Theorem sylow1lem3 18015

Description: Lemma for sylow1 18018. 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 18013	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
10	9	simpld 475	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ)
11		pcndvds 15570	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
12	1, 10, 11	syl2anc 693	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
13	9	simprd 479	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
14	13	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑆)) + 1) = (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 6666	. . . . . 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 18014	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 17741	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 8261	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 208	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23		ssrab2 3687	. . . . . . . . 9 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} ⊆ 𝒫 𝑋
24	8, 23	eqsstri 3635	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
25		ssfi 8180	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
26	22, 24, 25	sylancl 694	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
27	20, 26	qshash 14559	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
28	15, 27	breq12d 4666	. . . . 5 ⊢ (𝜑 → ((𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧)))
29	12, 28	mtbid 314	. . . 4 ⊢ (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
30		pwfi 8261	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
31	26, 30	sylib 208	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
32	20	qsss 7808	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
33		ssfi 8180	. . . . . . 7 ⊢ ((𝒫 𝑆 ∈ Fin ∧ (𝑆 / ∼ ) ⊆ 𝒫 𝑆) → (𝑆 / ∼ ) ∈ Fin)
34	31, 32, 33	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
35	34	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
36		prmnn 15388	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
37	1, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
38	1, 10	pccld 15555	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) ∈ ℕ0)
39	13, 38	eqeltrrd 2702	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0)
40		peano2nn0 11333	. . . . . . . . 9 ⊢ (((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0 → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
41	39, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
42	37, 41	nnexpcld 13030	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
43	42	nnzd 11481	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
44	43	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
45		erdm 7752	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
46	20, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
47		elqsn0 7816	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
48	46, 47	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
49	26	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
50	32	sselda 3603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
51	50	elpwid 4170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
52		ssfi 8180	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Fin ∧ 𝑧 ⊆ 𝑆) → 𝑧 ∈ Fin)
53	49, 51, 52	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
54		hashnncl 13157	. . . . . . . . 9 ⊢ (𝑧 ∈ Fin → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
56	48, 55	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℕ)
57	56	adantlr 751	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℕ)
58	57	nnzd 11481	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℤ)
59		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (#‘𝑎) = (#‘𝑧))
60	59	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (#‘𝑎)) = (𝑃 pCnt (#‘𝑧)))
61	60	breq1d 4663	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
62	61	notbid 308	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
63	62	rspccva 3308	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
64	63	adantll 750	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
65	2	grpbn0 17451	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
66	3, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
67		hashnncl 13157	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
68	4, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
69	66, 68	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
70	1, 69	pccld 15555	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
71	70	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
72	5	nn0zd 11480	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
73	71, 72	zsubcld 11487	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
74	73	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
75	74	zred 11482	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℝ)
76	1	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
77	76, 57	pccld 15555	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℕ0)
78	77	nn0zd 11480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℤ)
79	78	zred 11482	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℝ)
80	75, 79	ltnled 10184	. . . . . . . 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)))
81	64, 80	mpbird 247	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) < (𝑃 pCnt (#‘𝑧)))
82		zltp1le 11427	. . . . . . . 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))))
83	74, 78, 82	syl2anc 693	. . . . . . 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))))
84	81, 83	mpbid 222	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (#‘𝑧)))
85	41	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
86		pcdvdsb 15573	. . . . . . 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)))
87	76, 58, 85, 86	syl3anc 1326	. . . . . 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)))
88	84, 87	mpbid 222	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ (#‘𝑧))
89	35, 44, 58, 88	fsumdvds 15030	. . . 4 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
90	29, 89	mtand 691	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
91		dfrex2 2996	. . 3 ⊢ (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
92	90, 91	sylibr 224	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
93		eqid 2622	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
94		fveq2 6191	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (#‘[𝑧] ∼ ) = (#‘𝑎))
95	94	oveq2d 6666	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (#‘[𝑧] ∼ )) = (𝑃 pCnt (#‘𝑎)))
96	95	breq1d 4663	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → ((𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
97	96	imbi1d 331	. . . 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))))
98		eceq1 7782	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
99	98	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (#‘[𝑤] ∼ ) = (#‘[𝑧] ∼ ))
100	99	oveq2d 6666	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (#‘[𝑤] ∼ )) = (𝑃 pCnt (#‘[𝑧] ∼ )))
101	100	breq1d 4663	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
102	101	rspcev 3309	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
103	102	ex 450	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
104	103	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
105	93, 97, 104	ectocld 7814	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
106	105	rexlimdva 3031	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
107	92, 106	mpd 15	1 ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  {cpr 4179   class class class wbr 4653  {copab 4712   ↦ cmpt 4729  dom cdm 5114  ran crn 5115  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   Er wer 7739  [cec 7740   / cqs 7741  Fincfn 7955  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ↑cexp 12860  #chash 13117  Σcsu 14416   ∥ cdvds 14983  ℙcprime 15385   pCnt cpc 15541  Basecbs 15857  +_gcplusg 15941  Grpcgrp 17422   GrpAct cga 17722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ga 17723

This theorem is referenced by:  sylow1  18018