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Theorem sylow1lem1 17836

Description: Lemma for sylow1 17841. The p-adic valuation of the size of 𝑆 is equal to the number of excess powers of 𝑃 in (#‘𝑋) / (𝑃↑𝑁). (Contributed by Mario Carneiro, 15-Jan-2015.)

**Hypotheses**
Ref	Expression
sylow1.x	⊢ 𝑋 = (Base‘𝐺)
sylow1.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow1.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow1.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow1.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
sylow1.d	⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝑋))
sylow1lem.a	⊢ + = (+_g‘𝐺)
sylow1lem.s	⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}

**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.

Step	Hyp	Ref	Expression
1		sylow1.f	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Fin)
2		sylow1.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
3		prmnn 15226	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ ℕ)
5		sylow1.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	4, 5	nnexpcld 12892	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ)
7	6	nnzd 11357	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℤ)
8		hashbc 13094	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑃↑𝑁) ∈ ℤ) → ((#‘𝑋)C(𝑃↑𝑁)) = (#‘{𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}))
9	1, 7, 8	syl2anc 691	. . . 4 ⊢ (𝜑 → ((#‘𝑋)C(𝑃↑𝑁)) = (#‘{𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}))
10		sylow1lem.s	. . . . 5 ⊢ 𝑆 = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}
11	10	fveq2i 6106	. . . 4 ⊢ (#‘𝑆) = (#‘{𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)})
12	9, 11	syl6eqr 2662	. . 3 ⊢ (𝜑 → ((#‘𝑋)C(𝑃↑𝑁)) = (#‘𝑆))
13		sylow1.d	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (#‘𝑋))
14		sylow1.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ Grp)
15		sylow1.x	. . . . . . . . . . 11 ⊢ 𝑋 = (Base‘𝐺)
16	15	grpbn0 17274	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
18		hasheq0 13015	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) = 0 ↔ 𝑋 = ∅))
19	1, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝑋) = 0 ↔ 𝑋 = ∅))
20	19	necon3bbid 2819	. . . . . . . . 9 ⊢ (𝜑 → (¬ (#‘𝑋) = 0 ↔ 𝑋 ≠ ∅))
21	17, 20	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → ¬ (#‘𝑋) = 0)
22		hashcl 13009	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (#‘𝑋) ∈ ℕ₀)
23	1, 22	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ₀)
24		elnn0 11171	. . . . . . . . . 10 ⊢ ((#‘𝑋) ∈ ℕ₀ ↔ ((#‘𝑋) ∈ ℕ ∨ (#‘𝑋) = 0))
25	23, 24	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ∨ (#‘𝑋) = 0))
26	25	ord 391	. . . . . . . 8 ⊢ (𝜑 → (¬ (#‘𝑋) ∈ ℕ → (#‘𝑋) = 0))
27	21, 26	mt3d 139	. . . . . . 7 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
28		dvdsle 14870	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℕ) → ((𝑃↑𝑁) ∥ (#‘𝑋) → (𝑃↑𝑁) ≤ (#‘𝑋)))
29	7, 27, 28	syl2anc 691	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) → (𝑃↑𝑁) ≤ (#‘𝑋)))
30	13, 29	mpd 15	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ≤ (#‘𝑋))
31	6	nnnn0d 11228	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ₀)
32		nn0uz 11598	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	syl6eleq 2698	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ_≥‘0))
34	23	nn0zd 11356	. . . . . 6 ⊢ (𝜑 → (#‘𝑋) ∈ ℤ)
35		elfz5 12205	. . . . . 6 ⊢ (((𝑃↑𝑁) ∈ (ℤ_≥‘0) ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (0...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
36	33, 34, 35	syl2anc 691	. . . . 5 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (0...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
37	30, 36	mpbird 246	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (0...(#‘𝑋)))
38		bccl2 12972	. . . 4 ⊢ ((𝑃↑𝑁) ∈ (0...(#‘𝑋)) → ((#‘𝑋)C(𝑃↑𝑁)) ∈ ℕ)
39	37, 38	syl 17	. . 3 ⊢ (𝜑 → ((#‘𝑋)C(𝑃↑𝑁)) ∈ ℕ)
40	12, 39	eqeltrrd 2689	. 2 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ)
41		nnuz 11599	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
42	6, 41	syl6eleq 2698	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ_≥‘1))
43		elfz5 12205	. . . . . . . . . 10 ⊢ (((𝑃↑𝑁) ∈ (ℤ_≥‘1) ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
44	42, 34, 43	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
45	30, 44	mpbird 246	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (1...(#‘𝑋)))
46		1zzd 11285	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
47		fzsubel 12248	. . . . . . . . 9 ⊢ (((1 ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) ∧ ((𝑃↑𝑁) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((#‘𝑋) − 1))))
48	46, 34, 7, 46, 47	syl22anc 1319	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((#‘𝑋) − 1))))
49	45, 48	mpbid 221	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((#‘𝑋) − 1)))
50		1m1e0 10966	. . . . . . . 8 ⊢ (1 − 1) = 0
51	50	oveq1i 6559	. . . . . . 7 ⊢ ((1 − 1)...((#‘𝑋) − 1)) = (0...((#‘𝑋) − 1))
52	49, 51	syl6eleq 2698	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ (0...((#‘𝑋) − 1)))
53		bcp1nk 12966	. . . . . 6 ⊢ (((𝑃↑𝑁) − 1) ∈ (0...((#‘𝑋) − 1)) → ((((#‘𝑋) − 1) + 1)C(((𝑃↑𝑁) − 1) + 1)) = ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((((#‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1))))
54	52, 53	syl 17	. . . . 5 ⊢ (𝜑 → ((((#‘𝑋) − 1) + 1)C(((𝑃↑𝑁) − 1) + 1)) = ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((((#‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1))))
55	23	nn0cnd 11230	. . . . . . 7 ⊢ (𝜑 → (#‘𝑋) ∈ ℂ)
