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Theorem sylow1lem1 18702
Description: Lemma for sylow1 18707. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (♯‘𝑋) / (𝑃𝑁). (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)}
Assertion
Ref Expression
sylow1lem1 (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
Distinct variable groups:   𝑁,𝑠   𝑋,𝑠   + ,𝑠   𝐺,𝑠   𝑃,𝑠
Allowed substitution hints:   𝜑(𝑠)   𝑆(𝑠)

Proof of Theorem sylow1lem1
Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow1.f . . . . 5 (𝜑𝑋 ∈ Fin)
2 sylow1.p . . . . . . . 8 (𝜑𝑃 ∈ ℙ)
3 prmnn 15996 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
42, 3syl 17 . . . . . . 7 (𝜑𝑃 ∈ ℕ)
5 sylow1.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
64, 5nnexpcld 13591 . . . . . 6 (𝜑 → (𝑃𝑁) ∈ ℕ)
76nnzd 12065 . . . . 5 (𝜑 → (𝑃𝑁) ∈ ℤ)
8 hashbc 13796 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑃𝑁) ∈ ℤ) → ((♯‘𝑋)C(𝑃𝑁)) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
91, 7, 8syl2anc 586 . . . 4 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
10 sylow1lem.s . . . . 5 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
1110fveq2i 6649 . . . 4 (♯‘𝑆) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)})
129, 11syl6eqr 2873 . . 3 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) = (♯‘𝑆))
13 sylow1.d . . . . . 6 (𝜑 → (𝑃𝑁) ∥ (♯‘𝑋))
14 sylow1.g . . . . . . . . . 10 (𝜑𝐺 ∈ Grp)
15 sylow1.x . . . . . . . . . . 11 𝑋 = (Base‘𝐺)
1615grpbn0 18111 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
1714, 16syl 17 . . . . . . . . 9 (𝜑𝑋 ≠ ∅)
18 hasheq0 13709 . . . . . . . . . . 11 (𝑋 ∈ Fin → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
191, 18syl 17 . . . . . . . . . 10 (𝜑 → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
2019necon3bbid 3043 . . . . . . . . 9 (𝜑 → (¬ (♯‘𝑋) = 0 ↔ 𝑋 ≠ ∅))
2117, 20mpbird 259 . . . . . . . 8 (𝜑 → ¬ (♯‘𝑋) = 0)
22 hashcl 13702 . . . . . . . . . . 11 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
231, 22syl 17 . . . . . . . . . 10 (𝜑 → (♯‘𝑋) ∈ ℕ0)
24 elnn0 11878 . . . . . . . . . 10 ((♯‘𝑋) ∈ ℕ0 ↔ ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
2523, 24sylib 220 . . . . . . . . 9 (𝜑 → ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
2625ord 860 . . . . . . . 8 (𝜑 → (¬ (♯‘𝑋) ∈ ℕ → (♯‘𝑋) = 0))
2721, 26mt3d 150 . . . . . . 7 (𝜑 → (♯‘𝑋) ∈ ℕ)
28 dvdsle 15640 . . . . . . 7 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℕ) → ((𝑃𝑁) ∥ (♯‘𝑋) → (𝑃𝑁) ≤ (♯‘𝑋)))
297, 27, 28syl2anc 586 . . . . . 6 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) → (𝑃𝑁) ≤ (♯‘𝑋)))
3013, 29mpd 15 . . . . 5 (𝜑 → (𝑃𝑁) ≤ (♯‘𝑋))
316nnnn0d 11934 . . . . . . 7 (𝜑 → (𝑃𝑁) ∈ ℕ0)
32 nn0uz 12259 . . . . . . 7 0 = (ℤ‘0)
3331, 32eleqtrdi 2921 . . . . . 6 (𝜑 → (𝑃𝑁) ∈ (ℤ‘0))
3423nn0zd 12064 . . . . . 6 (𝜑 → (♯‘𝑋) ∈ ℤ)
35 elfz5 12884 . . . . . 6 (((𝑃𝑁) ∈ (ℤ‘0) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
3633, 34, 35syl2anc 586 . . . . 5 (𝜑 → ((𝑃𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
3730, 36mpbird 259 . . . 4 (𝜑 → (𝑃𝑁) ∈ (0...(♯‘𝑋)))
38 bccl2 13668 . . . 4 ((𝑃𝑁) ∈ (0...(♯‘𝑋)) → ((♯‘𝑋)C(𝑃𝑁)) ∈ ℕ)
3937, 38syl 17 . . 3 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) ∈ ℕ)
4012, 39eqeltrrd 2912 . 2 (𝜑 → (♯‘𝑆) ∈ ℕ)
41 nnuz 12260 . . . . . . . . . . 11 ℕ = (ℤ‘1)
426, 41eleqtrdi 2921 . . . . . . . . . 10 (𝜑 → (𝑃𝑁) ∈ (ℤ‘1))
43 elfz5 12884 . . . . . . . . . 10 (((𝑃𝑁) ∈ (ℤ‘1) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
4442, 34, 43syl2anc 586 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
4530, 44mpbird 259 . . . . . . . 8 (𝜑 → (𝑃𝑁) ∈ (1...(♯‘𝑋)))
46 1zzd 11992 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
47 fzsubel 12927 . . . . . . . . 9 (((1 ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) ∧ ((𝑃𝑁) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1))))
4846, 34, 7, 46, 47syl22anc 836 . . . . . . . 8 (𝜑 → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1))))
4945, 48mpbid 234 . . . . . . 7 (𝜑 → ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1)))
50 1m1e0 11688 . . . . . . . 8 (1 − 1) = 0
5150oveq1i 7143 . . . . . . 7 ((1 − 1)...((♯‘𝑋) − 1)) = (0...((♯‘𝑋) − 1))
5249, 51eleqtrdi 2921 . . . . . 6 (𝜑 → ((𝑃𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)))
53 bcp1nk 13662 . . . . . 6 (((𝑃𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)) → ((((♯‘𝑋) − 1) + 1)C(((𝑃𝑁) − 1) + 1)) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1))))
