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Theorem sylow1lem1 18277
Description: Lemma for sylow1 18282. 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 15668 . . . . . . . 8 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
42, 3syl 17 . . . . . . 7 (𝜑𝑃 ∈ ℕ)
5 sylow1.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
64, 5nnexpcld 13237 . . . . . 6 (𝜑 → (𝑃𝑁) ∈ ℕ)
76nnzd 11728 . . . . 5 (𝜑 → (𝑃𝑁) ∈ ℤ)
8 hashbc 13438 . . . . 5 ((𝑋 ∈ Fin ∧ (𝑃𝑁) ∈ ℤ) → ((♯‘𝑋)C(𝑃𝑁)) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
91, 7, 8syl2anc 579 . . . 4 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}))
10 sylow1lem.s . . . . 5 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)}
1110fveq2i 6378 . . . 4 (♯‘𝑆) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃𝑁)})
129, 11syl6eqr 2817 . . 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 17718 . . . . . . . . . 10 (𝐺 ∈ Grp → 𝑋 ≠ ∅)
1714, 16syl 17 . . . . . . . . 9 (𝜑𝑋 ≠ ∅)
18 hasheq0 13356 . . . . . . . . . . 11 (𝑋 ∈ Fin → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
191, 18syl 17 . . . . . . . . . 10 (𝜑 → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
2019necon3bbid 2974 . . . . . . . . 9 (𝜑 → (¬ (♯‘𝑋) = 0 ↔ 𝑋 ≠ ∅))
2117, 20mpbird 248 . . . . . . . 8 (𝜑 → ¬ (♯‘𝑋) = 0)
22 hashcl 13349 . . . . . . . . . . 11 (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ0)
231, 22syl 17 . . . . . . . . . 10 (𝜑 → (♯‘𝑋) ∈ ℕ0)
24 elnn0 11540 . . . . . . . . . 10 ((♯‘𝑋) ∈ ℕ0 ↔ ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
2523, 24sylib 209 . . . . . . . . 9 (𝜑 → ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
2625ord 890 . . . . . . . 8 (𝜑 → (¬ (♯‘𝑋) ∈ ℕ → (♯‘𝑋) = 0))
2721, 26mt3d 142 . . . . . . 7 (𝜑 → (♯‘𝑋) ∈ ℕ)
28 dvdsle 15317 . . . . . . 7 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℕ) → ((𝑃𝑁) ∥ (♯‘𝑋) → (𝑃𝑁) ≤ (♯‘𝑋)))
297, 27, 28syl2anc 579 . . . . . 6 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) → (𝑃𝑁) ≤ (♯‘𝑋)))
3013, 29mpd 15 . . . . 5 (𝜑 → (𝑃𝑁) ≤ (♯‘𝑋))
316nnnn0d 11598 . . . . . . 7 (𝜑 → (𝑃𝑁) ∈ ℕ0)
32 nn0uz 11922 . . . . . . 7 0 = (ℤ‘0)
3331, 32syl6eleq 2854 . . . . . 6 (𝜑 → (𝑃𝑁) ∈ (ℤ‘0))
3423nn0zd 11727 . . . . . 6 (𝜑 → (♯‘𝑋) ∈ ℤ)
35 elfz5 12541 . . . . . 6 (((𝑃𝑁) ∈ (ℤ‘0) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
3633, 34, 35syl2anc 579 . . . . 5 (𝜑 → ((𝑃𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
3730, 36mpbird 248 . . . 4 (𝜑 → (𝑃𝑁) ∈ (0...(♯‘𝑋)))
38 bccl2 13314 . . . 4 ((𝑃𝑁) ∈ (0...(♯‘𝑋)) → ((♯‘𝑋)C(𝑃𝑁)) ∈ ℕ)
3937, 38syl 17 . . 3 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) ∈ ℕ)
4012, 39eqeltrrd 2845 . 2 (𝜑 → (♯‘𝑆) ∈ ℕ)
41 nnuz 11923 . . . . . . . . . . 11 ℕ = (ℤ‘1)
426, 41syl6eleq 2854 . . . . . . . . . 10 (𝜑 → (𝑃𝑁) ∈ (ℤ‘1))
43 elfz5 12541 . . . . . . . . . 10 (((𝑃𝑁) ∈ (ℤ‘1) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
4442, 34, 43syl2anc 579 . . . . . . . . 9 (𝜑 → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃𝑁) ≤ (♯‘𝑋)))
4530, 44mpbird 248 . . . . . . . 8 (𝜑 → (𝑃𝑁) ∈ (1...(♯‘𝑋)))
46 1zzd 11655 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
47 fzsubel 12584 . . . . . . . . 9 (((1 ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) ∧ ((𝑃𝑁) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1))))
4846, 34, 7, 46, 47syl22anc 867 . . . . . . . 8 (𝜑 → ((𝑃𝑁) ∈ (1...(♯‘𝑋)) ↔ ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1))))
4945, 48mpbid 223 . . . . . . 7 (𝜑 → ((𝑃𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1)))
50 1m1e0 11344 . . . . . . . 8 (1 − 1) = 0
5150oveq1i 6852 . . . . . . 7 ((1 − 1)...((♯‘𝑋) − 1)) = (0...((♯‘𝑋) − 1))
5249, 51syl6eleq 2854 . . . . . 6 (𝜑 → ((𝑃𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)))
53 bcp1nk 13308 . . . . . 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 11600 . . . . . . 7 (𝜑 → (♯‘𝑋) ∈ ℂ)
56 ax-1cn 10247 . . . . . . 7 1 ∈ ℂ
57 npcan 10544 . . . . . . 7 (((♯‘𝑋) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
5855, 56, 57sylancl 580 . . . . . 6 (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
596nncnd 11292 . . . . . . 7 (𝜑 → (𝑃𝑁) ∈ ℂ)
60 npcan 10544 . . . . . . 7 (((𝑃𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑃𝑁) − 1) + 1) = (𝑃𝑁))
6159, 56, 60sylancl 580 . . . . . 6 (𝜑 → (((𝑃𝑁) − 1) + 1) = (𝑃𝑁))
6258, 61oveq12d 6860 . . . . 5 (𝜑 → ((((♯‘𝑋) − 1) + 1)C(((𝑃𝑁) − 1) + 1)) = ((♯‘𝑋)C(𝑃𝑁)))
6358, 61oveq12d 6860 . . . . . 6 (𝜑 → ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1)) = ((♯‘𝑋) / (𝑃𝑁)))
6463oveq2d 6858 . . . . 5 (𝜑 → ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((((♯‘𝑋) − 1) + 1) / (((𝑃𝑁) − 1) + 1))) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁))))
6554, 62, 643eqtr3d 2807 . . . 4 (𝜑 → ((♯‘𝑋)C(𝑃𝑁)) = ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁))))
