prmdvdsbc - Metamath Proof Explorer

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Theorem prmdvdsbc 16655

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 16603	. . . . . . . . . 10 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4	3	nnzd 12516	. . . . . . . . 9 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
5		1nn0 12418	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
6		eluzmn 12760	. . . . . . . . 9 ⊢ ((𝑃 ∈ ℤ ∧ 1 ∈ ℕ₀) → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
7	4, 5, 6	sylancl 586	. . . . . . . 8 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘(𝑃 − 1)))
8		fzss2 13485	. . . . . . . 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 13544	. . . . . . 7 ⊢ (1...𝑃) ⊆ (0...𝑃)
11	9, 10	sstrdi 3950	. . . . . 6 ⊢ (𝑃 ∈ ℙ → (1...(𝑃 − 1)) ⊆ (0...𝑃))
12	11	sselda 3937	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ (0...𝑃))
13		bcval2 14230	. . . . 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 12463	. . . . . . 7 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ₀)
16	15	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℕ₀)
17		elfzelz 13445	. . . . . . 7 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℤ)
18	17	adantl 481	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℤ)
19		bccl 14247	. . . . . 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 12515	. . . 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 13474	. . . . . . . . 9 ⊢ (𝑁 ∈ (1...(𝑃 − 1)) → 𝑁 ∈ ℕ)
24	23	adantl 481	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ)
25	24	nnnn0d 12463	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℕ₀)
26		1zzd 12524	. . . . . . . 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 13516	. . . . . . . . . 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 12453	. . . . . . . 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 14209	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℕ)
37	36	nnzd 12516	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℤ)
38	25	faccld 14209	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℕ)
39	38	nnzd 12516	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℤ)
40	37, 39	zmulcld 12604	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ∈ ℤ)
41	37	zcnd 12599	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ∈ ℂ)
42	39	zcnd 12599	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘𝑁) ∈ ℂ)
43		facne0 14211	. . . . 5 ⊢ ((𝑃 − 𝑁) ∈ ℕ₀ → (!‘(𝑃 − 𝑁)) ≠ 0)
44	35, 43	syl 17	. . . 4 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (!‘(𝑃 − 𝑁)) ≠ 0)
45		facne0 14211	. . . . 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 11790	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → ((!‘(𝑃 − 𝑁)) · (!‘𝑁)) ≠ 0)
48		uzid 12768	. . . . . 6 ⊢ (𝑃 ∈ ℤ → 𝑃 ∈ (ℤ_≥‘𝑃))
49	4, 48	syl 17	. . . . 5 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ (ℤ_≥‘𝑃))
50		dvdsfac 16255	. . . . 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 12464	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∈ ℝ)
54	24	nnrpd 12953	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑁 ∈ ℝ⁺)
55	53, 54	ltsubrpd 12987	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → (𝑃 − 𝑁) < 𝑃)
56		prmndvdsfaclt 16654	. . . . . 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 16654	. . . . . 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 16642	. . . . . . . 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 16650	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ (1...(𝑃 − 1))) → 𝑃 ∥ ((!‘𝑃) / ((!‘(𝑃 − 𝑁)) · (!‘𝑁))))
70	69, 14	breqtrrd 5123	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 3905 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 0cc0 11028 1c1 11029 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 ℕcn 12146 ℕ₀cn0 12402 ℤcz 12489 ℤ_≥cuz 12753 ...cfz 13428 !cfa 14198 Ccbc 14227 ∥ cdvds 16181 ℙcprime 16600

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 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fz 13429 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424 df-prm 16601

This theorem is referenced by: freshmansdream 21499

W3C validator