Theorem sylow1lem1 18013

Description: Lemma for sylow1 18018. 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 15388	. . . . . . . 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 13030	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ)
7	6	nnzd 11481	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℤ)
8		hashbc 13237	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑃↑𝑁) ∈ ℤ) → ((#‘𝑋)C(𝑃↑𝑁)) = (#‘{𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}))
9	1, 7, 8	syl2anc 693	. . . 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 6194	. . . 4 ⊢ (#‘𝑆) = (#‘{𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)})
12	9, 11	syl6eqr 2674	. . 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 17451	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
18		hasheq0 13154	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → ((#‘𝑋) = 0 ↔ 𝑋 = ∅))
19	1, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((#‘𝑋) = 0 ↔ 𝑋 = ∅))
20	19	necon3bbid 2831	. . . . . . . . 9 ⊢ (𝜑 → (¬ (#‘𝑋) = 0 ↔ 𝑋 ≠ ∅))
21	17, 20	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → ¬ (#‘𝑋) = 0)
22		hashcl 13147	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (#‘𝑋) ∈ ℕ0)
23	1, 22	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ0)
24		elnn0 11294	. . . . . . . . . 10 ⊢ ((#‘𝑋) ∈ ℕ0 ↔ ((#‘𝑋) ∈ ℕ ∨ (#‘𝑋) = 0))
25	23, 24	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → ((#‘𝑋) ∈ ℕ ∨ (#‘𝑋) = 0))
26	25	ord 392	. . . . . . . 8 ⊢ (𝜑 → (¬ (#‘𝑋) ∈ ℕ → (#‘𝑋) = 0))
27	21, 26	mt3d 140	. . . . . . 7 ⊢ (𝜑 → (#‘𝑋) ∈ ℕ)
28		dvdsle 15032	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℕ) → ((𝑃↑𝑁) ∥ (#‘𝑋) → (𝑃↑𝑁) ≤ (#‘𝑋)))
29	7, 27, 28	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) → (𝑃↑𝑁) ≤ (#‘𝑋)))
30	13, 29	mpd 15	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ≤ (#‘𝑋))
31	6	nnnn0d 11351	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ0)
32		nn0uz 11722	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
33	31, 32	syl6eleq 2711	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ≥‘0))
34	23	nn0zd 11480	. . . . . 6 ⊢ (𝜑 → (#‘𝑋) ∈ ℤ)
35		elfz5 12334	. . . . . 6 ⊢ (((𝑃↑𝑁) ∈ (ℤ≥‘0) ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (0...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
36	33, 34, 35	syl2anc 693	. . . . 5 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (0...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
37	30, 36	mpbird 247	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (0...(#‘𝑋)))
38		bccl2 13110	. . . 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 2702	. 2 ⊢ (𝜑 → (#‘𝑆) ∈ ℕ)
41		nnuz 11723	. . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1)
42	6, 41	syl6eleq 2711	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ≥‘1))
43		elfz5 12334	. . . . . . . . . 10 ⊢ (((𝑃↑𝑁) ∈ (ℤ≥‘1) ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
44	42, 34, 43	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ (𝑃↑𝑁) ≤ (#‘𝑋)))
45	30, 44	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (1...(#‘𝑋)))
46		1zzd 11408	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
47		fzsubel 12377	. . . . . . . . 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 1327	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(#‘𝑋)) ↔ ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((#‘𝑋) − 1))))
49	45, 48	mpbid 222	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((#‘𝑋) − 1)))
50		1m1e0 11089	. . . . . . . 8 ⊢ (1 − 1) = 0
51	50	oveq1i 6660	. . . . . . 7 ⊢ ((1 − 1)...((#‘𝑋) − 1)) = (0...((#‘𝑋) − 1))
52	49, 51	syl6eleq 2711	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ (0...((#‘𝑋) − 1)))
53		bcp1nk 13104	. . . . . 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 11353	. . . . . . 7 ⊢ (𝜑 → (#‘𝑋) ∈ ℂ)
56		ax-1cn 9994	. . . . . . 7 ⊢ 1 ∈ ℂ
57		npcan 10290	. . . . . . 7 ⊢ (((#‘𝑋) ∈ ℂ ∧ 1 ∈ ℂ) → (((#‘𝑋) − 1) + 1) = (#‘𝑋))
58	55, 56, 57	sylancl 694	. . . . . 6 ⊢ (𝜑 → (((#‘𝑋) − 1) + 1) = (#‘𝑋))
59	6	nncnd 11036	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℂ)
60		npcan 10290	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
61	59, 56, 60	sylancl 694	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
62	58, 61	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((((#‘𝑋) − 1) + 1)C(((𝑃↑𝑁) − 1) + 1)) = ((#‘𝑋)C(𝑃↑𝑁)))
63	58, 61	oveq12d 6668	. . . . . 6 ⊢ (𝜑 → ((((#‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1)) = ((#‘𝑋) / (𝑃↑𝑁)))
64	63	oveq2d 6666	. . . . 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 2664	. . . 4 ⊢ (𝜑 → ((#‘𝑋)C(𝑃↑𝑁)) = ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁))))
66	65	oveq2d 6666	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))))
67	12	oveq2d 6666	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt (#‘𝑆)))
68		bccl2 13110	. . . . . . 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 11481	. . . . 5 ⊢ (𝜑 → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℤ)
71	69	nnne0d 11065	. . . . 5 ⊢ (𝜑 → (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ≠ 0)
72	6	nnne0d 11065	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ≠ 0)
73		dvdsval2 14986	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑃↑𝑁) ≠ 0 ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
74	7, 72, 34, 73	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
