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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 (𝜑𝐺 ∈ Grp)
sylow1.f (𝜑𝑋 ∈ Fin)
sylow1.p (𝜑𝑃 ∈ ℙ)
sylow1.n (𝜑𝑁 ∈ ℕ0)
sylow1.d (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))
sylow1lem.a + = (+g𝐺)
sylow1lem.s 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}
sylow1lem.m = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))
sylow1lem3.1 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
Assertion
Ref Expression
sylow1lem3 (𝜑 → ∃𝑤𝑆 (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.
StepHypRef Expression
1 sylow1.p . . . . . 6 (𝜑𝑃 ∈ ℙ)
2 sylow1.x . . . . . . . 8 𝑋 = (Base‘𝐺)
3 sylow1.g . . . . . . . 8 (𝜑𝐺 ∈ Grp)
4 sylow1.f . . . . . . . 8 (𝜑𝑋 ∈ Fin)
5 sylow1.n . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
6 sylow1.d . . . . . . . 8 (𝜑 → (𝑃𝑁) ∥ (#‘𝑋))
7 sylow1lem.a . . . . . . . 8 + = (+g𝐺)
8 sylow1lem.s . . . . . . . 8 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)}
92, 3, 4, 1, 5, 6, 7, 8sylow1lem1 17836 . . . . . . 7 (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
109simpld 474 . . . . . 6 (𝜑 → (#‘𝑆) ∈ ℕ)
11 pcndvds 15408 . . . . . 6 ((𝑃 ∈ ℙ ∧ (#‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
121, 10, 11syl2anc 691 . . . . 5 (𝜑 → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
139simprd 478 . . . . . . . 8 (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
1413oveq1d 6564 . . . . . . 7 (𝜑 → ((𝑃 pCnt (#‘𝑆)) + 1) = (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1))
1514oveq2d 6565 . . . . . 6 (𝜑 → (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) = (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)))
16 sylow1lem.m . . . . . . . . 9 = (𝑥𝑋, 𝑦𝑆 ↦ ran (𝑧𝑦 ↦ (𝑥 + 𝑧)))
172, 3, 4, 1, 5, 6, 7, 8, 16sylow1lem2 17837 . . . . . . . 8 (𝜑 ∈ (𝐺 GrpAct 𝑆))
18 sylow1lem3.1 . . . . . . . . 9 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
1918, 2gaorber 17564 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑆) → Er 𝑆)
2017, 19syl 17 . . . . . . 7 (𝜑 Er 𝑆)
21 pwfi 8144 . . . . . . . . 9 (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
224, 21sylib 207 . . . . . . . 8 (𝜑 → 𝒫 𝑋 ∈ Fin)
23 ssrab2 3650 . . . . . . . . 9 {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)} ⊆ 𝒫 𝑋
248, 23eqsstri 3598 . . . . . . . 8 𝑆 ⊆ 𝒫 𝑋
25 ssfi 8065 . . . . . . . 8 ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
2622, 24, 25sylancl 693 . . . . . . 7 (𝜑𝑆 ∈ Fin)
2720, 26qshash 14398 . . . . . 6 (𝜑 → (#‘𝑆) = Σ𝑧 ∈ (𝑆 / )(#‘𝑧))
2815, 27breq12d 4596 . . . . 5 (𝜑 → ((𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆) ↔ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / )(#‘𝑧)))
2912, 28mtbid 313 . . . 4 (𝜑 → ¬ (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / )(#‘𝑧))
30 pwfi 8144 . . . . . . . 8 (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
3126, 30sylib 207 . . . . . . 7 (𝜑 → 𝒫 𝑆 ∈ Fin)
3220qsss 7695 . . . . . . 7 (𝜑 → (𝑆 / ) ⊆ 𝒫 𝑆)
33 ssfi 8065 . . . . . . 7 ((𝒫 𝑆 ∈ Fin ∧ (𝑆 / ) ⊆ 𝒫 𝑆) → (𝑆 / ) ∈ Fin)
3431, 32, 33syl2anc 691 . . . . . 6 (𝜑 → (𝑆 / ) ∈ Fin)
3534adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑆 / ) ∈ Fin)
36 prmnn 15226 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
371, 36syl 17 . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
381, 10pccld 15393 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt (#‘𝑆)) ∈ ℕ0)
3913, 38eqeltrrd 2689 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0)
40 peano2nn0 11210 . . . . . . . . 9 (((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0 → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
4139, 40syl 17 . . . . . . . 8 (𝜑 → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
4237, 41nnexpcld 12892 . . . . . . 7 (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
4342nnzd 11357 . . . . . 6 (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℤ)
4443adantr 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)
4620, 45syl 17 . . . . . . . . 9 (𝜑 → dom = 𝑆)
47 elqsn0 7703 . . . . . . . . 9 ((dom = 𝑆𝑧 ∈ (𝑆 / )) → 𝑧 ≠ ∅)
4846, 47sylan 487 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ≠ ∅)
4926adantr 480 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑆 ∈ Fin)
5032sselda 3568 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ∈ 𝒫 𝑆)
5150elpwid 4118 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧𝑆)
52 ssfi 8065 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑧𝑆) → 𝑧 ∈ Fin)
