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Theorem sylow1lem1 19466

Description: Lemma for sylow1 19471. 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 16611	. . . . . . . 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 14208	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ)
7	6	nnzd 12585	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℤ)
8		hashbc 14412	. . . . 5 ⊢ ((𝑋 ∈ Fin ∧ (𝑃↑𝑁) ∈ ℤ) → ((♯‘𝑋)C(𝑃↑𝑁)) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}))
9	1, 7, 8	syl2anc 585	. . . 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 6895	. . . 4 ⊢ (♯‘𝑆) = (♯‘{𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)})
12	9, 11	eqtr4di 2791	. . 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 18851	. . . . . . . . . 10 ⊢ (𝐺 ∈ Grp → 𝑋 ≠ ∅)
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ≠ ∅)
18		hasheq0 14323	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
19	1, 18	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((♯‘𝑋) = 0 ↔ 𝑋 = ∅))
20	19	necon3bbid 2979	. . . . . . . . 9 ⊢ (𝜑 → (¬ (♯‘𝑋) = 0 ↔ 𝑋 ≠ ∅))
21	17, 20	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → ¬ (♯‘𝑋) = 0)
22		hashcl 14316	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
23	1, 22	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ₀)
24		elnn0 12474	. . . . . . . . . 10 ⊢ ((♯‘𝑋) ∈ ℕ₀ ↔ ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
25	23, 24	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ((♯‘𝑋) ∈ ℕ ∨ (♯‘𝑋) = 0))
26	25	ord 863	. . . . . . . 8 ⊢ (𝜑 → (¬ (♯‘𝑋) ∈ ℕ → (♯‘𝑋) = 0))
27	21, 26	mt3d 148	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑋) ∈ ℕ)
28		dvdsle 16253	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℕ) → ((𝑃↑𝑁) ∥ (♯‘𝑋) → (𝑃↑𝑁) ≤ (♯‘𝑋)))
29	7, 27, 28	syl2anc 585	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (♯‘𝑋) → (𝑃↑𝑁) ≤ (♯‘𝑋)))
30	13, 29	mpd 15	. . . . 5 ⊢ (𝜑 → (𝑃↑𝑁) ≤ (♯‘𝑋))
31	6	nnnn0d 12532	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℕ₀)
32		nn0uz 12864	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
33	31, 32	eleqtrdi 2844	. . . . . 6 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ_≥‘0))
34	23	nn0zd 12584	. . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ∈ ℤ)
35		elfz5 13493	. . . . . 6 ⊢ (((𝑃↑𝑁) ∈ (ℤ_≥‘0) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃↑𝑁) ≤ (♯‘𝑋)))
36	33, 34, 35	syl2anc 585	. . . . 5 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (0...(♯‘𝑋)) ↔ (𝑃↑𝑁) ≤ (♯‘𝑋)))
37	30, 36	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (0...(♯‘𝑋)))
38		bccl2 14283	. . . 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 2835	. 2 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ)
41		nnuz 12865	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
42	6, 41	eleqtrdi 2844	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (ℤ_≥‘1))
43		elfz5 13493	. . . . . . . . . 10 ⊢ (((𝑃↑𝑁) ∈ (ℤ_≥‘1) ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃↑𝑁) ≤ (♯‘𝑋)))
44	42, 34, 43	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(♯‘𝑋)) ↔ (𝑃↑𝑁) ≤ (♯‘𝑋)))
45	30, 44	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ (1...(♯‘𝑋)))
46		1zzd 12593	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
47		fzsubel 13537	. . . . . . . . 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 838	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) ∈ (1...(♯‘𝑋)) ↔ ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1))))
49	45, 48	mpbid 231	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ((1 − 1)...((♯‘𝑋) − 1)))
50		1m1e0 12284	. . . . . . . 8 ⊢ (1 − 1) = 0
51	50	oveq1i 7419	. . . . . . 7 ⊢ ((1 − 1)...((♯‘𝑋) − 1)) = (0...((♯‘𝑋) − 1))
52	49, 51	eleqtrdi 2844	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ (0...((♯‘𝑋) − 1)))
53		bcp1nk 14277	. . . . . 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 12534	. . . . . . 7 ⊢ (𝜑 → (♯‘𝑋) ∈ ℂ)
56		ax-1cn 11168	. . . . . . 7 ⊢ 1 ∈ ℂ
57		npcan 11469	. . . . . . 7 ⊢ (((♯‘𝑋) ∈ ℂ ∧ 1 ∈ ℂ) → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
58	55, 56, 57	sylancl 587	. . . . . 6 ⊢ (𝜑 → (((♯‘𝑋) − 1) + 1) = (♯‘𝑋))
59	6	nncnd 12228	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℂ)
60		npcan 11469	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
61	59, 56, 60	sylancl 587	. . . . . 6 ⊢ (𝜑 → (((𝑃↑𝑁) − 1) + 1) = (𝑃↑𝑁))
62	58, 61	oveq12d 7427	. . . . 5 ⊢ (𝜑 → ((((♯‘𝑋) − 1) + 1)C(((𝑃↑𝑁) − 1) + 1)) = ((♯‘𝑋)C(𝑃↑𝑁)))
63	58, 61	oveq12d 7427	. . . . . 6 ⊢ (𝜑 → ((((♯‘𝑋) − 1) + 1) / (((𝑃↑𝑁) − 1) + 1)) = ((♯‘𝑋) / (𝑃↑𝑁)))
64	63	oveq2d 7425	. . . . 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 2781	. . . 4 ⊢ (𝜑 → ((♯‘𝑋)C(𝑃↑𝑁)) = ((((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((♯‘𝑋) / (𝑃↑𝑁))))
66	65	oveq2d 7425	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((♯‘𝑋) / (𝑃↑𝑁)))))
67	12	oveq2d 7425	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((♯‘𝑋)C(𝑃↑𝑁))) = (𝑃 pCnt (♯‘𝑆)))
68		bccl2 14283	. . . . . . 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 12585	. . . . 5 ⊢ (𝜑 → (((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ∈ ℤ)