75	13, 74	mpbid 222	. . . . 5 ⊢ (𝜑 → ((#‘𝑋) / (𝑃↑𝑁)) ∈ ℤ)
76	27	nnne0d 11065	. . . . . 6 ⊢ (𝜑 → (#‘𝑋) ≠ 0)
77	55, 59, 76, 72	divne0d 10817	. . . . 5 ⊢ (𝜑 → ((#‘𝑋) / (𝑃↑𝑁)) ≠ 0)
78		pcmul 15556	. . . . 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 1335	. . . 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 10056	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
81	55, 59, 80	npncand 10416	. . . . . . . 8 ⊢ (𝜑 → (((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1)) = ((#‘𝑋) − 1))
82	81	oveq1d 6665	. . . . . . 7 ⊢ (𝜑 → ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)) = (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)))
83	82	oveq2d 6666	. . . . . 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 11035	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℝ)
85	84	ltm1d 10956	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) < (𝑃↑𝑁))
86		nnm1nn0 11334	. . . . . . . . 9 ⊢ ((𝑃↑𝑁) ∈ ℕ → ((𝑃↑𝑁) − 1) ∈ ℕ0)
87	6, 86	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ℕ0)
88		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 < (𝑃↑𝑁) ↔ 0 < (𝑃↑𝑁)))
89		bcxmaslem1 14566	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0))
90	89	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)))
91	90	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0))
92	88, 91	imbi12d 334	. . . . . . . . . 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 330	. . . . . . . . 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 4656	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 < (𝑃↑𝑁) ↔ 𝑛 < (𝑃↑𝑁)))
95		bcxmaslem1 14566	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛))
96	95	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)))
97	96	eqeq1d 2624	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))
98	94, 97	imbi12d 334	. . . . . . . . . 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 330	. . . . . . . . 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 4656	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 < (𝑃↑𝑁) ↔ (𝑛 + 1) < (𝑃↑𝑁)))
101		bcxmaslem1 14566	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 + 1) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102	101	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103	102	eqeq1d 2624	. . . . . . . . . . 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 334	. . . . . . . . . 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 330	. . . . . . . . 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 4656	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → (𝑥 < (𝑃↑𝑁) ↔ ((𝑃↑𝑁) − 1) < (𝑃↑𝑁)))
107		bcxmaslem1 14566	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((#‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)))
108	107	oveq2d 6666	. . . . . . . . . . . 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 2624	. . . . . . . . . . 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 334	. . . . . . . . . 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 330	. . . . . . . . 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 11424	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ≤ (#‘𝑋) ↔ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0))
113	7, 34, 112	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑃↑𝑁) ≤ (#‘𝑋) ↔ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0))
114	30, 113	mpbid 222	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0)
115		0nn0 11307	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
116		nn0addcl 11328	. . . . . . . . . . . . . 14 ⊢ ((((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0 ∧ 0 ∈ ℕ0) → (((#‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ0)
117	114, 115, 116	sylancl 694	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((#‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ0)
118		bcn0 13097	. . . . . . . . . . . . 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 6666	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121		pc1 15560	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
122	2, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt 1) = 0)
123	120, 122	eqtrd 2656	. . . . . . . . . 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 11301	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℝ)
126	125	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℝ)
127		nn0p1nn 11332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ0 → (𝑛 + 1) ∈ ℕ)
128	127	ad2antrl 764	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℕ)
129	128	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℝ)
130	6	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℕ)
131	130	nnred 11035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℝ)
132	126	ltp1d 10954	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑛 + 1))
133		simprr 796	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) < (𝑃↑𝑁))
134	126, 129, 131, 132, 133	lttrd 10198	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑃↑𝑁))
135	134	expr 643	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → ((𝑛 + 1) < (𝑃↑𝑁) → 𝑛 < (𝑃↑𝑁)))
136	135	imim1d 82	. . . . . . . . . . . 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 6657	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0)
139	138	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℂ)