56		ax-1cn 9873	. . . . . . 7 ⊢ 1 ∈ ℂ
57		npcan 10169	. . . . . . 7 ⊢ (((#‘𝑋) ∈ ℂ ∧ 1 ∈ ℂ) → (((#‘𝑋) − 1) + 1) = (#‘𝑋))
58	55, 56, 57	sylancl 693	. . . . . 6 ⊢ (𝜑 → (((#‘𝑋) − 1) + 1) = (#‘𝑋))
59	6	nncnd 10913	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℂ)
60		npcan 10169	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
61	59, 56, 60	sylancl 693	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
62	58, 61	oveq12d 6567	. . . . 5 ⊢ (𝜑 → ((((#‘𝑋) − 1) + 1)C(((𝑃↑𝑁) − 1) + 1)) = ((#‘𝑋)C(𝑃↑𝑁)))
63	58, 61	oveq12d 6567	. . . . . 6 ⊢ (𝜑 → ((((#‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1)) = ((#‘𝑋) / (𝑃↑𝑁)))
64	63	oveq2d 6565	. . . . 5 ⊢ (𝜑 → ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((((#‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1))) = ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁))))
65	54, 62, 64	3eqtr3d 2652	. . . 4 ⊢ (𝜑 → ((#‘𝑋)C(𝑃↑𝑁)) = ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁))))
66	65	oveq2d 6565	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))))
67	12	oveq2d 6565	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt (#‘𝑆)))
68		bccl2 12972	. . . . . . 7 ⊢ (((𝑃↑𝑁) − 1) ∈ (0...((#‘𝑋) − 1)) → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℕ)
69	52, 68	syl 17	. . . . . 6 ⊢ (𝜑 → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℕ)
70	69	nnzd 11357	. . . . 5 ⊢ (𝜑 → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℤ)
71	69	nnne0d 10942	. . . . 5 ⊢ (𝜑 → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ≠ 0)
72	6	nnne0d 10942	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ≠ 0)
73		dvdsval2 14824	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑃↑𝑁) ≠ 0 ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
74	7, 72, 34, 73	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
75	13, 74	mpbid 221	. . . . 5 ⊢ (𝜑 → ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ)
76	27	nnne0d 10942	. . . . . 6 ⊢ (𝜑 → (#‘𝑋) ≠ 0)
77	55, 59, 76, 72	divne0d 10696	. . . . 5 ⊢ (𝜑 → ((#‘𝑋) / (𝑃↑𝑁)) ≠ 0)
78		pcmul 15394	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℤ ∧ (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ≠ 0) ∧ (((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ ∧ ((#‘𝑋) / (𝑃↑𝑁)) ≠ 0)) → (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))) = ((𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) + (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁)))))
79	2, 70, 71, 75, 77, 78	syl122anc 1327	. . . 4 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))) = ((𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) + (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁)))))
80		1cnd 9935	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
81	55, 59, 80	npncand 10295	. . . . . . . 8 ⊢ (𝜑 → (((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1)) = ((#‘𝑋) − 1))
82	81	oveq1d 6564	. . . . . . 7 ⊢ (𝜑 → ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)) = (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)))
83	82	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = (𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))))
84	6	nnred 10912	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℝ)
85	84	ltm1d 10835	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) < (𝑃↑𝑁))
86		nnm1nn0 11211	. . . . . . . . 9 ⊢ ((𝑃↑𝑁) ∈ ℕ → ((𝑃↑𝑁) − 1) ∈ ℕ₀)
87	6, 86	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ℕ₀)
88		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 < (𝑃↑𝑁) ↔ 0 < (𝑃↑𝑁)))
89		bcxmaslem1 14405	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0))
90	89	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)))
91	90	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0))
92	88, 91	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 0 → ((𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0) ↔ (0 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0)))
93	92	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝜑 → (𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (0 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0))))
94		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 < (𝑃↑𝑁) ↔ 𝑛 < (𝑃↑𝑁)))
95		bcxmaslem1 14405	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛))
96	95	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)))
97	96	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))
98	94, 97	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = 𝑛 → ((𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0) ↔ (𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)))
99	98	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))))
100		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 < (𝑃↑𝑁) ↔ (𝑛 + 1) < (𝑃↑𝑁)))
101		bcxmaslem1 14405	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 + 1) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102	101	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103	102	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
104	100, 103	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0) ↔ ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
105	104	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
106		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → (𝑥 < (𝑃↑𝑁) ↔ ((𝑃↑𝑁) − 1) < (𝑃↑𝑁)))
107		bcxmaslem1 14405	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)))
108	107	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))))
109	108	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0))
