prmdvdsbc - Metamath Proof Explorer

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Theorem prmdvdsbc 16696

Description: Condition for a prime number to divide a binomial coefficient. (Contributed by Thierry Arnoux, 17-Sep-2023.)

**Assertion**
Ref	Expression
prmdvdsbc	⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑁))

**Proof of Theorem prmdvdsbc**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))) = ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁)))
2		simpl 482	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
3		prmnn 16644	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	nnzd 12556	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
5		1nn0 12458	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
6		eluzmn 12800	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
7	4, 5, 6	sylancl 586	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
8		fzss2 13525	. . . . . . . 8 ⊢ (𝑃 ∈ (ℤ_≥‘(𝑃 − 1)) → (1...(𝑃 − 1)) ⊆ (1...𝑃))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ⊆ (1...𝑃))
10		fz1ssfz0 13584	. . . . . . 7 ⊢ (1...𝑃) ⊆ (0...𝑃)
11	9, 10	sstrdi 3959	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ⊆ (0...𝑃))
12	11	sselda 3946	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...𝑃))
13		bcval2 14270	. . . . 5 ⊢ (𝑁 ∈ (0...𝑃) → (𝑃C𝑁) = ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
14	12, 13	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃C𝑁) = ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
15	3	nnnn0d 12503	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℕ₀)
17		elfzelz 13485	. . . . . . 7 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
19		bccl 14287	. . . . . 6 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝑁 ∈ ℤ) → (𝑃C𝑁) ∈ ℕ₀)
20	16, 18, 19	syl2anc 584	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃C𝑁) ∈ ℕ₀)
21	20	nn0zd 12555	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃C𝑁) ∈ ℤ)
22	14, 21	eqeltrrd 2829	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))) ∈ ℤ)
23		elfznn 13514	. . . . . . . . 9 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
24	23	adantl 481	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
25	24	nnnn0d 12503	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
26		1zzd 12564	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 1 ∈ ℤ)
27	4	adantr 480	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℤ)
28		simpr 484	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (1...(𝑃 − 1)))
29		elfzm11 13556	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 ∈ (1...(𝑃 − 1)) ↔ (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁 ∧ 𝑁 < 𝑃)))
30	29	biimpa 476	. . . . . . . . 9 ⊢ (((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑁 ∈ ℤ ∧ 1 ≤ 𝑁 ∧ 𝑁 < 𝑃))
31	30	simp3d 1144	. . . . . . . 8 ⊢ (((1 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 < 𝑃)
32	26, 27, 28, 31	syl21anc 837	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 < 𝑃)
33		ltsubnn0 12493	. . . . . . . 8 ⊢ ((𝑃 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑁 < 𝑃 → (𝑃 − 𝑁) ∈ ℕ₀))
34	33	imp 406	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 < 𝑃) → (𝑃 − 𝑁) ∈ ℕ₀)
35	16, 25, 32, 34	syl21anc 837	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑁) ∈ ℕ₀)
36	35	faccld 14249	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℕ)
37	36	nnzd 12556	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℤ)
38	25	faccld 14249	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℕ)
39	38	nnzd 12556	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℤ)
40	37, 39	zmulcld 12644	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ∈ ℤ)