5452, 53syl 17 . . . . 5 (𝜑 → ((((♯‘𝑋) − 1) + 1)C(((𝑃𝑁) − 1) + 1)) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1))))
5523nn0cnd 11936 . . . . . . 7 (𝜑 → (♯‘𝑋) ∈ ℂ)
56 ax-1cn 10573 . . . . . . 7 1 ∈ ℂ
57 npcan 10873 . . . . . . 7 (((♯‘𝑋) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
5855, 56, 57sylancl 588 . . . . . 6 (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
596nncnd 11632 . . . . . . 7 (𝜑 → (𝑃𝑁) ∈ ℂ)
60 npcan 10873 . . . . . . 7 (((𝑃𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑃𝑁) − 1) + 1) = (𝑃𝑁))
6159, 56, 60sylancl 588 . . . . . 6 (𝜑 → (((𝑃𝑁) − 1) + 1) = (𝑃𝑁))
6258, 61oveq12d 7151 . . . . 5 (𝜑 → ((((♯‘𝑋) − 1) + 1)C(((𝑃𝑁) − 1) + 1)) = ((♯‘𝑋)C(𝑃𝑁)))
6358, 61oveq12d 7151 . . . . . 6 (𝜑 → ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1)) = ((♯‘𝑋) / (𝑃𝑁)))
6463oveq2d 7149 . . . . 5 (𝜑 → ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1))) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁))))
6554, 62, 643eqtr3d 2863 . . . 4 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁))))
6665oveq2d 7149 . . 3 (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃𝑁))) = (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))))
6712oveq2d 7149 . . 3 (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃𝑁))) = (𝑃 pCnt (♯‘𝑆)))
68 bccl2 13668 . . . . . . 7 (((𝑃𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)) → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℕ)
6952, 68syl 17 . . . . . 6 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℕ)
7069nnzd 12065 . . . . 5 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℤ)
7169nnne0d 11666 . . . . 5 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ≠ 0)
726nnne0d 11666 . . . . . . 7 (𝜑 → (𝑃𝑁) ≠ 0)
73 dvdsval2 15590 . . . . . . 7 (((𝑃𝑁) ∈ ℤ ∧ (𝑃𝑁) ≠ 0 ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ))
747, 72, 34, 73syl3anc 1367 . . . . . 6 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ))
7513, 74mpbid 234 . . . . 5 (𝜑 → ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ)
7627nnne0d 11666 . . . . . 6 (𝜑 → (♯‘𝑋) ≠ 0)
7755, 59, 76, 72divne0d 11410 . . . . 5 (𝜑 → ((♯‘𝑋) / (𝑃𝑁)) ≠ 0)
78 pcmul 16166 . . . . 5 ((𝑃 ∈ ℙ ∧ ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℤ ∧ (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ≠ 0) ∧ (((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ ∧ ((♯‘𝑋) / (𝑃𝑁)) ≠ 0)) → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))) = ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁)))))
792, 70, 71, 75, 77, 78syl122anc 1375 . . . 4 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))) = ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁)))))
80 1cnd 10614 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
8155, 59, 80npncand 10999 . . . . . . . 8 (𝜑 → (((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1)) = ((♯‘𝑋) − 1))
8281oveq1d 7148 . . . . . . 7 (𝜑 → ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1)) = (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)))
8382oveq2d 7149 . . . . . 6 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = (𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))))
846nnred 11631 . . . . . . . 8 (𝜑 → (𝑃𝑁) ∈ ℝ)
8584ltm1d 11550 . . . . . . 7 (𝜑 → ((𝑃𝑁) − 1) < (𝑃𝑁))
86 nnm1nn0 11917 . . . . . . . . 9 ((𝑃𝑁) ∈ ℕ → ((𝑃𝑁) − 1) ∈ ℕ0)
876, 86syl 17 . . . . . . . 8 (𝜑 → ((𝑃𝑁) − 1) ∈ ℕ0)
88 breq1 5045 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 < (𝑃𝑁) ↔ 0 < (𝑃𝑁)))
89 bcxmaslem1 15169 . . . . . . . . . . . . 13 (𝑥 = 0 → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0))
9089oveq2d 7149 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)))
9190eqeq1d 2822 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0))
9288, 91imbi12d 347 . . . . . . . . . 10 (𝑥 = 0 → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ (0 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0)))
9392imbi2d 343 . . . . . . . . 9 (𝑥 = 0 → ((𝜑 → (𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (0 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0))))
94 breq1 5045 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥 < (𝑃𝑁) ↔ 𝑛 < (𝑃𝑁)))
95 bcxmaslem1 15169 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛))
9695oveq2d 7149 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)))
9796eqeq1d 2822 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0))