110	106, 109	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0) ↔ (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0)))
111	110	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((𝜑 → (𝑥 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0)) ↔ (𝜑 → (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0))))
112		znn0sub 11301	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ≤ (#‘𝑋) ↔ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀))
113	7, 34, 112	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑃↑𝑁) ≤ (#‘𝑋) ↔ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀))
114	30, 113	mpbid 221	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
115		0nn0 11184	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
116		nn0addcl 11205	. . . . . . . . . . . . . 14 ⊢ ((((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → (((#‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ₀)
117	114, 115, 116	sylancl 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((#‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ₀)
118		bcn0 12959	. . . . . . . . . . . . 13 ⊢ ((((#‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ₀ → ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0) = 1)
119	117, 118	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0) = 1)
120	119	oveq2d 6565	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121		pc1 15398	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
122	2, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt 1) = 0)
123	120, 122	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0)
124	123	a1d 25	. . . . . . . . 9 ⊢ (𝜑 → (0 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0))
125		nn0re 11178	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
126	125	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℝ)
127		nn0p1nn 11209	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
128	127	ad2antrl 760	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℕ)
129	128	nnred 10912	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℝ)
130	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℕ)
131	130	nnred 10912	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℝ)
132	126	ltp1d 10833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑛 + 1))
133		simprr 792	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) < (𝑃↑𝑁))
134	126, 129, 131, 132, 133	lttrd 10077	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑃↑𝑁))
135	134	expr 641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 + 1) < (𝑃↑𝑁) → 𝑛 < (𝑃↑𝑁)))
136	135	imim1d 80	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0) → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)))
137		oveq1 6556	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0 → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
138	114	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
139	138	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℂ)
140		nn0cn 11179	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
141	140	ad2antrl 760	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℂ)
142		1cnd 9935	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 1 ∈ ℂ)
143	139, 141, 142	addassd 9941	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = (((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1)))
144	143	oveq1d 6564	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
145		nn0addge2 11217	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℝ ∧ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀) → 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛))
146	126, 138, 145	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛))
147		simprl 790	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℕ₀)
148	147, 32	syl6eleq 2698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (ℤ_≥‘0))
149	138, 147	nn0addcld 11232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℕ₀)
150	149	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℤ)
151		elfz5 12205	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ (ℤ_≥‘0) ∧ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℤ) → (𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
152	148, 150, 151	syl2anc 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
153	146, 152	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
154		bcp1nk 12966	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
155	153, 154	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)C(𝑛 + 1)) = (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
156	144, 155	eqtr3d 2646	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)) = (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))
157	156	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
158	2	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑃 ∈ ℙ)
159		bccl2 12972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ)
160	153, 159	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ)
161		nnq 11677	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℚ)
162	160, 161	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℚ)
163	160	nnne0d 10942	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ≠ 0)
164	150	peano2zd 11361	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ)
165		znq 11668	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
166	164, 128, 165	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
167		nn0p1nn 11209	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℕ₀ → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ)
