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Theorem sylow1lem3 18061
Description: Lemma for sylow1 18064. 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 18059 . . . . . . 7 (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
109simpld 474 . . . . . 6 (𝜑 → (#‘𝑆) ∈ ℕ)
11 pcndvds 15617 . . . . . 6 ((𝑃 ∈ ℙ ∧ (#‘𝑆) ∈ ℕ) → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
121, 10, 11syl2anc 694 . . . . 5 (𝜑 → ¬ (𝑃↑((𝑃 pCnt (#‘𝑆)) + 1)) ∥ (#‘𝑆))
139simprd 478 . . . . . . . 8 (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
1413oveq1d 6705 . . . . . . 7 (𝜑 → ((𝑃 pCnt (#‘𝑆)) + 1) = (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1))
1514oveq2d 6706 . . . . . 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 18060 . . . . . . . 8 (𝜑 ∈ (𝐺 GrpAct 𝑆))
18 sylow1lem3.1 . . . . . . . . 9 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑆 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}
1918, 2gaorber 17787 . . . . . . . 8 ( ∈ (𝐺 GrpAct 𝑆) → Er 𝑆)
2017, 19syl 17 . . . . . . 7 (𝜑 Er 𝑆)
21 pwfi 8302 . . . . . . . . 9 (𝑋 ∈ Fin ↔ 𝒫 𝑋 ∈ Fin)
224, 21sylib 208 . . . . . . . 8 (𝜑 → 𝒫 𝑋 ∈ Fin)
23 ssrab2 3720 . . . . . . . . 9 {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃𝑁)} ⊆ 𝒫 𝑋
248, 23eqsstri 3668 . . . . . . . 8 𝑆 ⊆ 𝒫 𝑋
25 ssfi 8221 . . . . . . . 8 ((𝒫 𝑋 ∈ Fin ∧ 𝑆 ⊆ 𝒫 𝑋) → 𝑆 ∈ Fin)
2622, 24, 25sylancl 695 . . . . . . 7 (𝜑𝑆 ∈ Fin)
2720, 26qshash 14603 . . . . . 6 (𝜑 → (#‘𝑆) = Σ𝑧 ∈ (𝑆 / )(#‘𝑧))
2815, 27breq12d 4698 . . . . 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 8302 . . . . . . . 8 (𝑆 ∈ Fin ↔ 𝒫 𝑆 ∈ Fin)
3126, 30sylib 208 . . . . . . 7 (𝜑 → 𝒫 𝑆 ∈ Fin)
3220qsss 7851 . . . . . . 7 (𝜑 → (𝑆 / ) ⊆ 𝒫 𝑆)
33 ssfi 8221 . . . . . . 7 ((𝒫 𝑆 ∈ Fin ∧ (𝑆 / ) ⊆ 𝒫 𝑆) → (𝑆 / ) ∈ Fin)
3431, 32, 33syl2anc 694 . . . . . 6 (𝜑 → (𝑆 / ) ∈ Fin)
3534adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑆 / ) ∈ Fin)
36 prmnn 15435 . . . . . . . . 9 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
371, 36syl 17 . . . . . . . 8 (𝜑𝑃 ∈ ℕ)
381, 10pccld 15602 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt (#‘𝑆)) ∈ ℕ0)
3913, 38eqeltrrd 2731 . . . . . . . . 9 (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℕ0)
40 peano2nn0 11371 . . . . . . . . 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 13070 . . . . . . 7 (𝜑 → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∈ ℕ)
4342nnzd 11519 . . . . . 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 7797 . . . . . . . . . 10 ( Er 𝑆 → dom = 𝑆)
4620, 45syl 17 . . . . . . . . 9 (𝜑 → dom = 𝑆)
47 elqsn0 7859 . . . . . . . . 9 ((dom = 𝑆𝑧 ∈ (𝑆 / )) → 𝑧 ≠ ∅)
4846, 47sylan 487 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ≠ ∅)
4926adantr 480 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑆 ∈ Fin)
5032sselda 3636 . . . . . . . . . . 11 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ∈ 𝒫 𝑆)
5150elpwid 4203 . . . . . . . . . 10 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧𝑆)
52 ssfi 8221 . . . . . . . . . 10 ((𝑆 ∈ Fin ∧ 𝑧𝑆) → 𝑧 ∈ Fin)
5349, 51, 52syl2anc 694 . . . . . . . . 9 ((𝜑𝑧 ∈ (𝑆 / )) → 𝑧 ∈ Fin)
54 hashnncl 13195 . . . . . . . . 9 (𝑧 ∈ Fin → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
5553, 54syl 17 . . . . . . . 8 ((𝜑𝑧 ∈ (𝑆 / )) → ((#‘𝑧) ∈ ℕ ↔ 𝑧 ≠ ∅))
5648, 55mpbird 247 . . . . . . 7 ((𝜑𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℕ)
5756adantlr 751 . . . . . 6 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℕ)
5857nnzd 11519 . . . . 5 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (#‘𝑧) ∈ ℤ)
59 fveq2 6229 . . . . . . . . . . . . 13 (𝑎 = 𝑧 → (#‘𝑎) = (#‘𝑧))
6059oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = 𝑧 → (𝑃 pCnt (#‘𝑎)) = (𝑃 pCnt (#‘𝑧)))
6160breq1d 4695 . . . . . . . . . . 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 3339 . . . . . . . . 9 ((∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∧ 𝑧 ∈ (𝑆 / )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
6463adantll 750 . . . . . . . 8 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ¬ (𝑃 pCnt (#‘𝑧)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
652grpbn0 17498 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