9894, 97imbi12d 347 . . . . . . . . . 10 (𝑥 = 𝑛 → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ (𝑛 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0)))
9998imbi2d 343 . . . . . . . . 9 (𝑥 = 𝑛 → ((𝜑 → (𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (𝑛 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0))))
100 breq1 5045 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥 < (𝑃𝑁) ↔ (𝑛 + 1) < (𝑃𝑁)))
101 bcxmaslem1 15169 . . . . . . . . . . . . 13 (𝑥 = (𝑛 + 1) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102101oveq2d 7149 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103102eqeq1d 2822 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
104100, 103imbi12d 347 . . . . . . . . . 10 (𝑥 = (𝑛 + 1) → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ ((𝑛 + 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
105104imbi2d 343 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → ((𝑛 + 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
106 breq1 5045 . . . . . . . . . . 11 (𝑥 = ((𝑃𝑁) − 1) → (𝑥 < (𝑃𝑁) ↔ ((𝑃𝑁) − 1) < (𝑃𝑁)))
107 bcxmaslem1 15169 . . . . . . . . . . . . 13 (𝑥 = ((𝑃𝑁) − 1) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1)))
108107oveq2d 7149 . . . . . . . . . . . 12 (𝑥 = ((𝑃𝑁) − 1) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))))
109108eqeq1d 2822 . . . . . . . . . . 11 (𝑥 = ((𝑃𝑁) − 1) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0))
110106, 109imbi12d 347 . . . . . . . . . 10 (𝑥 = ((𝑃𝑁) − 1) → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0)))
111110imbi2d 343 . . . . . . . . 9 (𝑥 = ((𝑃𝑁) − 1) → ((𝜑 → (𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0))))
112 znn0sub 12008 . . . . . . . . . . . . . . . 16 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0))
1137, 34, 112syl2anc 586 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑃𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0))
11430, 113mpbid 234 . . . . . . . . . . . . . 14 (𝜑 → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
115 0nn0 11891 . . . . . . . . . . . . . 14 0 ∈ ℕ0
116 nn0addcl 11911 . . . . . . . . . . . . . 14 ((((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0 ∧ 0 ∈ ℕ0) → (((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0)
117114, 115, 116sylancl 588 . . . . . . . . . . . . 13 (𝜑 → (((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0)
118 bcn0 13655 . . . . . . . . . . . . 13 ((((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0 → ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0) = 1)
119117, 118syl 17 . . . . . . . . . . . 12 (𝜑 → ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0) = 1)
120119oveq2d 7149 . . . . . . . . . . 11 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121 pc1 16170 . . . . . . . . . . . 12 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
1222, 121syl 17 . . . . . . . . . . 11 (𝜑 → (𝑃 pCnt 1) = 0)
123120, 122eqtrd 2855 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0)
124123a1d 25 . . . . . . . . 9 (𝜑 → (0 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0))
125 nn0re 11885 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
126125ad2antrl 726 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℝ)
127 nn0p1nn 11915 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
128127ad2antrl 726 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℕ)
129128nnred 11631 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℝ)
1306adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℕ)
131130nnred 11631 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℝ)
132126ltp1d 11548 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 < (𝑛 + 1))
133 simprr 771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) < (𝑃𝑁))
134126, 129, 131, 132, 133lttrd 10779 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 < (𝑃𝑁))
135134expr 459 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 + 1) < (𝑃𝑁) → 𝑛 < (𝑃𝑁)))
136135imim1d 82 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ0) → ((𝑛 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0) → ((𝑛 + 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0)))
137 oveq1 7140 . . . . . . . . . . 11 ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
138114adantr 483 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
139138nn0cnd 11936 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℂ)