140		nn0cn 11302	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ)
141	140	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℂ)
142		1cnd 10056	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 1 ∈ ℂ)
143	139, 141, 142	addassd 10062	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = (((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1)))
144	143	oveq1d 6665	. . . . . . . . . . . . . . . . . . 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 11340	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ ℝ ∧ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0) → 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛))
146	126, 138, 145	syl2anc 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛))
147		simprl 794	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℕ0)
148	147, 32	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (ℤ≥‘0))
149	138, 147	nn0addcld 11355	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℕ0)
150	149	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℤ)
151		elfz5 12334	. . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
153	146, 152	mpbird 247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (0...(((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
154		bcp1nk 13104	. . . . . . . . . . . . . . . . . . . 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 2658	. . . . . . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑃 ∈ ℙ)
159		bccl2 13110	. . . . . . . . . . . . . . . . . . . 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 11801	. . . . . . . . . . . . . . . . . . 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 11065	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ≠ 0)
164	150	peano2zd 11485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ)
165		znq 11792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
166	164, 128, 165	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
167		nn0p1nn 11332	. . . . . . . . . . . . . . . . . . . . 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 11842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ+)
170		nnrp 11842	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℝ+)
171		rpdivcl 11856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
172	169, 170, 171	syl2an 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
173	168, 128, 172	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ+)
174	173	rpne0d 11877	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)
175		pcqmul 15558	. . . . . . . . . . . . . . . . . 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 1335	. . . . . . . . . . . . . . . . 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 2656	. . . . . . . . . . . . . . . 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 11065	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ≠ 0)
179		pcdiv 15557	. . . . . . . . . . . . . . . . . . . 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 1331	. . . . . . . . . . . . . . . . . . 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 11036	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℂ)
182	139, 181	addcomd 10238	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((#‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1)) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
183	143, 182	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
184	183	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
185		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((#‘𝑋) − (𝑃↑𝑁)) = 0)
186	185	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))) = ((𝑛 + 1) + 0))
187	181	addid1d 10236	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑛 + 1) + 0) = (𝑛 + 1))
188	187	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + 0) = (𝑛 + 1))
189	186, 188	eqtr2d 2657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) = 0) → (𝑛 + 1) = ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁))))
190	189	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑃 ∈ ℙ)
192		nnq 11801	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑛 + 1) ∈ ℚ)
195	138	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
196		zq 11794	. . . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
199	158, 128	pccld 15555	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ0)
200	199	nn0red 11352	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
201	200	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
202	5	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℕ0)
203	202	nn0red 11352	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℝ)
204	203	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℝ)
205		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0)
206	205	neneqd 2799	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ¬ ((#‘𝑋) − (𝑃↑𝑁)) = 0)
207	114	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0)
208		elnn0 11294	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ0 ↔ (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((#‘𝑋) − (𝑃↑𝑁)) = 0))
209	207, 208	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((#‘𝑋) − (𝑃↑𝑁)) = 0))
210	209	ord 392	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (¬ ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ → ((#‘𝑋) − (𝑃↑𝑁)) = 0))
211	206, 210	mt3d 140	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℕ)