6665oveq2d 6858 . . 3 (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃𝑁))) = (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))))
6712oveq2d 6858 . . 3 (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃𝑁))) = (𝑃 pCnt (♯‘𝑆)))
68 bccl2 13314 . . . . . . 7 (((𝑃𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)) → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℕ)
6952, 68syl 17 . . . . . 6 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℕ)
7069nnzd 11728 . . . . 5 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ∈ ℤ)
7169nnne0d 11322 . . . . 5 (𝜑 → (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) ≠ 0)
726nnne0d 11322 . . . . . . 7 (𝜑 → (𝑃𝑁) ≠ 0)
73 dvdsval2 15268 . . . . . . 7 (((𝑃𝑁) ∈ ℤ ∧ (𝑃𝑁) ≠ 0 ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ))
747, 72, 34, 73syl3anc 1490 . . . . . 6 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ))
7513, 74mpbid 223 . . . . 5 (𝜑 → ((♯‘𝑋) / (𝑃𝑁)) ∈ ℤ)
7627nnne0d 11322 . . . . . 6 (𝜑 → (♯‘𝑋) ≠ 0)
7755, 59, 76, 72divne0d 11071 . . . . 5 (𝜑 → ((♯‘𝑋) / (𝑃𝑁)) ≠ 0)
78 pcmul 15835 . . . . 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 1498 . . . 4 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))) = ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁)))))
80 1cnd 10288 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
8155, 59, 80npncand 10670 . . . . . . . 8 (𝜑 → (((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1)) = ((♯‘𝑋) − 1))
8281oveq1d 6857 . . . . . . 7 (𝜑 → ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1)) = (((♯‘𝑋) − 1)C((𝑃𝑁) − 1)))
8382oveq2d 6858 . . . . . 6 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = (𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))))
846nnred 11291 . . . . . . . 8 (𝜑 → (𝑃𝑁) ∈ ℝ)
8584ltm1d 11210 . . . . . . 7 (𝜑 → ((𝑃𝑁) − 1) < (𝑃𝑁))
86 nnm1nn0 11581 . . . . . . . . 9 ((𝑃𝑁) ∈ ℕ → ((𝑃𝑁) − 1) ∈ ℕ0)
876, 86syl 17 . . . . . . . 8 (𝜑 → ((𝑃𝑁) − 1) ∈ ℕ0)
88 breq1 4812 . . . . . . . . . . 11 (𝑥 = 0 → (𝑥 < (𝑃𝑁) ↔ 0 < (𝑃𝑁)))
89 bcxmaslem1 14850 . . . . . . . . . . . . 13 (𝑥 = 0 → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0))
9089oveq2d 6858 . . . . . . . . . . . 12 (𝑥 = 0 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)))
9190eqeq1d 2767 . . . . . . . . . . 11 (𝑥 = 0 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0))
9288, 91imbi12d 335 . . . . . . . . . 10 (𝑥 = 0 → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ (0 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0)))
9392imbi2d 331 . . . . . . . . 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 4812 . . . . . . . . . . 11 (𝑥 = 𝑛 → (𝑥 < (𝑃𝑁) ↔ 𝑛 < (𝑃𝑁)))
95 bcxmaslem1 14850 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛))
9695oveq2d 6858 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)))
9796eqeq1d 2767 . . . . . . . . . . 11 (𝑥 = 𝑛 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0))
9894, 97imbi12d 335 . . . . . . . . . 10 (𝑥 = 𝑛 → ((𝑥 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0) ↔ (𝑛 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0)))
9998imbi2d 331 . . . . . . . . 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 4812 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → (𝑥 < (𝑃𝑁) ↔ (𝑛 + 1) < (𝑃𝑁)))
101 bcxmaslem1 14850 . . . . . . . . . . . . 13 (𝑥 = (𝑛 + 1) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102101oveq2d 6858 . . . . . . . . . . . 12 (𝑥 = (𝑛 + 1) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103102eqeq1d 2767 . . . . . . . . . . 11 (𝑥 = (𝑛 + 1) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
104100, 103imbi12d 335 . . . . . . . . . 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 331 . . . . . . . . 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 4812 . . . . . . . . . . 11 (𝑥 = ((𝑃𝑁) − 1) → (𝑥 < (𝑃𝑁) ↔ ((𝑃𝑁) − 1) < (𝑃𝑁)))
107 bcxmaslem1 14850 . . . . . . . . . . . . 13 (𝑥 = ((𝑃𝑁) − 1) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1)))
108107oveq2d 6858 . . . . . . . . . . . 12 (𝑥 = ((𝑃𝑁) − 1) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))))
109108eqeq1d 2767 . . . . . . . . . . 11 (𝑥 = ((𝑃𝑁) − 1) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0))
110106, 109imbi12d 335 . . . . . . . . . 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 331 . . . . . . . . 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 11671 . . . . . . . . . . . . . . . 16 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0))
1137, 34, 112syl2anc 579 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑃𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0))
11430, 113mpbid 223 . . . . . . . . . . . . . 14 (𝜑 → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
115 0nn0 11555 . . . . . . . . . . . . . 14 0 ∈ ℕ0
116 nn0addcl 11575 . . . . . . . . . . . . . 14 ((((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0 ∧ 0 ∈ ℕ0) → (((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0)
