Theorem sylow1lem3 19669

Description: Lemma for sylow1 19672. 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 19667	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
10	9	simpld 499	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ)
11		pcndvds 16925	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
12	1, 10, 11	syl2anc 595	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
13	9	simprd 500	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
14	13	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑆)) + 1) = (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 7427	. . . . . 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 19668	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 19377	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 9277	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23	8	ssrab3 4044	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
24		ssfi 9156	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
25	22, 23, 24	sylancl 597	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
26	20, 25	qshash 15878	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
27	15, 26	breq12d 5126	. . . . 5 ⊢ (𝜑 → ((𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧)))
28	12, 27	mtbid 327	. . . 4 ⊢ (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
29		pwfi 9277	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
30	25, 29	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
31	20	qsss 8772	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
32	30, 31	ssfid 9228	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
33	32	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
34		prmnn 16731	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
35	1, 34	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
36	1, 10	pccld 16909	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) ∈ ℕ0)
37	13, 36	eqeltrrd 2870	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0)
38		peano2nn0 12543	. . . . . . . . 9 ⊢ (((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0 → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
39	37, 38	syl 18	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
40	35, 39	nnexpcld 14280	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
41	40	nnzd 12616	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
42	41	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
43		erdm 8704	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
44	20, 43	syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
45		elqsn0 8781	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
46	44, 45	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
47	25	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
48	31	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
49	48	elpwid 4576	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
50	47, 49	ssfid 9228	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
51		hashnncl 14401	. . . . . . . . 9 ⊢ (𝑧 ∈ Fin → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
52	50, 51	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
53	46, 52	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
54	53	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
55	54	nnzd 12616	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℤ)
56		fveq2 6882	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (♯‘𝑎) = (♯‘𝑧))
57	56	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (♯‘𝑎)) = (𝑃 pCnt (♯‘𝑧)))
58	57	breq1d 5123	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
59	58	notbid 321	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
60	59	rspccva 3589	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
61	60	adantll 726	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
62	2	grpbn0 19032	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
63	3, 62	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
64		hashnncl 14401	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
65	4, 64	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
66	63, 65	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
67	1, 66	pccld 16909	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
68	67	nn0zd 12615	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
69	5	nn0zd 12615	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	68, 69	zsubcld 12704	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
71	70	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
72	71	zred 12699	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℝ)
73	1	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
74	73, 54	pccld 16909	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℕ0)
75	74	nn0zd 12615	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℤ)
76	75	zred 12699	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℝ)
77	72, 76	ltnled 11356	. . . . . . . 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 260	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)))
79		zltp1le 12643	. . . . . . . 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 595	. . . . . . 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 235	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧)))
82	39	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
83		pcdvdsb 16928	. . . . . . 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 1396	. . . . . 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 235	. . . . 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 16365	. . . 4 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
87	28, 86	mtand 827	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
88		dfrex2 3098	. . 3 ⊢ (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
89	87, 88	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
90		eqid 2769	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
91		fveq2 6882	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (♯‘[𝑧] ∼ ) = (♯‘𝑎))
92	91	oveq2d 7427	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (♯‘[𝑧] ∼ )) = (𝑃 pCnt (♯‘𝑎)))
93	92	breq1d 5123	. . . . 5 ⊢ ([𝑧] ∼ = 𝑎 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
94	93	imbi1d 344	. . . 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 8733	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
96	95	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘[𝑧] ∼ ))
97	96	oveq2d 7427	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (♯‘[𝑤] ∼ )) = (𝑃 pCnt (♯‘[𝑧] ∼ )))
98	97	breq1d 5123	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
99	98	rspcev 3590	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
100	99	ex 417	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
101	100	adantl 486	. . . 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 8779	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
103	102	rexlimdva 3172	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
104	89, 103	mpd 16	1 ⊢ (𝜑 → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {cpr 4596   class class class wbr 5113  {copab 5177   ↦ cmpt 5196  dom cdm 5662  ran crn 5663  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   Er wer 8690  [cec 8691   / cqs 8692  Fincfn 8942  1c1 11100   + caddc 11102   < clt 11242   ≤ cle 11243   − cmin 11440  ℕcn 12232  ℕ₀cn0 12503  ℤcz 12590  ↑cexp 14096  ♯chash 14365  Σcsu 15736   ∥ cdvds 16309  ℙcprime 16728   pCnt cpc 16895  Basecbs 17268  +_gcplusg 17309  Grpcgrp 18999   GrpAct cga 19358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-er 8693  df-ec 8695  df-qs 8699  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-fz 13535  df-fzo 13682  df-fl 13824  df-mod 13902  df-seq 14037  df-exp 14097  df-fac 14309  df-bc 14338  df-hash 14366  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-clim 15538  df-sum 15737  df-dvds 16310  df-gcd 16552  df-prm 16729  df-pc 16896  df-0g 17493  df-mgm 18697  df-sgrp 18776  df-mnd 18792  df-grp 19002  df-minusg 19003  df-ga 19359

This theorem is referenced by:  sylow1  19672