Theorem sylow1lem3 19462

Description: Lemma for sylow1 19465. One of the orbits of the group action has p-adic valuation less than the prime count of the set 𝑆. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
sylow1.x	⊢ 𝑋 = (Base‘𝐺)
sylow1.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow1.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow1.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow1.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sylow1.d	⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))
sylow1lem.a	⊢ + = (+g‘𝐺)
sylow1lem.s	⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
sylow1lem.m	⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
sylow1lem3.1	⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}

Assertion

Ref	Expression
sylow1lem3	⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))

Distinct variable groups:   𝑔,𝑠,𝑥,𝑦,𝑧,𝑤   𝑆,𝑔   𝑥,𝑤,𝑦,𝑧,𝑆   𝑔,𝑁   𝑤,𝑠,𝑁,𝑥,𝑦,𝑧   𝑔,𝑋,𝑠,𝑤,𝑥,𝑦,𝑧   + ,𝑠,𝑤,𝑥,𝑦,𝑧   𝑤, ∼ ,𝑧   ⊕ ,𝑔,𝑤,𝑥,𝑦,𝑧   𝑔,𝐺,𝑠,𝑥,𝑦,𝑧   𝑃,𝑔,𝑠,𝑤,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤,𝑔,𝑠)   + (𝑔)   ⊕ (𝑠)   ∼ (𝑥,𝑦,𝑔,𝑠)   𝑆(𝑠)   𝐺(𝑤)

Proof of Theorem sylow1lem3

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow1.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ ℙ)
2		sylow1.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
3		sylow1.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Grp)
4		sylow1.f	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
5		sylow1.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
6		sylow1.d	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))
7		sylow1lem.a	. . . . . . . 8 ⊢ + = (+g‘𝐺)
8		sylow1lem.s	. . . . . . . 8 ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
9	2, 3, 4, 1, 5, 6, 7, 8	sylow1lem1 19460	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
10	9	simpld 495	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ)
11		pcndvds 16795	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
12	1, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
13	9	simprd 496	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
14	13	oveq1d 7420	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑆)) + 1) = (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 7421	. . . . . 6 ⊢ (𝜑 → (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) = (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)))
16		sylow1lem.m	. . . . . . . . 9 ⊢ ⊕ = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑆 ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥 + 𝑧)))
17	2, 3, 4, 1, 5, 6, 7, 8, 16	sylow1lem2 19461	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 19166	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 9174	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23	8	ssrab3 4079	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
24		ssfi 9169	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
25	22, 23, 24	sylancl 586	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
26	20, 25	qshash 15769	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
27	15, 26	breq12d 5160	. . . . 5 ⊢ (𝜑 → ((𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧)))
28	12, 27	mtbid 323	. . . 4 ⊢ (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
29		pwfi 9174	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
30	25, 29	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
31	20	qsss 8768	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
32	30, 31	ssfid 9263	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
33	32	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
34		prmnn 16607	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
35	1, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
36	1, 10	pccld 16779	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) ∈ ℕ0)
37	13, 36	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0)
38		peano2nn0 12508	. . . . . . . . 9 ⊢ (((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0 → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
39	37, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
40	35, 39	nnexpcld 14204	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
41	40	nnzd 12581	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
42	41	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
43		erdm 8709	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
44	20, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
45		elqsn0 8776	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
46	44, 45	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
47	25	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
48	31	sselda 3981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
49	48	elpwid 4610	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
50	47, 49	ssfid 9263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
51		hashnncl 14322	. . . . . . . . 9 ⊢ (𝑧 ∈ Fin → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
52	50, 51	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
53	46, 52	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
54	53	adantlr 713	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
55	54	nnzd 12581	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℤ)
56		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (♯‘𝑎) = (♯‘𝑧))
57	56	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (♯‘𝑎)) = (𝑃 pCnt (♯‘𝑧)))
58	57	breq1d 5157	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
59	58	notbid 317	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
60	59	rspccva 3611	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
61	60	adantll 712	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
62	2	grpbn0 18847	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
63	3, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
64		hashnncl 14322	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
65	4, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
66	63, 65	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
67	1, 66	pccld 16779	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
68	67	nn0zd 12580	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
69	5	nn0zd 12580	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	68, 69	zsubcld 12667	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
71	70	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
72	71	zred 12662	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℝ)
73	1	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
74	73, 54	pccld 16779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℕ0)
75	74	nn0zd 12580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℤ)
76	75	zred 12662	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℝ)
77	72, 76	ltnled 11357	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)) ↔ ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
78	61, 77	mpbird 256	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)))
79		zltp1le 12608	. . . . . . . 8 ⊢ ((((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ ∧ (𝑃 pCnt (♯‘𝑧)) ∈ ℤ) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)) ↔ (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧))))
80	71, 75, 79	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)) ↔ (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧))))
81	78, 80	mpbid 231	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧)))
82	39	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
83		pcdvdsb 16798	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑧) ∈ ℤ ∧ (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0) → ((((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧)) ↔ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ (♯‘𝑧)))
84	73, 55, 82, 83	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧)) ↔ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ (♯‘𝑧)))
85	81, 84	mpbid 231	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ (♯‘𝑧))
86	33, 42, 55, 85	fsumdvds 16247	. . . 4 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
87	28, 86	mtand 814	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
88		dfrex2 3073	. . 3 ⊢ (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
89	87, 88	sylibr 233	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
90		eqid 2732	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
91		fveq2 6888	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (♯‘[𝑧] ∼ ) = (♯‘𝑎))
92	91	oveq2d 7421	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (♯‘[𝑧] ∼ )) = (𝑃 pCnt (♯‘𝑎)))
93	92	breq1d 5157	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
94	93	imbi1d 341	. . . 4 ⊢ ([𝑧] ∼ = 𝑎 → (((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ↔ ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))))
95		eceq1 8737	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
96	95	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘[𝑧] ∼ ))
97	96	oveq2d 7421	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (♯‘[𝑤] ∼ )) = (𝑃 pCnt (♯‘[𝑧] ∼ )))
98	97	breq1d 5157	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
99	98	rspcev 3612	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
100	99	ex 413	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
101	100	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
102	90, 94, 101	ectocld 8774	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
103	102	rexlimdva 3155	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
104	89, 103	mpd 15	1 ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3432   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {cpr 4629   class class class wbr 5147  {copab 5209   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   Er wer 8696  [cec 8697   / cqs 8698  Fincfn 8935  1c1 11107   + caddc 11109   < clt 11244   ≤ cle 11245   − cmin 11440  ℕcn 12208  ℕ₀cn0 12468  ℤcz 12554  ↑cexp 14023  ♯chash 14286  Σcsu 15628   ∥ cdvds 16193  ℙcprime 16604   pCnt cpc 16765  Basecbs 17140  +_gcplusg 17193  Grpcgrp 18815   GrpAct cga 19147

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-er 8699  df-ec 8701  df-qs 8705  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-fz 13481  df-fzo 13624  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-fac 14230  df-bc 14259  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-dvds 16194  df-gcd 16432  df-prm 16605  df-pc 16766  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-ga 19148

This theorem is referenced by:  sylow1  19465