117114, 115, 116sylancl 580 . . . . . . . . . . . . 13 (𝜑 → (((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0)
118 bcn0 13301 . . . . . . . . . . . . 13 ((((♯‘𝑋) − (𝑃𝑁)) + 0) ∈ ℕ0 → ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0) = 1)
119117, 118syl 17 . . . . . . . . . . . 12 (𝜑 → ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0) = 1)
120119oveq2d 6858 . . . . . . . . . . 11 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121 pc1 15839 . . . . . . . . . . . 12 (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
1222, 121syl 17 . . . . . . . . . . 11 (𝜑 → (𝑃 pCnt 1) = 0)
123120, 122eqtrd 2799 . . . . . . . . . 10 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0)
124123a1d 25 . . . . . . . . 9 (𝜑 → (0 < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 0)C0)) = 0))
125 nn0re 11548 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0𝑛 ∈ ℝ)
126125ad2antrl 719 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℝ)
127 nn0p1nn 11579 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
128127ad2antrl 719 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℕ)
129128nnred 11291 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℝ)
1306adantr 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℕ)
131130nnred 11291 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℝ)
132126ltp1d 11208 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 < (𝑛 + 1))
133 simprr 789 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) < (𝑃𝑁))
134126, 129, 131, 132, 133lttrd 10452 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 < (𝑃𝑁))
135134expr 448 . . . . . . . . . . 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 6849 . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
139138nn0cnd 11600 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℂ)
140 nn0cn 11549 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ0𝑛 ∈ ℂ)
141140ad2antrl 719 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℂ)
142 1cnd 10288 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 1 ∈ ℂ)
143139, 141, 142addassd 10316 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) = (((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1)))
144143oveq1d 6857 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
145 nn0addge2 11587 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0) → 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛))
146126, 138, 145syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛))
147 simprl 787 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ ℕ0)
148147, 32syl6eleq 2854 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ (ℤ‘0))
149138, 147nn0addcld 11602 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℕ0)
150149nn0zd 11727 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℤ)
151 elfz5 12541 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ (ℤ‘0) ∧ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛) ∈ ℤ) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
152148, 150, 151syl2anc 579 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
153146, 152mpbird 248 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑛 ∈ (0...(((♯‘𝑋) − (𝑃𝑁)) + 𝑛)))
154 bcp1nk 13308 . . . . . . . . . . . . . . . 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 2801 . . . . . . . . . . . . . 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 6858 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑃 ∈ ℙ)
159 bccl2 13314 . . . . . . . . . . . . . . . 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 12002 . . . . . . . . . . . . . . 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 11322 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛) ≠ 0)
164150peano2zd 11732 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℤ)
165 znq 11993 . . . . . . . . . . . . . . 15 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
166164, 128, 165syl2anc 579 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
167 nn0p1nn 11579 . . . . . . . . . . . . . . . . 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 12041 . . . . . . . . . . . . . . . . 17 (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℝ+)
170 nnrp 12041 . . . . . . . . . . . . . . . . 17 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
171 rpdivcl 12054 . . . . . . . . . . . . . . . . 17 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
172169, 170, 171syl2an 589 . . . . . . . . . . . . . . . 16 ((((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
173168, 128, 172syl2anc 579 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
174173rpne0d 12075 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)
