Theorem sylow1lem3 19823

Description: Lemma for sylow1 19826. 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 19821	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
10	9	simpld 500	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ)
11		pcndvds 17053	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
12	1, 10, 11	syl2anc 596	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
13	9	simprd 501	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
14	13	oveq1d 7439	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑆)) + 1) = (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 7440	. . . . . 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 19822	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 19531	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 9317	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23	8	ssrab3 4006	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
24		ssfi 9195	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
25	22, 23, 24	sylancl 598	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
26	20, 25	qshash 16003	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
27	15, 26	breq12d 5115	. . . . 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 9317	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
30	25, 29	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
31	20	qsss 8803	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
32	30, 31	ssfid 9267	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
33	32	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
34		prmnn 16858	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
35	1, 34	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
36	1, 10	pccld 17037	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) ∈ ℕ0)
37	13, 36	eqeltrrd 2861	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0)
38		peano2nn0 12653	. . . . . . . . 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 14396	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
41	40	nnzd 12726	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
42	41	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
43		erdm 8735	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
44	20, 43	syl 18	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
45		elqsn0 8812	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
46	44, 45	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
47	25	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
48	31	sselda 3909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
49	48	elpwid 4565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
50	47, 49	ssfid 9267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
51		hashnncl 14517	. . . . . . . . 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 728	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
55	54	nnzd 12726	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℤ)
56		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (♯‘𝑎) = (♯‘𝑧))
57	56	oveq2d 7440	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (♯‘𝑎)) = (𝑃 pCnt (♯‘𝑧)))
58	57	breq1d 5112	. . . . . . . . . . 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 3575	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
61	60	adantll 727	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
62	2	grpbn0 19186	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
63	3, 62	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
64		hashnncl 14517	. . . . . . . . . . . . . . . 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 17037	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
68	67	nn0zd 12725	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
69	5	nn0zd 12725	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	68, 69	zsubcld 12815	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
71	70	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
72	71	zred 12810	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℝ)
73	1	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
74	73, 54	pccld 17037	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℕ0)
75	74	nn0zd 12725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℤ)
76	75	zred 12810	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℝ)
77	72, 76	ltnled 11464	. . . . . . . 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 12753	. . . . . . . 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 596	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
83		pcdvdsb 17056	. . . . . . 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 1398	. . . . . 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 16487	. . . 4 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
87	28, 86	mtand 828	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
88		dfrex2 3089	. . 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 2760	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
91		fveq2 6890	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (♯‘[𝑧] ∼ ) = (♯‘𝑎))
92	91	oveq2d 7440	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (♯‘[𝑧] ∼ )) = (𝑃 pCnt (♯‘𝑎)))
93	92	breq1d 5112	. . . . 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 8764	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
96	95	fveq2d 6894	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘[𝑧] ∼ ))
97	96	oveq2d 7440	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (♯‘[𝑤] ∼ )) = (𝑃 pCnt (♯‘[𝑧] ∼ )))
98	97	breq1d 5112	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
99	98	rspcev 3576	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
100	99	ex 418	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
101	100	adantl 487	. . . 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 8810	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
103	102	rexlimdva 3163	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412   ⊆ wss 3896  ∅c0 4277  𝒫 cpw 4556  {cpr 4586   class class class wbr 5102  {copab 5166   ↦ cmpt 5185  dom cdm 5651  ran crn 5652  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426   Er wer 8721  [cec 8722   / cqs 8723  Fincfn 8980  1c1 11208   + caddc 11210   < clt 11350   ≤ cle 11351   − cmin 11548  ℕcn 12342  ℕ₀cn0 12613  ℤcz 12700  ↑cexp 14211  ♯chash 14481  Σcsu 15862   ∥ cdvds 16431  ℙcprime 16855   pCnt cpc 17023  Basecbs 17396  +_gcplusg 17437  Grpcgrp 19153   GrpAct cga 19512

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284  ax-pre-sup 11285

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-disj 5070  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-2o 8484  df-oadd 8487  df-er 8724  df-ec 8726  df-qs 8730  df-map 8856  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-sup 9441  df-inf 9442  df-oi 9511  df-dju 9989  df-card 10027  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-div 11981  df-nn 12343  df-2 12412  df-3 12413  df-n0 12614  df-z 12701  df-uz 12973  df-q 13083  df-rp 13128  df-fz 13647  df-fzo 13796  df-fl 13939  df-mod 14017  df-seq 14152  df-exp 14212  df-fac 14425  df-bc 14454  df-hash 14482  df-cj 15275  df-re 15276  df-im 15277  df-sqrt 15411  df-abs 15412  df-clim 15664  df-sum 15863  df-dvds 16432  df-gcd 16674  df-prm 16856  df-pc 17024  df-0g 17621  df-mgm 18825  df-sgrp 18917  df-mnd 18933  df-grp 19156  df-minusg 19157  df-ga 19513

This theorem is used by:  sylow1  19826