↔ 𝑧 ≈ 1_o))
45	43, 44	imbi12d 347	. . . . . . . . . . . . 13 ⊢ ([𝑤] ∼ = 𝑧 → (([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_o) ↔ (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_o)))
46	21	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ∼ Er 𝑌)
47		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
48	46, 47	erref 8769	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∼ 𝑤)
49		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑤 ∈ V
50	49, 49	elec 8795	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ [𝑤] ∼ ↔ 𝑤 ∼ 𝑤)
51	48, 50	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ [𝑤] ∼ )
52		ssel 3907	. . . . . . . . . . . . . . 15 ⊢ ([𝑤] ∼ ⊆ 𝑍 → (𝑤 ∈ [𝑤] ∼ → 𝑤 ∈ 𝑍))
53	51, 52	syl5com 32	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → 𝑤 ∈ 𝑍))
54	1, 2, 3, 4, 5, 6, 7	sylow2alem1 19864	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ = {𝑤})
55	49	ensn1 9079	. . . . . . . . . . . . . . . . 17 ⊢ {𝑤} ≈ 1_o
56	54, 55	eqbrtrdi 5148	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ≈ 1_o)
57	56	ex 418	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_o))
58	57	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → (𝑤 ∈ 𝑍 → [𝑤] ∼ ≈ 1_o))
59	53, 58	syld 48	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑌) → ([𝑤] ∼ ⊆ 𝑍 → [𝑤] ∼ ≈ 1_o))
60	40, 45, 59	ectocld 8834	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑌 / ∼ )) → (𝑧 ∈ 𝒫 𝑍 → 𝑧 ≈ 1_o))
61	60	impr 460	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑧 ∈ (𝑌 / ∼ ) ∧ 𝑧 ∈ 𝒫 𝑍)) → 𝑧 ≈ 1_o)
62	39, 61	sylan2b 606	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ≈ 1_o)
63		en1b 9083	. . . . . . . . . 10 ⊢ (𝑧 ≈ 1_o ↔ 𝑧 = {∪ 𝑧})
64	62, 63	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 = {∪ 𝑧})
65	64	fveq2d 6915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = (♯‘{∪ 𝑧}))
66		vuniex 7783	. . . . . . . . 9 ⊢ ∪ 𝑧 ∈ V
67		hashsng 14543	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ V → (♯‘{∪ 𝑧}) = 1)
68	66, 67	ax-mp 5	. . . . . . . 8 ⊢ (♯‘{∪ 𝑧}) = 1
69	65, 68	eqtrdi 2814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (♯‘𝑧) = 1)
70	69	sumeq2dv 15901	. . . . . 6 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)1)
71	6	ssrab3 4010	. . . . . . . . . . 11 ⊢ 𝑍 ⊆ 𝑌
72		ssfi 9219	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Fin ∧ 𝑍 ⊆ 𝑌) → 𝑍 ∈ Fin)
73	5, 71, 72	sylancl 598	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ Fin)
74		hashcl 14530	. . . . . . . . . 10 ⊢ (𝑍 ∈ Fin → (♯‘𝑍) ∈ ℕ₀)
75	73, 74	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝑍) ∈ ℕ₀)
76	75	nn0cnd 12699	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) ∈ ℂ)
77	76	mulridd 11356	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = (♯‘𝑍))
78	6, 5	rabexd 5306	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ V)
79		eqid 2763	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑍 ↦ {𝑤}) = (𝑤 ∈ 𝑍 ↦ {𝑤})
80	7	relopabi 5806	. . . . . . . . . . . . . . 15 ⊢ Rel ∼
81		relssdmrn 6290	. . . . . . . . . . . . . . 15 ⊢ (Rel ∼ → ∼ ⊆ (dom ∼ × ran ∼ ))
82	80, 81	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ∼ ⊆ (dom ∼ × ran ∼ )
83		erdm 8759	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → dom ∼ = 𝑌)
84	21, 83	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ∼ = 𝑌)
85	84, 5	eqeltrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ∼ ∈ Fin)
86		errn 8771	. . . . . . . . . . . . . . . . 17 ⊢ ( ∼ Er 𝑌 → ran ∼ = 𝑌)
87	21, 86	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran ∼ = 𝑌)
88	87, 5	eqeltrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ran ∼ ∈ Fin)
89	85, 88	xpexd 7792	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (dom ∼ × ran ∼ ) ∈ V)
90		ssexg 5286	. . . . . . . . . . . . . 14 ⊢ (( ∼ ⊆ (dom ∼ × ran ∼ ) ∧ (dom ∼ × ran ∼ ) ∈ V) → ∼ ∈ V)
91	82, 89, 90	sylancr 599	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∼ ∈ V)
92		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑍)
93	71, 92	sselid 3911	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → 𝑤 ∈ 𝑌)
94		ecelqsw 8820	. . . . . . . . . . . . 13 ⊢ (( ∼ ∈ V ∧ 𝑤 ∈ 𝑌) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
95	91, 93, 94	syl2an2r 698	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → [𝑤] ∼ ∈ (𝑌 / ∼ ))
96	54, 95	eqeltrrd 2865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ (𝑌 / ∼ ))
97		snelpwi 5417	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑍 → {𝑤} ∈ 𝒫 𝑍)
98	97	adantl 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ 𝒫 𝑍)