175 pcqmul 15837 . . . . . . . . . . . . . 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 1498 . . . . . . . . . . . . 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 2799 . . . . . . . . . . . 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 11322 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) ≠ 0)
179 pcdiv 15836 . . . . . . . . . . . . . . . 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 1494 . . . . . . . . . . . . . . 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 11292 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℂ)
182139, 181addcomd 10492 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1)) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))))
183143, 182eqtrd 2799 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))))
184183oveq2d 6858 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
185 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((♯‘𝑋) − (𝑃𝑁)) = 0)
186185oveq2d 6858 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))) = ((𝑛 + 1) + 0))
187181addid1d 10490 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑛 + 1) + 0) = (𝑛 + 1))
188187adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → ((𝑛 + 1) + 0) = (𝑛 + 1))
189186, 188eqtr2d 2800 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → (𝑛 + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁))))
190189oveq2d 6858 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) = 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
1912ad2antrr 717 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑃 ∈ ℙ)
192 nnq 12002 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℚ)
193128, 192syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℚ)
194193adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑛 + 1) ∈ ℚ)
195138nn0zd 11727 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ)
196 zq 11995 . . . . . . . . . . . . . . . . . . . . . 22 (((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
197195, 196syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
198197adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℚ)
199158, 128pccld 15834 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
200199nn0red 11599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
201200adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
2025adantr 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑁 ∈ ℕ0)
203202nn0red 11599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 𝑁 ∈ ℝ)
204203adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ∈ ℝ)
205 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ≠ 0)
206205neneqd 2942 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ¬ ((♯‘𝑋) − (𝑃𝑁)) = 0)
207114ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0)
208 elnn0 11540 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ0 ↔ (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃𝑁)) = 0))
209207, 208sylib 209 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃𝑁)) = 0))
210209ord 890 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (¬ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ → ((♯‘𝑋) − (𝑃𝑁)) = 0))
211206, 210mt3d 142 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℕ)
212191, 211pccld 15834 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ∈ ℕ0)
213212nn0red 11599 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ∈ ℝ)
214128nnzd 11728 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑛 + 1) ∈ ℤ)
215 pcdvdsb 15852 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑁) ∥ (𝑛 + 1)))
216158, 214, 202, 215syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃𝑁) ∥ (𝑛 + 1)))
2177adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃𝑁) ∈ ℤ)
218 dvdsle 15317 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑃𝑁) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃𝑁) ∥ (𝑛 + 1) → (𝑃𝑁) ≤ (𝑛 + 1)))
219217, 128, 218syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃𝑁) ∥ (𝑛 + 1) → (𝑃𝑁) ≤ (𝑛 + 1)))
220216, 219sylbid 231 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) → (𝑃𝑁) ≤ (𝑛 + 1)))
221203, 200lenltd 10437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ ¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
222131, 129lenltd 10437 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃𝑁) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃𝑁)))
223220, 221, 2223imtr3d 284 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 → ¬ (𝑛 + 1) < (𝑃𝑁)))
224133, 223mt4d 153 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225224adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
226 dvdssubr 15312 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑃𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
2277, 34, 226syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑃𝑁) ∥ (♯‘𝑋) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
22813, 227mpbid 223 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁)))
229228ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁)))