71	69	nnne0d 12262	. . . . 5 ⊢ (𝜑 → (((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)) ≠ 0)
72	6	nnne0d 12262	. . . . . . 7 ⊢ (𝜑 → (𝑃↑𝑁) ≠ 0)
73		dvdsval2 16200	. . . . . . 7 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑃↑𝑁) ≠ 0 ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
74	7, 72, 34, 73	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (♯‘𝑋) ↔ ((♯‘𝑋) / (𝑃↑𝑁)) ∈ ℤ))
75	13, 74	mpbid 231	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) / (𝑃↑𝑁)) ∈ ℤ)
76	27	nnne0d 12262	. . . . . 6 ⊢ (𝜑 → (♯‘𝑋) ≠ 0)
77	55, 59, 76, 72	divne0d 12006	. . . . 5 ⊢ (𝜑 → ((♯‘𝑋) / (𝑃↑𝑁)) ≠ 0)
78		pcmul 16784	. . . . 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 1380	. . . 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 11209	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℂ)
81	55, 59, 80	npncand 11595	. . . . . . . 8 ⊢ (𝜑 → (((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1)) = ((♯‘𝑋) − 1))
82	81	oveq1d 7424	. . . . . . 7 ⊢ (𝜑 → ((((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)) = (((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)))
83	82	oveq2d 7425	. . . . . 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 12227	. . . . . . . 8 ⊢ (𝜑 → (𝑃↑𝑁) ∈ ℝ)
85	84	ltm1d 12146	. . . . . . 7 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) < (𝑃↑𝑁))
86		nnm1nn0 12513	. . . . . . . . 9 ⊢ ((𝑃↑𝑁) ∈ ℕ → ((𝑃↑𝑁) − 1) ∈ ℕ₀)
87	6, 86	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((𝑃↑𝑁) − 1) ∈ ℕ₀)
88		breq1 5152	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (𝑥 < (𝑃↑𝑁) ↔ 0 < (𝑃↑𝑁)))
89		bcxmaslem1 15780	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃↑𝑁)) + 0)C0))
90	89	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 0)C0)))
91	90	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = 0))
92	88, 91	imbi12d 345	. . . . . . . . . 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 341	. . . . . . . . 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 5152	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (𝑥 < (𝑃↑𝑁) ↔ 𝑛 < (𝑃↑𝑁)))
95		bcxmaslem1 15780	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛))
96	95	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)))
97	96	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = 0 ↔ (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛)) = 0))
98	94, 97	imbi12d 345	. . . . . . . . . 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 341	. . . . . . . . 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 5152	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 < (𝑃↑𝑁) ↔ (𝑛 + 1) < (𝑃↑𝑁)))
101		bcxmaslem1 15780	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑛 + 1) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1)))
102	101	oveq2d 7425	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑛 + 1) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥)) = (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1))C(𝑛 + 1))))
103	102	eqeq1d 2735	. . . . . . . . . . 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 345	. . . . . . . . . 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 341	. . . . . . . . 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 5152	. . . . . . . . . . 11 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → (𝑥 < (𝑃↑𝑁) ↔ ((𝑃↑𝑁) − 1) < (𝑃↑𝑁)))
107		bcxmaslem1 15780	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑃↑𝑁) − 1) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑥)C𝑥) = ((((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1)))
108	107	oveq2d 7425	. . . . . . . . . . . 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 2735	. . . . . . . . . . 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 345	. . . . . . . . . 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 341	. . . . . . . . 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 12609	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀))
113	7, 34, 112	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑃↑𝑁) ≤ (♯‘𝑋) ↔ ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀))
114	30, 113	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
115		0nn0 12487	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
116		nn0addcl 12507	. . . . . . . . . . . . . 14 ⊢ ((((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀ ∧ 0 ∈ ℕ₀) → (((♯‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ₀)
117	114, 115, 116	sylancl 587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((♯‘𝑋) − (𝑃↑𝑁)) + 0) ∈ ℕ₀)
118		bcn0 14270	. . . . . . . . . . . . 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 7425	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 0)C0)) = (𝑃 pCnt 1))
121		pc1 16788	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ ℙ → (𝑃 pCnt 1) = 0)
122	2, 121	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 pCnt 1) = 0)
123	120, 122	eqtrd 2773	. . . . . . . . . 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 12481	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℝ)