140 nn0cn 11886 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ0𝑛 ∈ ℂ)
141140ad2antrl 726 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℂ)
142 1cnd 10614 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 1 ∈ ℂ)
143139, 141, 142addassd 10641 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) = (((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1)))
144143oveq1d 7148 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
145 nn0addge2 11923 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0) → 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛))
146126, 138, 145syl2anc 586 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛))
147 simprl 769 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℕ0)
148147, 32eleqtrdi 2921 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ (ℤ‘0))
149138, 147nn0addcld 11938 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℕ0)
150149nn0zd 12064 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℤ)
151 elfz5 12884 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (ℤ‘0) ∧ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℤ) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
152148, 150, 151syl2anc 586 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
153146, 152mpbird 259 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
154 bcp1nk 13662 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
155153, 154syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
156144, 155eqtr3d 2857 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1)) = (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
157156oveq2d 7149 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
1582adantr 483 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑃 ∈ ℙ)
159 bccl2 13668 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℕ)
160153, 159syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℕ)
161 nnq 12340 . . . . . . . . . . . . . . 15 (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℕ → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℚ)
162160, 161syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℚ)
163160nnne0d 11666 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ≠ 0)
164150peano2zd 12069 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℤ)
165 znq 12331 . . . . . . . . . . . . . . 15 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
166164, 128, 165syl2anc 586 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
167 nn0p1nn 11915 . . . . . . . . . . . . . . . . 17 ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℕ0 → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ)
168149, 167syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ)
169 nnrp 12379 . . . . . . . . . . . . . . . . 17 (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℝ+)
170 nnrp 12379 . . . . . . . . . . . . . . . . 17 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
171 rpdivcl 12393 . . . . . . . . . . . . . . . . 17 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
172169, 170, 171syl2an 597 . . . . . . . . . . . . . . . 16 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
173168, 128, 172syl2anc 586 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
174173rpne0d 12415 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)
175 pcqmul 16168 . . . . . . . . . . . . . 14 ((𝑃 ∈ ℙ ∧ (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ∈ ℚ ∧ ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ≠ 0) ∧ ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ ∧ (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
176158, 162, 163, 166, 174, 175syl122anc 1375 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) · (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
177157, 176eqtrd 2855 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
178168nnne0d 11666 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ≠ 0)
179 pcdiv 16167 . . . . . . . . . . . . . . . 16 ((𝑃 ∈ ℙ ∧ (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℤ ∧ ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ≠ 0) ∧ (𝑛 + 1) ∈ ℕ) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))))
180158, 164, 178, 128, 179syl121anc 1371 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))))
181128nncnd 11632 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℂ)
182139, 181, 143comraddd 10832 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))))
