Theorem sylow1lem3 19541

Description: Lemma for sylow1 19544. 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 19539	. . . . . . 7 ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
10	9	simpld 494	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ)
11		pcndvds 16806	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (♯‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
12	1, 10, 11	syl2anc 585	. . . . 5 ⊢ (𝜑 → ¬ (𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆))
13	9	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
14	13	oveq1d 7383	. . . . . . 7 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑆)) + 1) = (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1))
15	14	oveq2d 7384	. . . . . 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 19540	. . . . . . . 8 ⊢ (𝜑 → ⊕ ∈ (𝐺 GrpAct 𝑆))
18		sylow1lem3.1	. . . . . . . . 9 ⊢ ∼ = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}
19	18, 2	gaorber 19249	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑆) → ∼ Er 𝑆)
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → ∼ Er 𝑆)
21		pwfi 9231	. . . . . . . . 9 ⊢ (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
22	4, 21	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝒫 𝑋 ∈ Fin)
23	8	ssrab3 4036	. . . . . . . 8 ⊢ 𝑆 ⊆ 𝒫 𝑋
24		ssfi 9109	. . . . . . . 8 ⊢ ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
25	22, 23, 24	sylancl 587	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Fin)
26	20, 25	qshash 15762	. . . . . 6 ⊢ (𝜑 → (♯‘𝑆) = Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
27	15, 26	breq12d 5113	. . . . 5 ⊢ (𝜑 → ((𝑃↑((𝑃 pCnt (♯‘𝑆)) + 1)) ∥ (♯‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧)))
28	12, 27	mtbid 324	. . . 4 ⊢ (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / ∼ )(♯‘𝑧))
29		pwfi 9231	. . . . . . . 8 ⊢ (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
30	25, 29	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝒫 𝑆 ∈ Fin)
31	20	qsss 8724	. . . . . . 7 ⊢ (𝜑 → (𝑆 / ∼ ) ⊆ 𝒫 𝑆)
32	30, 31	ssfid 9181	. . . . . 6 ⊢ (𝜑 → (𝑆 / ∼ ) ∈ Fin)
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑆 / ∼ ) ∈ Fin)
34		prmnn 16613	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
35	1, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℕ)
36	1, 10	pccld 16790	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) ∈ ℕ0)
37	13, 36	eqeltrrd 2838	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℕ0)
38		peano2nn0 12453	. . . . . . . . 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 14180	. . . . . . 7 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
41	40	nnzd 12526	. . . . . 6 ⊢ (𝜑 → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
42	41	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
43		erdm 8656	. . . . . . . . . 10 ⊢ ( ∼ Er 𝑆 → dom ∼ = 𝑆)
44	20, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom ∼ = 𝑆)
45		elqsn0 8733	. . . . . . . . 9 ⊢ ((dom ∼ = 𝑆 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
46	44, 45	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ≠ ∅)
47	25	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑆 ∈ Fin)
48	31	sselda 3935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ 𝒫 𝑆)
49	48	elpwid 4565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ⊆ 𝑆)
50	47, 49	ssfid 9181	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑧 ∈ Fin)
51		hashnncl 14301	. . . . . . . . 9 ⊢ (𝑧 ∈ Fin → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
52	50, 51	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((♯‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
53	46, 52	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
54	53	adantlr 716	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℕ)
55	54	nnzd 12526	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (♯‘𝑧) ∈ ℤ)
56		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑧 → (♯‘𝑎) = (♯‘𝑧))
57	56	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑧 → (𝑃 pCnt (♯‘𝑎)) = (𝑃 pCnt (♯‘𝑧)))
58	57	breq1d 5110	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑧 → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
59	58	notbid 318	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
60	59	rspccva 3577	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
61	60	adantll 715	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ¬ (𝑃 pCnt (♯‘𝑧)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
62	2	grpbn0 18908	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
63	3, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ≠ ∅)
64		hashnncl 14301	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
65	4, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
66	63, 65	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
67	1, 66	pccld 16790	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
68	67	nn0zd 12525	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
69	5	nn0zd 12525	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
70	68, 69	zsubcld 12613	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
71	70	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
72	71	zred 12608	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℝ)
73	1	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → 𝑃 ∈ ℙ)
74	73, 54	pccld 16790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℕ0)
75	74	nn0zd 12525	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℤ)
76	75	zred 12608	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (𝑃 pCnt (♯‘𝑧)) ∈ ℝ)
77	72, 76	ltnled 11292	. . . . . . . 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 257	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) < (𝑃 pCnt (♯‘𝑧)))
79		zltp1le 12553	. . . . . . . 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 585	. . . . . . 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 232	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (♯‘𝑧)))
82	39	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / ∼ )) → (((𝑃 pCnt (♯‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
83		pcdvdsb 16809	. . . . . . 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 1374	. . . . . 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 232	. . . . 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 816	. . 3 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
88		dfrex2 3065	. . 3 ⊢ (∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ∼ ) ¬ (𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
89	87, 88	sylibr 234	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑆 / ∼ )(𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
90		eqid 2737	. . . 4 ⊢ (𝑆 / ∼ ) = (𝑆 / ∼ )
91		fveq2 6842	. . . . . . 7 ⊢ ([𝑧] ∼ = 𝑎 → (♯‘[𝑧] ∼ ) = (♯‘𝑎))
92	91	oveq2d 7384	. . . . . 6 ⊢ ([𝑧] ∼ = 𝑎 → (𝑃 pCnt (♯‘[𝑧] ∼ )) = (𝑃 pCnt (♯‘𝑎)))
93	92	breq1d 5110	. . . . 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 8685	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → [𝑤] ∼ = [𝑧] ∼ )
96	95	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (♯‘[𝑤] ∼ ) = (♯‘[𝑧] ∼ ))
97	96	oveq2d 7384	. . . . . . . 8 ⊢ (𝑤 = 𝑧 → (𝑃 pCnt (♯‘[𝑤] ∼ )) = (𝑃 pCnt (♯‘[𝑧] ∼ )))
98	97	breq1d 5110	. . . . . . 7 ⊢ (𝑤 = 𝑧 → ((𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
99	98	rspcev 3578	. . . . . 6 ⊢ ((𝑧 ∈ 𝑆 ∧ (𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
100	99	ex 412	. . . . 5 ⊢ (𝑧 ∈ 𝑆 → ((𝑃 pCnt (♯‘[𝑧] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
101	100	adantl 481	. . . 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 8731	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑆 / ∼ )) → ((𝑃 pCnt (♯‘𝑎)) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁) → ∃𝑤 ∈ 𝑆 (𝑃 pCnt (♯‘[𝑤] ∼ )) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
103	102	rexlimdva 3139	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {cpr 4584   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  dom cdm 5632  ran crn 5633  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370   Er wer 8642  [cec 8643   / cqs 8644  Fincfn 8895  1c1 11039   + caddc 11041   < clt 11178   ≤ cle 11179   − cmin 11376  ℕcn 12157  ℕ₀cn0 12413  ℤcz 12500  ↑cexp 13996  ♯chash 14265  Σcsu 15621   ∥ cdvds 16191  ℙcprime 16610   pCnt cpc 16776  Basecbs 17148  +_gcplusg 17189  Grpcgrp 18875   GrpAct cga 19230

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-ec 8647  df-qs 8651  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-fac 14209  df-bc 14238  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-dvds 16192  df-gcd 16434  df-prm 16611  df-pc 16777  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-ga 19231

This theorem is referenced by:  sylow1  19544