Theorem sylow1lem3 17838

Description: Lemma for sylow1 17841. 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 17836	. . . . . . 7 ⊢ (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
10	9	simpld 474	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ)
11		pcndvds 15408	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (#‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
12	1, 10, 11	syl2anc 691	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
13	9	simprd 478	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
14	13	oveq1d 6564	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑆)) + 1) = (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 6565	. . . . . 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 17837	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 17564	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 8144	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 207	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23		ssrab2 3650	. . . . . . . . 9 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} ⊆ 𝒫 𝑋
24	8, 23	eqsstri 3598	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
25		ssfi 8065	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
26	22, 24, 25	sylancl 693	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
27	20, 26	qshash 14398	. . . . . 6 ⊢ (𝜑 → (#‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
28	15, 27	breq12d 4596	. . . . 5 ⊢ (𝜑 → ((𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧)))
29	12, 28	mtbid 313	. . . 4 ⊢ (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
30		pwfi 8144	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
31	26, 30	sylib 207	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
32	20	qsss 7695	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
33		ssfi 8065	. . . . . . 7 ⊢ ((𝒫 𝑆 ∈ Fin ∧ (𝑆 / ∼ ) ⊆ 𝒫 𝑆) → (𝑆 / ∼ ) ∈ Fin)
34	31, 32, 33	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
36		prmnn 15226	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
37	1, 36	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
38	1, 10	pccld 15393	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) ∈ ℕ0)
39	13, 38	eqeltrrd 2689	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0)
40		peano2nn0 11210	. . . . . . . . 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 12892	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
43	42	nnzd 11357	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
44	43	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
45		erdm 7639	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
46	20, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
47		elqsn0 7703	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
48	46, 47	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
49	26	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
50	32	sselda 3568	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
51	50	elpwid 4118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
52		ssfi 8065	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Fin ∧ 𝑧 ⊆ 𝑆) → 𝑧 ∈ Fin)
53	49, 51, 52	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
54		hashnncl 13018	. . . . . . . . 9 ⊢ (𝑧 ∈ Fin → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
55	53, 54	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
56	48, 55	mpbird 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℕ)
57	56	adantlr 747	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℕ)
58	57	nnzd 11357	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (#‘𝑧) ∈ ℤ)
59		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (#‘𝑎) = (#‘𝑧))
60	59	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (#‘𝑎)) = (𝑃 pCnt (#‘𝑧)))
61	60	breq1d 4593	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
62	61	notbid 307	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
63	62	rspccva 3281	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
64	63	adantll 746	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
65	2	grpbn0 17274	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
66	3, 65	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
67		hashnncl 13018	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
68	4, 67	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
69	66, 68	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
70	1, 69	pccld 15393	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
71	70	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
72	5	nn0zd 11356	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
73	71, 72	zsubcld 11363	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
74	73	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
75	74	zred 11358	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℝ)
76	1	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
77	76, 57	pccld 15393	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℕ0)
78	77	nn0zd 11356	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℤ)
79	78	zred 11358	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (#‘𝑧)) ∈ ℝ)
80	75, 79	ltnled 10063	. . . . . . . 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 246	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) < (𝑃 pCnt (#‘𝑧)))
82		zltp1le 11304	. . . . . . . 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 691	. . . . . . 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 221	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (#‘𝑧)))
85	41	ad2antrr 758	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
86		pcdvdsb 15411	. . . . . . 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 1318	. . . . . 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 221	. . . . 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 14868	. . . 4 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(#‘𝑧))
90	29, 89	mtand 689	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
91		dfrex2 2979	. . 3 ⊢ (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
92	90, 91	sylibr 223	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
93		eqid 2610	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
94		fveq2 6103	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (#‘[𝑧] ∼ ) = (#‘𝑎))
95	94	oveq2d 6565	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (#‘[𝑧] ∼ )) = (𝑃 pCnt (#‘𝑎)))
96	95	breq1d 4593	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → ((𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
97	96	imbi1d 330	. . . 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 7669	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
99	98	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (#‘[𝑤] ∼ ) = (#‘[𝑧] ∼ ))
100	99	oveq2d 6565	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (#‘[𝑤] ∼ )) = (𝑃 pCnt (#‘[𝑧] ∼ )))
101	100	breq1d 4593	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
102	101	rspcev 3282	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
103	102	ex 449	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (#‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
104	103	adantl 481	. . . 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 7701	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (#‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
106	105	rexlimdva 3013	. 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   Er wer 7626  [cec 7627   / cqs 7628  Fincfn 7841  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ↑cexp 12722  #chash 12979  Σcsu 14264   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695  +_gcplusg 15768  Grpcgrp 17245   GrpAct cga 17545

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-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-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-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-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-ga 17546

This theorem is referenced by:  sylow1  17841