126	125	ad2antrl 727	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℝ)
127		nn0p1nn 12511	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
128	127	ad2antrl 727	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℕ)
129	128	nnred 12227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℝ)
130	6	adantr 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℕ)
131	130	nnred 12227	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℝ)
132	126	ltp1d 12144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑛 + 1))
133		simprr 772	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) < (𝑃↑𝑁))
134	126, 129, 131, 132, 133	lttrd 11375	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 < (𝑃↑𝑁))
135	134	expr 458	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ₀) → ((𝑛 + 1) < (𝑃↑𝑁) → 𝑛 < (𝑃↑𝑁)))
136	135	imim1d 82	. . . . . . . . . 10 ⊢ ((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 7416	. . . . . . . . . . 11 ⊢ ((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 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
139	138	nn0cnd 12534	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℂ)
140		nn0cn 12482	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℂ)
141	140	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℂ)
142		1cnd 11209	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 1 ∈ ℂ)
143	139, 141, 142	addassd 11236	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = (((♯‘𝑋) − (𝑃↑𝑁)) + (𝑛 + 1)))
144	143	oveq1d 7424	. . . . . . . . . . . . . . 15 ⊢ ((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 12519	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ ℝ ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀) → 𝑛 ≤ (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛))
146	126, 138, 145	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ≤ (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛))
147		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ ℕ₀)
148	147, 32	eleqtrdi 2844	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (ℤ_≥‘0))
149	138, 147	nn0addcld 12536	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℕ₀)
150	149	nn0zd 12584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℤ)
151		elfz5 13493	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (ℤ_≥‘0) ∧ (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℤ) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
152	148, 150, 151	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)) ↔ 𝑛 ≤ (((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
153	146, 152	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑛 ∈ (0...(((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)))
154		bcp1nk 14277	. . . . . . . . . . . . . . . 16 ⊢ (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	. . . . . . . . . . . . . . 15 ⊢ ((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 2775	. . . . . . . . . . . . . 14 ⊢ ((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 7425	. . . . . . . . . . . . 13 ⊢ ((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 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑃 ∈ ℙ)
159		bccl2 14283	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...(((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ)
160	153, 159	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ)
161		nnq 12946	. . . . . . . . . . . . . . 15 ⊢ (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℕ → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℚ)
162	160, 161	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ∈ ℚ)
163	160	nnne0d 12262	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛)C𝑛) ≠ 0)
164	150	peano2zd 12669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ)
165		znq 12936	. . . . . . . . . . . . . . 15 ⊢ ((((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
166	164, 128, 165	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℚ)
167		nn0p1nn 12511	. . . . . . . . . . . . . . . . 17 ⊢ ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) ∈ ℕ₀ → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ)
168	149, 167	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ)
169		nnrp 12985	. . . . . . . . . . . . . . . . 17 ⊢ (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ⁺)
170		nnrp 12985	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℝ⁺)
171		rpdivcl 12999	. . . . . . . . . . . . . . . . 17 ⊢ ((((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℝ⁺ ∧ (𝑛 + 1) ∈ ℝ⁺) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
172	169, 170, 171	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ∈ ℕ ∧ (𝑛 + 1) ∈ ℕ) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
173	168, 128, 172	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ∈ ℝ⁺)
174	173	rpne0d 13021	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)) ≠ 0)