230207nn0zd 11727 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ)
2315ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ∈ ℕ0)
232 pcdvdsb 15852 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) − (𝑃𝑁)) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
233191, 230, 231, 232syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))) ↔ (𝑃𝑁) ∥ ((♯‘𝑋) − (𝑃𝑁))))
234229, 233mpbird 248 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → 𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))))
235201, 204, 213, 225, 234ltletrd 10451 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < (𝑃 pCnt ((♯‘𝑋) − (𝑃𝑁))))
236191, 194, 198, 235pcadd2 15873 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) ∧ ((♯‘𝑋) − (𝑃𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
237190, 236pm2.61dane 3024 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃𝑁)))))
238184, 237eqtr4d 2802 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
239199nn0cnd 11600 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
240238, 239eqeltrd 2844 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) ∈ ℂ)
241240, 238subeq0bd 10710 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))) = 0)
242180, 241eqtrd 2799 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
243242oveq2d 6858 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
244 00id 10465 . . . . . . . . . . . . 13 (0 + 0) = 0
245243, 244syl6req 2816 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → 0 = (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
246177, 245eqeq12d 2780 . . . . . . . . . . 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))))))
247137, 246syl5ibr 237 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + 𝑛)C𝑛)) = 0 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
248136, 247animpimp2impd 872 . . . . . . . . 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))))
24993, 99, 105, 111, 124, 248nn0ind 11719 . . . . . . . 8 (((𝑃𝑁) − 1) ∈ ℕ0 → (𝜑 → (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0)))
25087, 249mpcom 38 . . . . . . 7 (𝜑 → (((𝑃𝑁) − 1) < (𝑃𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0))
25185, 250mpd 15 . . . . . 6 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃𝑁)) + ((𝑃𝑁) − 1))C((𝑃𝑁) − 1))) = 0)
25283, 251eqtr3d 2801 . . . . 5 (𝜑 → (𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) = 0)
253 pcdiv 15836 . . . . . . 7 ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) ∈ ℤ ∧ (♯‘𝑋) ≠ 0) ∧ (𝑃𝑁) ∈ ℕ) → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))))
2542, 34, 76, 6, 253syl121anc 1494 . . . . . 6 (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))))
2555nn0zd 11727 . . . . . . . 8 (𝜑𝑁 ∈ ℤ)
256 pcid 15856 . . . . . . . 8 ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝑃𝑁)) = 𝑁)
2572, 255, 256syl2anc 579 . . . . . . 7 (𝜑 → (𝑃 pCnt (𝑃𝑁)) = 𝑁)
258257oveq2d 6858 . . . . . 6 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
259254, 258eqtrd 2799 . . . . 5 (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
260252, 259oveq12d 6860 . . . 4 (𝜑 → ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃𝑁)))) = (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
2612, 27pccld 15834 . . . . . . . 8 (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ0)
262261nn0zd 11727 . . . . . . 7 (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
263262, 255zsubcld 11734 . . . . . 6 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
264263zcnd 11730 . . . . 5 (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℂ)
265264addid2d 10491 . . . 4 (𝜑 → (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26679, 260, 2653eqtrd 2803 . . 3 (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃𝑁) − 1)) · ((♯‘𝑋) / (𝑃𝑁)))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26766, 67, 2663eqtr3d 2807 . 2 (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
26840, 267jca 507 1 (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2937  {crab 3059  c0 4079  𝒫 cpw 4315   class class class wbr 4809  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194   < clt 10328  cle 10329  cmin 10520   / cdiv 10938  cn 11274  0cn0 11538  cz 11624  cuz 11886  cq 11989  +crp 12028  ...cfz 12533  cexp 13067  Ccbc 13293  chash 13321  cdvds 15265  cprime 15665   pCnt cpc 15820  Basecbs 16130  +gcplusg 16214  Grpcgrp 17689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-q 11990  df-rp 12029  df-fz 12534  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-fac 13265  df-bc 13294  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-dvds 15266  df-gcd 15498  df-prm 15666  df-pc 15821  df-0g 16368  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-grp 17692
This theorem is referenced by:  sylow1lem3  18279
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