663, 65syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑋 ≠ ∅)
67 hashnncl 13195 . . . . . . . . . . . . . . . 16 (𝑋 ∈ Fin → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
684, 67syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ((#‘𝑋) ∈ ℕ ↔ 𝑋 ≠ ∅))
6966, 68mpbird 247 . . . . . . . . . . . . . 14 (𝜑 → (#‘𝑋) ∈ ℕ)
701, 69pccld 15602 . . . . . . . . . . . . 13 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
7170nn0zd 11518 . . . . . . . . . . . 12 (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
725nn0zd 11518 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
7371, 72zsubcld 11525 . . . . . . . . . . 11 (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
7473ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
7574zred 11520 . . . . . . . . 9 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℝ)
761ad2antrr 762 . . . . . . . . . . . 12 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → 𝑃 ∈ ℙ)
7776, 57pccld 15602 . . . . . . . . . . 11 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℕ0)
7877nn0zd 11518 . . . . . . . . . 10 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℤ)
7978zred 11520 . . . . . . . . 9 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃 pCnt (#‘𝑧)) ∈ ℝ)
8075, 79ltnled 10222 . . . . . . . 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 247 . . . . . . 7 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑋)) − 𝑁) < (𝑃 pCnt (#‘𝑧)))
82 zltp1le 11465 . . . . . . . 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 694 . . . . . . 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 222 . . . . . 6 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ≤ (𝑃 pCnt (#‘𝑧)))
8541ad2antrr 762 . . . . . . 7 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1) ∈ ℕ0)
86 pcdvdsb 15620 . . . . . . 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 1366 . . . . . 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 222 . . . . 5 (((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) ∧ 𝑧 ∈ (𝑆 / )) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ (#‘𝑧))
8935, 44, 58, 88fsumdvds 15077 . . . 4 ((𝜑 ∧ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)) → (𝑃↑(((𝑃 pCnt (#‘𝑋)) − 𝑁) + 1)) ∥ Σ𝑧 ∈ (𝑆 / )(#‘𝑧))
9029, 89mtand 692 . . 3 (𝜑 → ¬ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
91 dfrex2 3025 . . 3 (∃𝑎 ∈ (𝑆 / )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ ¬ ∀𝑎 ∈ (𝑆 / ) ¬ (𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
9290, 91sylibr 224 . 2 (𝜑 → ∃𝑎 ∈ (𝑆 / )(𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁))
93 eqid 2651 . . . 4 (𝑆 / ) = (𝑆 / )
94 fveq2 6229 . . . . . . 7 ([𝑧] = 𝑎 → (#‘[𝑧] ) = (#‘𝑎))
9594oveq2d 6706 . . . . . 6 ([𝑧] = 𝑎 → (𝑃 pCnt (#‘[𝑧] )) = (𝑃 pCnt (#‘𝑎)))
9695breq1d 4695 . . . . 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 7825 . . . . . . . . . 10 (𝑤 = 𝑧 → [𝑤] = [𝑧] )
9998fveq2d 6233 . . . . . . . . 9 (𝑤 = 𝑧 → (#‘[𝑤] ) = (#‘[𝑧] ))
10099oveq2d 6706 . . . . . . . 8 (𝑤 = 𝑧 → (𝑃 pCnt (#‘[𝑤] )) = (𝑃 pCnt (#‘[𝑧] )))
101100breq1d 4695 . . . . . . 7 (𝑤 = 𝑧 → ((𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) ↔ (𝑃 pCnt (#‘[𝑧] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
102101rspcev 3340 . . . . . 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 7857 . . 3 ((𝜑𝑎 ∈ (𝑆 / )) → ((𝑃 pCnt (#‘𝑎)) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁) → ∃𝑤𝑆 (𝑃 pCnt (#‘[𝑤] )) ≤ ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
106105rexlimdva 3060 . 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 196  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  wss 3607  c0 3948  𝒫 cpw 4191  {cpr 4212   class class class wbr 4685  {copab 4745  cmpt 4762  dom cdm 5143  ran crn 5144  cfv 5926  (class class class)co 6690  cmpt2 6692   Er wer 7784  [cec 7785   / cqs 7786  Fincfn 7997  1c1 9975   + caddc 9977   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cexp 12900  #chash 13157  Σcsu 14460  cdvds 15027  cprime 15432   pCnt cpc 15588  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469   GrpAct cga 17768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-ga 17769
This theorem is referenced by:  sylow1  18064
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