99	96, 98	elind 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑍) → {𝑤} ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
100		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))
101	100	elin2d 4105	. . . . . . . . . . . . 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 3948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → {∪ 𝑧} ⊆ 𝑍)
104	66	snss 4745	. . . . . . . . . . 11 ⊢ (∪ 𝑧 ∈ 𝑍 ↔ {∪ 𝑧} ⊆ 𝑍)
105	103, 104	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → ∪ 𝑧 ∈ 𝑍)
106		sneq 4594	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ∪ 𝑧 → {𝑤} = {∪ 𝑧})
107	106	eqeq2d 2774	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∪ 𝑧 → (𝑧 = {𝑤} ↔ 𝑧 = {∪ 𝑧}))
108	64, 107	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
109	108	adantrl 729	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 → 𝑧 = {𝑤}))
110		unieq 4881	. . . . . . . . . . . 12 ⊢ (𝑧 = {𝑤} → ∪ 𝑧 = ∪ {𝑤})
111		unisnv 4890	. . . . . . . . . . . 12 ⊢ ∪ {𝑤} = 𝑤
112	110, 111	eqtr2di 2815	. . . . . . . . . . 11 ⊢ (𝑧 = {𝑤} → 𝑤 = ∪ 𝑧)
113	109, 112	impbid1 228	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑍 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍))) → (𝑤 = ∪ 𝑧 ↔ 𝑧 = {𝑤}))
114	79, 99, 105, 113	f1o2d 7702	. . . . . . . . 9 ⊢ (𝜑 → (𝑤 ∈ 𝑍 ↦ {𝑤}):𝑍⟶1-1onto((𝑌 / ∼ ) ∩ 𝒫 𝑍))
115	78, 114	hasheqf1od 14527	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝑍) = (♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)))
116	115	oveq1d 7461	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑍) · 1) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
117	77, 116	eqtr3d 2800	. . . . . 6 ⊢ (𝜑 → (♯‘𝑍) = ((♯‘((𝑌 / ∼ ) ∩ 𝒫 𝑍)) · 1))
118	38, 70, 117	3eqtr4rd 2809	. . . . 5 ⊢ (𝜑 → (♯‘𝑍) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧))
119	118	oveq1d 7461	. . . 4 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (Σ𝑧 ∈ ((𝑌 / ∼ ) ∩ 𝒫 𝑍)(♯‘𝑧) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)))
120	31, 32, 119	3eqtr4rd 2809	. . 3 ⊢ (𝜑 → ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌))
121		hashcl 14530	. . . . . 6 ⊢ (𝑌 ∈ Fin → (♯‘𝑌) ∈ ℕ₀)
122	5, 121	syl 18	. . . . 5 ⊢ (𝜑 → (♯‘𝑌) ∈ ℕ₀)
123	122	nn0cnd 12699	. . . 4 ⊢ (𝜑 → (♯‘𝑌) ∈ ℂ)
124		diffi 9221	. . . . . 6 ⊢ ((𝑌 / ∼ ) ∈ Fin → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
125	23, 124	syl 18	. . . . 5 ⊢ (𝜑 → ((𝑌 / ∼ ) ∖ 𝒫 𝑍) ∈ Fin)
126		eldifi 4070	. . . . . 6 ⊢ (𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍) → 𝑧 ∈ (𝑌 / ∼ ))
127	126, 30	sylan2 605	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)) → (♯‘𝑧) ∈ ℂ)
128	125, 127	fsumcl 15931	. . . 4 ⊢ (𝜑 → Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ∈ ℂ)
129	123, 76, 128	subaddd 11717	. . 3 ⊢ (𝜑 → (((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧) ↔ ((♯‘𝑍) + Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧)) = (♯‘𝑌)))
130	120, 129	mpbird 260	. 2 ⊢ (𝜑 → ((♯‘𝑌) − (♯‘𝑍)) = Σ𝑧 ∈ ((𝑌 / ∼ ) ∖ 𝒫 𝑍)(♯‘𝑧))
131	8, 130	breqtrrd 5137	1 ⊢ (𝜑 → 𝑃 ∥ ((♯‘𝑌) − (♯‘𝑍)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 𝒫 cpw 4560 {csn 4588 {cpr 4590 ∪ cuni 4870 class class class wbr 5106 {copab 5171 ↦ cmpt 5190 × cxp 5654 dom cdm 5656 ran crn 5657 Rel wrel 5661 ‘cfv 6561 (class class class)co 7446 1_oc1o 8500 Er wer 8745 [cec 8746 / cqs 8747 ≈ cen 9001 Fincfn 9004 ℂcc 11228 1c1 11231 + caddc 11233 · cmul 11235 − cmin 11571 ℕ₀cn0 12636 ♯chash 14504 Σcsu 15885 ∥ cdvds 16454 Basecbs 17419 GrpAct cga 19536 pGrp cpgp 19773

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-inf2 9672 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-disj 5074 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8748 df-ec 8750 df-qs 8754 df-map 8880 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-sup 9464 df-inf 9465 df-oi 9534 df-dju 10012 df-card 10050 df-acn 10053 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-n0 12637 df-xnn0 12710 df-z 12724 df-uz 12996 df-q 13106 df-rp 13151 df-fz 13670 df-fzo 13819 df-fl 13962 df-mod 14040 df-seq 14175 df-exp 14235 df-fac 14448 df-bc 14477 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-clim 15687 df-sum 15886 df-dvds 16455 df-gcd 16697 df-prm 16879 df-pc 17047 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-0g 17644 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-grp 19179 df-minusg 19180 df-sbg 19181 df-mulg 19310 df-subg 19365 df-eqg 19367 df-ga 19537 df-od 19775 df-pgp 19777

This theorem is used by: sylow2blem3 19869 sylow3lem6 19879

W3C validator