41	37	zcnd 12639	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℂ)
42	39	zcnd 12639	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℂ)
43		facne0 14251	. . . . 5 ⊢ ((𝑃 − 𝑁) ∈ ℕ₀ → (!‘(𝑃 − 𝑁)) ≠ 0)
44	35, 43	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ≠ 0)
45		facne0 14251	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (!‘𝑁) ≠ 0)
46	25, 45	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ≠ 0)
47	41, 42, 44, 46	mulne0d 11830	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ≠ 0)
48		uzid 12808	. . . . . 6 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ (ℤ_≥‘𝑃))
49	4, 48	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘𝑃))
50		dvdsfac 16296	. . . . 5 ⊢ ((𝑃 ∈ ℕ ∧ 𝑃 ∈ (ℤ_≥‘𝑃)) → 𝑃 ∥ (!‘𝑃))
51	3, 49, 50	syl2anc 584	. . . 4 ⊢ (𝑃 ∈ ℙ → 𝑃 ∥ (!‘𝑃))
52	51	adantr 480	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (!‘𝑃))
53	16	nn0red 12504	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
54	24	nnrpd 12993	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ⁺)
55	53, 54	ltsubrpd 13027	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑁) < 𝑃)
56		prmndvdsfaclt 16695	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (𝑃 − 𝑁) ∈ ℕ₀) → ((𝑃 − 𝑁) < 𝑃 → ¬ 𝑃 ∥ (!‘(𝑃 − 𝑁))))
57	56	imp 406	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ (𝑃 − 𝑁) ∈ ℕ₀) ∧ (𝑃 − 𝑁) < 𝑃) → ¬ 𝑃 ∥ (!‘(𝑃 − 𝑁)))
58	2, 35, 55, 57	syl21anc 837	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (!‘(𝑃 − 𝑁)))
59		prmndvdsfaclt 16695	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) → (𝑁 < 𝑃 → ¬ 𝑃 ∥ (!‘𝑁)))
60	59	imp 406	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℕ₀) ∧ 𝑁 < 𝑃) → ¬ 𝑃 ∥ (!‘𝑁))
61	2, 25, 32, 60	syl21anc 837	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ (!‘𝑁))
62		ioran 985	. . . . . 6 ⊢ (¬ (𝑃 ∥ (!‘(𝑃 − 𝑁)) ∨ 𝑃 ∥ (!‘𝑁)) ↔ (¬ 𝑃 ∥ (!‘(𝑃 − 𝑁)) ∧ ¬ 𝑃 ∥ (!‘𝑁)))
63		euclemma 16683	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ (!‘(𝑃 − 𝑁)) ∈ ℤ ∧ (!‘𝑁) ∈ ℤ) → (𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ↔ (𝑃 ∥ (!‘(𝑃 − 𝑁)) ∨ 𝑃 ∥ (!‘𝑁))))
64	63	biimpd 229	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ (!‘(𝑃 − 𝑁)) ∈ ℤ ∧ (!‘𝑁) ∈ ℤ) → (𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) → (𝑃 ∥ (!‘(𝑃 − 𝑁)) ∨ 𝑃 ∥ (!‘𝑁))))
65	64	con3d 152	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ (!‘(𝑃 − 𝑁)) ∈ ℤ ∧ (!‘𝑁) ∈ ℤ) → (¬ (𝑃 ∥ (!‘(𝑃 − 𝑁)) ∨ 𝑃 ∥ (!‘𝑁)) → ¬ 𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
66	62, 65	biimtrrid 243	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ (!‘(𝑃 − 𝑁)) ∈ ℤ ∧ (!‘𝑁) ∈ ℤ) → ((¬ 𝑃 ∥ (!‘(𝑃 − 𝑁)) ∧ ¬ 𝑃 ∥ (!‘𝑁)) → ¬ 𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
67	66	imp 406	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ (!‘(𝑃 − 𝑁)) ∈ ℤ ∧ (!‘𝑁) ∈ ℤ) ∧ (¬ 𝑃 ∥ (!‘(𝑃 − 𝑁)) ∧ ¬ 𝑃 ∥ (!‘𝑁))) → ¬ 𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁)))
68	2, 37, 39, 58, 61, 67	syl32anc 1380	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ¬ 𝑃 ∥ ((!‘(𝑃 − 𝑁)) · (!‘𝑁)))
69	1, 2, 22, 40, 47, 52, 68	dvdszzq 16691	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
70	69, 14	breqtrrd 5135	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ (𝑃C𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3914 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 0cc0 11068 1c1 11069 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ...cfz 13468 !cfa 14238 Ccbc 14267 ∥ cdvds 16222 ℙcprime 16641

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fz 13469 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-dvds 16223 df-gcd 16465 df-prm 16642

This theorem is referenced by: freshmansdream 21484

W3C validator