prmdvdsbc - Metamath Proof Explorer

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Theorem prmdvdsbc 16750

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 2736	. . 3 ⊢ ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))) = ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁)))
2		simpl 482	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℙ)
3		prmnn 16698	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	nnzd 12620	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
5		1nn0 12522	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
6		eluzmn 12864	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
7	4, 5, 6	sylancl 586	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
8		fzss2 13586	. . . . . . . 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 13645	. . . . . . 7 ⊢ (1...𝑃) ⊆ (0...𝑃)
11	9, 10	sstrdi 3976	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ⊆ (0...𝑃))
12	11	sselda 3963	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...𝑃))
13		bcval2 14328	. . . . 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 12567	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℕ₀)
17		elfzelz 13546	. . . . . . 7 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
19		bccl 14345	. . . . . 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 12619	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃C𝑁) ∈ ℤ)
22	14, 21	eqeltrrd 2836	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))) ∈ ℤ)
23		elfznn 13575	. . . . . . . . 9 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
24	23	adantl 481	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
25	24	nnnn0d 12567	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
26		1zzd 12628	. . . . . . . 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 13617	. . . . . . . . . 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 12557	. . . . . . . 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 14307	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℕ)
37	36	nnzd 12620	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℤ)
38	25	faccld 14307	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℕ)
39	38	nnzd 12620	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℤ)
40	37, 39	zmulcld 12708	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ∈ ℤ)
41	37	zcnd 12703	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℂ)
42	39	zcnd 12703	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℂ)
43		facne0 14309	. . . . 5 ⊢ ((𝑃 − 𝑁) ∈ ℕ₀ → (!‘(𝑃 − 𝑁)) ≠ 0)
44	35, 43	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ≠ 0)
45		facne0 14309	. . . . 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 11894	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ≠ 0)
48		uzid 12872	. . . . . 6 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ (ℤ_≥‘𝑃))
49	4, 48	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘𝑃))
50		dvdsfac 16350	. . . . 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 12568	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
54	24	nnrpd 13054	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ⁺)
55	53, 54	ltsubrpd 13088	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑁) < 𝑃)
56		prmndvdsfaclt 16749	. . . . . 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 16749	. . . . . 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 16737	. . . . . . . 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 16745	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
70	69, 14	breqtrrd 5152	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 2933 ⊆ wss 3931 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 0cc0 11134 1c1 11135 · cmul 11139 < clt 11274 ≤ cle 11275 − cmin 11471 / cdiv 11899 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 !cfa 14296 Ccbc 14325 ∥ cdvds 16277 ℙcprime 16695

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 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-fz 13530 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-prm 16696

This theorem is referenced by: freshmansdream 21540

W3C validator