183182oveq2d 7149 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
184 simpr 487 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((♯‘𝑋) − (𝑃𝑁)) = 0)
185184oveq2d 7149 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))) = ((𝑛 + 1) + 0))
186181addid1d 10818 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑛 + 1) + 0) = (𝑛 + 1))
187186adantr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((𝑛 + 1) + 0) = (𝑛 + 1))
188185, 187eqtr2d 2856 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → (𝑛 + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))))
189188oveq2d 7149 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
1902ad2antrr 724 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑃 ∈ ℙ)
191 nnq 12340 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℚ)
192128, 191syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℚ)
193192adantr 483 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑛 + 1) ∈ ℚ)
194138nn0zd 12064 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ)
195 zq 12333 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
196194, 195syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
197196adantr 483 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
198158, 128pccld 16165 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
199198nn0red 11935 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
200199adantr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
2015adantr 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑁 ∈ ℕ0)
202201nn0red 11935 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑁 ∈ ℝ)
203202adantr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ∈ ℝ)
204 simpr 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ≠ 0)
205204neneqd 3011 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ¬ ((♯‘𝑋) − (𝑃𝑁)) = 0)
206114ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
207 elnn0 11878 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0 ↔ (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃𝑁)) = 0))
208206, 207sylib 220 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃𝑁)) = 0))
209208ord 860 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (¬ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ → ((♯‘𝑋) − (𝑃𝑁)) = 0))
210205, 209mt3d 150 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ)
211190, 210pccld 16165 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ∈ ℕ0)
212211nn0red 11935 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ∈ ℝ)
213128nnzd 12065 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℤ)
214 pcdvdsb 16183 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑁) ∥ (𝑛 + 1)))
215158, 213, 201, 214syl3anc 1367 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑁) ∥ (𝑛 + 1)))
2167adantr 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℤ)
217 dvdsle 15640 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃𝑁) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃𝑁) ∥ (𝑛 + 1) → (𝑃𝑁) ≤ (𝑛 + 1)))
218216, 128, 217syl2anc 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃𝑁) ∥ (𝑛 + 1) → (𝑃𝑁) ≤ (𝑛 + 1)))
219215, 218sylbid 242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) → (𝑃𝑁) ≤ (𝑛 + 1)))
220202, 199lenltd 10764 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ ¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
221131, 129lenltd 10764 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃𝑁) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃𝑁)))
222219, 220, 2213imtr3d 295 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 → ¬ (𝑛 + 1) < (𝑃𝑁)))
223133, 222mt4d 117 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
224223adantr 483 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225 dvdssubr 15635 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
2267, 34, 225syl2anc 586 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
22713, 226mpbid 234 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁)))
228227ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁)))
229206nn0zd 12064 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ)
2305ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ∈ ℕ0)
231 pcdvdsb 16183 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
232190, 229, 230, 231syl3anc 1367 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
233228, 232mpbird 259 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))))