5349, 51, 52syl2anc 691 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ∈ Fin)
54 hashnncl 13018 . . . . . . . . 9 (𝑧 ∈ Fin → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
5553, 54syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑆 / )) → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
5648, 55mpbird 246 . . . . . . 7 ((𝜑𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℕ)
5756adantlr 747 . . . . . 6 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℕ)
5857nnzd 11357 . . . . 5 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℤ)
59 fveq2 6103 . . . . . . . . . . . . 13 (𝑎 = 𝑧 → (#‘𝑎) = (#‘𝑧))
6059oveq2d 6565 . . . . . . . . . . . 12 (𝑎 = 𝑧 → (𝑃 pCnt (#‘𝑎)) = (𝑃 pCnt (#‘𝑧)))
6160breq1d 4593 . . . . . . . . . . 11 (𝑎 = 𝑧 → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
6261notbid 307 . . . . . . . . . 10 (𝑎 = 𝑧 → (¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
6362rspccva 3281 . . . . . . . . 9 ((∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
6463adantll 746 . . . . . . . 8 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
652grpbn0 17274 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
663, 65syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 ≠ ∅)
67 hashnncl 13018 . . . . . . . . . . . . . . . 16 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
684, 67syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
6966, 68mpbird 246 . . . . . . . . . . . . . 14 (𝜑 → (#‘𝑋) ∈ ℕ)
701, 69pccld 15393 . . . . . . . . . . . . 13 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
7170nn0zd 11356 . . . . . . . . . . . 12 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
725nn0zd 11356 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
7371, 72zsubcld 11363 . . . . . . . . . . 11 (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
7473ad2antrr 758 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
7574zred 11358 . . . . . . . . 9 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℝ)
761ad2antrr 758 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → 𝑃 ∈ ℙ)
7776, 57pccld 15393 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℕ0)
7877nn0zd 11356 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℤ)
7978zred 11358 . . . . . . . . 9 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℝ)
8075, 79ltnled 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)))
8164, 80mpbird 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))))
8374, 78, 82syl2anc 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))))
8481, 83mpbid 221 . . . . . 6 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (#‘𝑧)))
8541ad2antrr 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)))
8776, 58, 85, 86syl3anc 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)))
8884, 87mpbid 221 . . . . 5 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ (#‘𝑧))
8935, 44, 58, 88fsumdvds 14868 . . . 4 ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / )(#‘𝑧))
9029, 89mtand 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))
9290, 91sylibr 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))
9594oveq2d 6565 . . . . . 6 ([𝑧] = 𝑎 → (𝑃 pCnt (#‘[𝑧] )) = (𝑃 pCnt (#‘𝑎)))
9695breq1d 4593 . . . . 5 ([𝑧] = 𝑎 → ((𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
9796imbi1d 330 . . . 4 ([𝑧] = 𝑎 → (((𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ↔ ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))))
98 eceq1 7669 . . . . . . . . . 10 (𝑤 = 𝑧 → [𝑤] = [𝑧] )
9998fveq2d 6107 . . . . . . . . 9 (𝑤 = 𝑧 → (#‘[𝑤] ) = (#‘[𝑧] ))
10099oveq2d 6565 . . . . . . . 8 (𝑤 = 𝑧 → (𝑃 pCnt (#‘[𝑤] )) = (𝑃 pCnt (#‘[𝑧] )))
101100breq1d 4593 . . . . . . 7 (𝑤 = 𝑧 → ((𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
102101rspcev 3282 . . . . . 6 ((𝑧𝑆 ∧ (𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
103102ex 449 . . . . 5 (𝑧𝑆 → ((𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
104103adantl 481 . . . 4 ((𝜑𝑧𝑆) → ((𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
10593, 97, 104ectocld 7701 . . 3 ((𝜑𝑎 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
106105rexlimdva 3013 . 2 (𝜑 → (∃𝑎 ∈ (𝑆 / )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
10792, 106mpd 15 1 (𝜑 → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
 Colors of variables: wff setvar class 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
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