168	149, 167	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ)
169		nnrp 11718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ⁺)
170		nnrp 11718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℝ⁺)
171		rpdivcl 11732	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
172	169, 170, 171	syl2an 493	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
173	168, 128, 172	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
174	173	rpne0d 11753	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)
175		pcqmul 15396	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 ∈ ℙ ∧ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℚ ∧ ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ≠ 0) ∧ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ ∧ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
176	158, 162, 163, 166, 174, 175	syl122anc 1327	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) · (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
177	157, 176	eqtrd 2644	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
178	168	nnne0d 10942	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ≠ 0)
179		pcdiv 15395	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑃 ∈ ℙ ∧ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ ∧ ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ≠ 0) ∧ (𝑛 + 1) ∈ ℕ) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))))
180	158, 164, 178, 128, 179	syl121anc 1323	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))))
181	128	nncnd 10913	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℂ)
182	139, 181	addcomd 10117	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1)) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
183	143, 182	eqtrd 2644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
184	183	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
185		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((#‘𝑋) − (𝑃↑𝑁)) = 0)
186	185	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))) = ((𝑛 + 1) + 0))
187	181	addid1d 10115	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑛 + 1) + 0) = (𝑛 + 1))
188	187	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + 0) = (𝑛 + 1))
189	186, 188	eqtr2d 2645	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → (𝑛 + 1) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
190	189	oveq2d 6565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
191	2	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑃 ∈ ℙ)
192		nnq 11677	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℚ)
193	128, 192	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℚ)
194	193	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑛 + 1) ∈ ℚ)
195	138	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
196		zq 11670	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
197	195, 196	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
198	197	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
199	158, 128	pccld 15393	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ₀)
200	199	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
201	200	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
202	5	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℕ₀)
203	202	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℝ)
204	203	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℝ)
205		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0)
206	205	neneqd 2787	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ¬ ((#‘𝑋) − (𝑃↑𝑁)) = 0)
207	114	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
208		elnn0 11171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀ ↔ (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((#‘𝑋) − (𝑃↑𝑁)) = 0))
209	207, 208	sylib 207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((#‘𝑋) − (𝑃↑𝑁)) = 0))
210	209	ord 391	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (¬ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ → ((#‘𝑋) − (𝑃↑𝑁)) = 0))
211	206, 210	mt3d 139	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ)
212	191, 211	pccld 15393	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ∈ ℕ₀)
213	212	nn0red 11229	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ∈ ℝ)
214	128	nnzd 11357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℤ)
215		pcdvdsb 15411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
216	158, 214, 202, 215	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
217	7	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℤ)
218		dvdsle 14870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
219	217, 128, 218	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
220	216, 219	sylbid 229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
221	203, 200	lenltd 10062	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ ¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
222	131, 129	lenltd 10062	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑𝑁)))
223	220, 221, 222	3imtr3d 281	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 → ¬ (𝑛 + 1) < (𝑃↑𝑁)))
224	133, 223	mt4d 151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225	224	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
226		dvdssubr 14865	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
227	7, 34, 226	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
228	13, 227	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁)))