234200, 203, 212, 224, 233ltletrd 10778 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))))
235190, 193, 197, 234pcadd2 16204 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
236189, 235pm2.61dane 3093 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
237183, 236eqtr4d 2858 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
238198nn0cnd 11936 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
239237, 238eqeltrd 2911 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) ∈ ℂ)
240239, 237subeq0bd 11044 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))) = 0)
241180, 240eqtrd 2855 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
242241oveq2d 7149 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
243 00id 10793 . . . . . . . . . . . . 13 (0 + 0) = 0
244242, 243syl6req 2872 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 0 = (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
245177, 244eqeq12d 2836 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0 ↔ ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))))))
246137, 245syl5ibr 248 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
247136, 246animpimp2impd 842 . . . . . . . . 9 (𝑛 ∈ ℕ0 → ((𝜑 → (𝑛 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0)) → (𝜑 → ((𝑛 + 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
24893, 99, 105, 111, 124, 247nn0ind 12056 . . . . . . . 8 (((𝑃𝑁) − 1) ∈ ℕ0 → (𝜑 → (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0)))
24987, 248mpcom 38 . . . . . . 7 (𝜑 → (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0))
25085, 249mpd 15 . . . . . 6 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0)
25183, 250eqtr3d 2857 . . . . 5 (𝜑 → (𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) = 0)
252 pcdiv 16167 . . . . . . 7 ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) ∈ ℤ ∧ (♯‘𝑋) ≠ 0) ∧ (𝑃𝑁) ∈ ℕ) → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))))
2532, 34, 76, 6, 252syl121anc 1371 . . . . . 6 (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))))
2545nn0zd 12064 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
255 pcid 16187 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝑃𝑁)) = 𝑁)
2562, 254, 255syl2anc 586 . . . . . . 7 (𝜑 → (𝑃 pCnt (𝑃𝑁)) = 𝑁)
257256oveq2d 7149 . . . . . 6 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
258253, 257eqtrd 2855 . . . . 5 (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
259251, 258oveq12d 7151 . . . 4 (𝜑 → ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁)))) = (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
2602, 27pccld 16165 . . . . . . . 8 (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
261260nn0zd 12064 . . . . . . 7 (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
262261, 254zsubcld 12071 . . . . . 6 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
263262zcnd 12067 . . . . 5 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℂ)
264263addid2d 10819 . . . 4 (𝜑 → (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26579, 259, 2643eqtrd 2859 . . 3 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26666, 67, 2653eqtr3d 2863 . 2 (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26740, 266jca 514 1 (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843   = wceq 1537  wcel 2114  wne 3006  {crab 3129  c0 4269  𝒫 cpw 4515   class class class wbr 5042  cfv 6331  (class class class)co 7133  Fincfn 8487  cc 10513  cr 10514  0cc0 10515  1c1 10516   + caddc 10518   · cmul 10520   < clt 10653  cle 10654  cmin 10848   / cdiv 11275  cn 11616  0cn0 11876  cz 11960  cuz 12222  cq 12327  +crp 12368  ...cfz 12876  cexp 13414  Ccbc 13647  chash 13675  cdvds 15587  cprime 15993   pCnt cpc 16151  Basecbs 16462  +gcplusg 16544  Grpcgrp 18082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592  ax-pre-sup 10593
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-sup 8884  df-inf 8885  df-dju 9308  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-q 12328  df-rp 12369  df-fz 12877  df-fl 13146  df-mod 13222  df-seq 13354  df-exp 13415  df-fac 13619  df-bc 13648  df-hash 13676  df-cj 14438  df-re 14439  df-im 14440  df-sqrt 14574  df-abs 14575  df-dvds 15588  df-gcd 15822  df-prm 15994  df-pc 16152  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085
This theorem is referenced by:  sylow1lem3  18704
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