175		pcqmul 16786	. . . . . . . . . . . . . 14 ⊢ ((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 1380	. . . . . . . . . . . . 13 ⊢ ((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 2773	. . . . . . . . . . . 12 ⊢ ((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 12262	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) ≠ 0)
179		pcdiv 16785	. . . . . . . . . . . . . . . 16 ⊢ ((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 1376	. . . . . . . . . . . . . . 15 ⊢ ((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 12228	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℂ)
182	139, 181, 143	comraddd 11428	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁))))
183	182	oveq2d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁)))))
184		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) = 0) → ((♯‘𝑋) − (𝑃↑𝑁)) = 0)
185	184	oveq2d 7425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁))) = ((𝑛 + 1) + 0))
186	181	addridd 11414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑛 + 1) + 0) = (𝑛 + 1))
187	186	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) = 0) → ((𝑛 + 1) + 0) = (𝑛 + 1))
188	185, 187	eqtr2d 2774	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) = 0) → (𝑛 + 1) = ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁))))
189	188	oveq2d 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) = 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁)))))
190	2	ad2antrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑃 ∈ ℙ)
191		nnq 12946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℚ)
192	128, 191	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℚ)
193	192	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑛 + 1) ∈ ℚ)
194	138	nn0zd 12584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
195		zq 12938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℤ → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
196	194, 195	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
197	196	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℚ)
198	158, 128	pccld 16783	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℕ₀)
199	198	nn0red 12533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
200	199	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) ∈ ℝ)
201	5	adantr 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℕ₀)
202	201	nn0red 12533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 𝑁 ∈ ℝ)
203	202	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℝ)
204		simpr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0)
205	204	neneqd 2946	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ¬ ((♯‘𝑋) − (𝑃↑𝑁)) = 0)
206	114	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀)
207		elnn0 12474	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ₀ ↔ (((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃↑𝑁)) = 0))
208	206, 207	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ ∨ ((♯‘𝑋) − (𝑃↑𝑁)) = 0))
209	208	ord 863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (¬ ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ → ((♯‘𝑋) − (𝑃↑𝑁)) = 0))
210	205, 209	mt3d 148	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℕ)
211	190, 210	pccld 16783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))) ∈ ℕ₀)
212	211	nn0red 12533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))) ∈ ℝ)
213	128	nnzd 12585	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑛 + 1) ∈ ℤ)
214		pcdvdsb 16802	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ ℙ ∧ (𝑛 + 1) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
215	158, 213, 201, 214	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ (𝑃↑𝑁) ∥ (𝑛 + 1)))
216	7	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃↑𝑁) ∈ ℤ)
217		dvdsle 16253	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (𝑛 + 1) ∈ ℕ) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
218	216, 128, 217	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ∥ (𝑛 + 1) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
219	215, 218	sylbid 239	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) → (𝑃↑𝑁) ≤ (𝑛 + 1)))
220	202, 199	lenltd 11360	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑁 ≤ (𝑃 pCnt (𝑛 + 1)) ↔ ¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁))
221	131, 129	lenltd 11360	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃↑𝑁) ≤ (𝑛 + 1) ↔ ¬ (𝑛 + 1) < (𝑃↑𝑁)))
222	219, 220, 221	3imtr3d 293	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (¬ (𝑃 pCnt (𝑛 + 1)) < 𝑁 → ¬ (𝑛 + 1) < (𝑃↑𝑁)))
223	133, 222	mt4d 117	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
224	223	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < 𝑁)
225		dvdssubr 16248	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑃↑𝑁) ∈ ℤ ∧ (♯‘𝑋) ∈ ℤ) → ((𝑃↑𝑁) ∥ (♯‘𝑋) ↔ (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁))))
226	7, 34, 225	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((𝑃↑𝑁) ∥ (♯‘𝑋) ↔ (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁))))
227	13, 226	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁)))
228	227	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁)))