212	191, 211	pccld 15555	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ∈ ℕ0)
213	212	nn0red 11352	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((#‘𝑋) − (𝑃↑𝑁))) ∈ ℝ)
214	128	nnzd 11481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℤ)
215		pcdvdsb 15573	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
216	158, 214, 202, 215	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
217	7	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℤ)
218		dvdsle 15032	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
219	217, 128, 218	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
220	216, 219	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
221	203, 200	lenltd 10183	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ ¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
222	131, 129	lenltd 10183	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑𝑁)))
223	220, 221, 222	3imtr3d 282	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 → ¬ (𝑛 + 1) < (𝑃↑𝑁)))
224	133, 223	mt4d 152	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225	224	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
226		dvdssubr 15027	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (#‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
227	7, 34, 226	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (#‘𝑋) ↔ (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁))))
228	13, 227	mpbid 222	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁)))
229	228	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃↑𝑁) ∥ ((#‘𝑋) − (𝑃↑𝑁)))
230	207	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((#‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
231	5	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((#‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℕ0)
232		pcdvdsb 15573	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . . . 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 10197	. . . . . . . . . . . . . . . . . . . . . . . 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 15594	. . . . . . . . . . . . . . . . . . . . . . 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 2881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((#‘𝑋) − (𝑃↑𝑁)))))
238	184, 237	eqtr4d 2659	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
239	199	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
240	238, 239	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) ∈ ℂ)
241	240, 238	subeq0bd 10456	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃 pCnt ((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))) = 0)
242	180, 241	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
243	242	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
244		00id 10211	. . . . . . . . . . . . . . . . 17 ⊢ (0 + 0) = 0
245	243, 244	syl6req 2673	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ0 ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 0 = (0 + (𝑃 pCnt (((((#‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
246	177, 245	eqeq12d 2637	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 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 47	. . . . . . . . . . 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 451	. . . . . . . . . 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 11472	. . . . . . . 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 38	. . . . . . 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 2658	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) = 0)
257		pcdiv 15557	. . . . . . 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 1331	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))))
259	5	nn0zd 11480	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
260		pcid 15577	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
261	2, 259, 260	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
262	261	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
263	258, 262	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
264	256, 263	oveq12d 6668	. . . 4 ⊢ (𝜑 → ((𝑃 pCnt (((#‘𝑋) − 1)C((𝑃↑𝑁) − 1))) + (𝑃 pCnt ((#‘𝑋) / (𝑃↑𝑁)))) = (0 + ((𝑃 pCnt (#‘𝑋)) − 𝑁)))
265	2, 27	pccld 15555	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℕ0)
266	265	nn0zd 11480	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑋)) ∈ ℤ)
267	266, 259	zsubcld 11487	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℤ)
268	267	zcnd 11483	. . . . 5 ⊢ (𝜑 → ((𝑃 pCnt (#‘𝑋)) − 𝑁) ∈ ℂ)
269	268	addid2d 10237	. . . 4 ⊢ (𝜑 → (0 + ((𝑃 pCnt (#‘𝑋)) − 𝑁)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
270	79, 264, 269	3eqtrd 2660	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((((#‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((#‘𝑋) / (𝑃↑𝑁)))) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
271	66, 67, 270	3eqtr3d 2664	. 2 ⊢ (𝜑 → (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁))
272	40, 271	jca 554	1 ⊢ (𝜑 → ((#‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (#‘𝑆)) = ((𝑃 pCnt (#‘𝑋)) − 𝑁)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ℚcq 11788  ℝ⁺crp 11832  ...cfz 12326  ↑cexp 12860  Ccbc 13089  #chash 13117   ∥ cdvds 14983  ℙcprime 15385   pCnt cpc 15541  Basecbs 15857  +_gcplusg 15941  Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by:  sylow1lem3  18015