229	228	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁)))
230	207	nn0zd 11356	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
231	5	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℕ₀)
232		pcdvdsb 15411	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ ℙ ∧ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
233	191, 230, 231, 232	syl3anc 1318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑁 ≤ (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
234	229, 233	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ≤ (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))))
235	201, 204, 213, 225, 234	ltletrd 10076	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))))
236	191, 194, 198, 235	pcadd2 15432	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
237	190, 236	pm2.61dane 2869	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
238	184, 237	eqtr4d 2647	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
239	199	nn0cnd 11230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
240	238, 239	eqeltrd 2688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) ∈ ℂ)
241	240, 238	subeq0bd 10335	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))) = 0)
242	180, 241	eqtrd 2644	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
243	242	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
244		00id 10090	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
245	243, 244	syl6req 2661	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 0 = (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
246	177, 245	eqeq12d 2625	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0 ↔ ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))))))
247	137, 246	syl5ibr 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))
248	247	expr 641	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 + 1) < (𝑃↑𝑁) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
249	248	a2d 29	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → (((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0) → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
250	136, 249	syld 46	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0) → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0)))
251	250	expcom 450	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝜑 → ((𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0) → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
252	251	a2d 29	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ₀ → ((𝜑 → (𝑛 < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0)) → (𝜑 → ((𝑛 + 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))) = 0))))
253	93, 99, 105, 111, 124, 252	nn0ind 11348	. . . . . . . 8 ⊢ (((𝑃↑𝑁) − 1) ∈ ℕ₀ → (𝜑 → (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0)))
254	87, 253	mpcom 37	. . . . . . 7 ⊢ (𝜑 → (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0))
255	85, 254	mpd 15	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0)
256	83, 255	eqtr3d 2646	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) = 0)
257		pcdiv 15395	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ((#‘𝑋) ∈ ℤ ∧ (#‘𝑋) ≠ 0) ∧ (𝑃↑𝑁) ∈ ℕ) → (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))))
258	2, 34, 76, 6, 257	syl121anc 1323	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))))
259	5	nn0zd 11356	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
260		pcid 15415	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
261	2, 259, 260	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
262	261	oveq2d 6565	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
263	258, 262	eqtrd 2644	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
264	256, 263	oveq12d 6567	. . . 4 ⊢ (𝜑 → ((𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) + (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁)))) = (0 + ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
265	2, 27	pccld 15393	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ₀)
266	265	nn0zd 11356	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
267	266, 259	zsubcld 11363	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
268	267	zcnd 11359	. . . . 5 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℂ)
269	268	addid2d 10116	. . . 4 ⊢ (𝜑 → (0 + ((𝑃 pCnt (#‘𝑋)) − 𝑁)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
270	79, 264, 269	3eqtrd 2648	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
271	66, 67, 270	3eqtr3d 2652	. 2 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
272	40, 271	jca 553	1 ⊢ (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {crab 2900 ∅c0 3874 𝒫 cpw 4108 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ℤ_≥cuz 11563 ℚcq 11664 ℝ⁺crp 11708 ...cfz 12197 ↑cexp 12722 Ccbc 12951 #chash 12979 ∥ cdvds 14821 ℙcprime 15223 pCnt cpc 15379 Basecbs 15695 +_gcplusg 15768 Grpcgrp 17245

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-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-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-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-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-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-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 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-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-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

This theorem is referenced by: sylow1lem3 17838

W3C validator