229	206	nn0zd 12584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℤ)
230	5	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ∈ ℕ₀)
231		pcdvdsb 16802	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ∈ ℤ ∧ 𝑁 ∈ ℕ₀) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))) ↔ (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁))))
232	190, 229, 230, 231	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))) ↔ (𝑃↑𝑁) ∥ ((♯‘𝑋) − (𝑃↑𝑁))))
233	228, 232	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → 𝑁 ≤ (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))))
234	200, 203, 212, 224, 233	ltletrd 11374	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) < (𝑃 pCnt ((♯‘𝑋) − (𝑃↑𝑁))))
235	190, 193, 197, 234	pcadd2 16823	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) ∧ ((♯‘𝑋) − (𝑃↑𝑁)) ≠ 0) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁)))))
236	189, 235	pm2.61dane 3030	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) = (𝑃 pCnt ((𝑛 + 1) + ((♯‘𝑋) − (𝑃↑𝑁)))))
237	183, 236	eqtr4d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) = (𝑃 pCnt (𝑛 + 1)))
238	198	nn0cnd 12534	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (𝑛 + 1)) ∈ ℂ)
239	237, 238	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) ∈ ℂ)
240	239, 237	subeq0bd 11640	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → ((𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1)) − (𝑃 pCnt (𝑛 + 1))) = 0)
241	180, 240	eqtrd 2773	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (𝑃 pCnt (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1))) = 0)
242	241	oveq2d 7425	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))) = (0 + 0))
243		00id 11389	. . . . . . . . . . . . 13 ⊢ (0 + 0) = 0
244	242, 243	eqtr2di 2790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ₀ ∧ (𝑛 + 1) < (𝑃↑𝑁))) → 0 = (0 + (𝑃 pCnt (((((♯‘𝑋) − (𝑃↑𝑁)) + 𝑛) + 1) / (𝑛 + 1)))))
245	177, 244	eqeq12d 2749	. . . . . . . . . . 11 ⊢ ((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))))))
246	137, 245	imbitrrid 245	. . . . . . . . . 10 ⊢ ((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))
247	136, 246	animpimp2impd 845	. . . . . . . . 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))))
248	93, 99, 105, 111, 124, 247	nn0ind 12657	. . . . . . . 8 ⊢ (((𝑃↑𝑁) − 1) ∈ ℕ₀ → (𝜑 → (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0)))
249	87, 248	mpcom 38	. . . . . . 7 ⊢ (𝜑 → (((𝑃↑𝑁) − 1) < (𝑃↑𝑁) → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0))
250	85, 249	mpd 15	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − (𝑃↑𝑁)) + ((𝑃↑𝑁) − 1))C((𝑃↑𝑁) − 1))) = 0)
251	83, 250	eqtr3d 2775	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1))) = 0)
252		pcdiv 16785	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ ((♯‘𝑋) ∈ ℤ ∧ (♯‘𝑋) ≠ 0) ∧ (𝑃↑𝑁) ∈ ℕ) → (𝑃 pCnt ((♯‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))))
253	2, 34, 76, 6, 252	syl121anc 1376	. . . . . 6 ⊢ (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))))
254	5	nn0zd 12584	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
255		pcid 16806	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
256	2, 254, 255	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (𝑃↑𝑁)) = 𝑁)
257	256	oveq2d 7425	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − (𝑃 pCnt (𝑃↑𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
258	253, 257	eqtrd 2773	. . . . 5 ⊢ (𝜑 → (𝑃 pCnt ((♯‘𝑋) / (𝑃↑𝑁))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
259	251, 258	oveq12d 7427	. . . 4 ⊢ (𝜑 → ((𝑃 pCnt (((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1))) + (𝑃 pCnt ((♯‘𝑋) / (𝑃↑𝑁)))) = (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))
260	2, 27	pccld 16783	. . . . . . . 8 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℕ₀)
261	260	nn0zd 12584	. . . . . . 7 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑋)) ∈ ℤ)
262	261, 254	zsubcld 12671	. . . . . 6 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℤ)
263	262	zcnd 12667	. . . . 5 ⊢ (𝜑 → ((𝑃 pCnt (♯‘𝑋)) − 𝑁) ∈ ℂ)
264	263	addlidd 11415	. . . 4 ⊢ (𝜑 → (0 + ((𝑃 pCnt (♯‘𝑋)) − 𝑁)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
265	79, 259, 264	3eqtrd 2777	. . 3 ⊢ (𝜑 → (𝑃 pCnt ((((♯‘𝑋) − 1)C((𝑃↑𝑁) − 1)) · ((♯‘𝑋) / (𝑃↑𝑁)))) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
266	66, 67, 265	3eqtr3d 2781	. 2 ⊢ (𝜑 → (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
267	40, 266	jca 513	1 ⊢ (𝜑 → ((♯‘𝑆) ∈ ℕ ∧ (𝑃 pCnt (♯‘𝑆)) = ((𝑃 pCnt (♯‘𝑋)) − 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 {crab 3433 ∅c0 4323 𝒫 cpw 4603 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11248 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ℚcq 12932 ℝ⁺crp 12974 ...cfz 13484 ↑cexp 14027 Ccbc 14262 ♯chash 14290 ∥ cdvds 16197 ℙcprime 16608 pCnt cpc 16769 Basecbs 17144 +_gcplusg 17197 Grpcgrp 18819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-fz 13485 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822

This theorem is referenced by: sylow1lem3 19468

W3C validator