Theorem dvdsnprmd 12847

Description: If a number is divisible by an integer greater than 1 and less then the number, the number is not prime. (Contributed by AV, 24-Jul-2021.)

Hypotheses

Ref	Expression
dvdsnprmd.g	|- ( ph -> 1 < A )
dvdsnprmd.l	|- ( ph -> A < N )
dvdsnprmd.d	|- ( ph -> A || N )

Assertion

Ref	Expression
dvdsnprmd	|- ( ph -> -. N e. Prime )

Proof of Theorem dvdsnprmd

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsnprmd.d	. 2 |- ( ph -> A || N )
2		dvdszrcl 12503	. . . 4 |- ( A || N -> ( A e. ZZ /\ N e. ZZ ) )
3		divides 12500	. . . 4 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( A || N <-> E. k e. ZZ ( k x. A ) = N ) )
4	1, 2, 3	3syl 17	. . 3 |- ( ph -> ( A || N <-> E. k e. ZZ ( k x. A ) = N ) )
5		2z 9622	. . . . . . . . 9 |- 2 e. ZZ
6	5	a1i 9	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> 2 e. ZZ )
7		simplr 529	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> k e. ZZ )
8		dvdsnprmd.l	. . . . . . . . . . . . 13 |- ( ph -> A < N )
9	8	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ZZ ) -> A < N )
10	9	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> A < N )
11		breq2 4118	. . . . . . . . . . . 12 |- ( ( k x. A ) = N -> ( A < ( k x. A ) <-> A < N ) )
12	11	adantl 277	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( A < ( k x. A ) <-> A < N ) )
13	10, 12	mpbird 167	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> A < ( k x. A ) )
14		dvdsnprmd.g	. . . . . . . . . . . . . 14 |- ( ph -> 1 < A )
15		zre 9598	. . . . . . . . . . . . . . . . . . 19 |- ( A e. ZZ -> A e. RR )
16	15	3ad2ant1 1045	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> A e. RR )
17		zre 9598	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ZZ -> k e. RR )
18	17	3ad2ant3 1047	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> k e. RR )
19		0lt1 8416	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 1
20		0red 8291	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 0 e. RR )
21		1red 8305	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 1 e. RR )
22		lttr 8363	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
23	20, 21, 15, 22	syl3anc 1274	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. ZZ -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
24	19, 23	mpani 430	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. ZZ -> ( 1 < A -> 0 < A ) )
25	24	imp 124	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ZZ /\ 1 < A ) -> 0 < A )
26	25	3adant3 1044	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> 0 < A )
27	16, 18, 26	3jca 1204	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
28	27	3exp 1229	. . . . . . . . . . . . . . . 16 |- ( A e. ZZ -> ( 1 < A -> ( k e. ZZ -> ( A e. RR /\ k e. RR /\ 0 < A ) ) ) )
29	28	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( 1 < A -> ( k e. ZZ -> ( A e. RR /\ k e. RR /\ 0 < A ) ) ) )
30	1, 2, 29	3syl 17	. . . . . . . . . . . . . 14 |- ( ph -> ( 1 < A -> ( k e. ZZ -> ( A e. RR /\ k e. RR /\ 0 < A ) ) ) )
31	14, 30	mpd 13	. . . . . . . . . . . . 13 |- ( ph -> ( k e. ZZ -> ( A e. RR /\ k e. RR /\ 0 < A ) ) )
32	31	imp 124	. . . . . . . . . . . 12 |- ( ( ph /\ k e. ZZ ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
33	32	adantr 276	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
34		ltmulgt12 9156	. . . . . . . . . . 11 |- ( ( A e. RR /\ k e. RR /\ 0 < A ) -> ( 1 < k <-> A < ( k x. A ) ) )
35	33, 34	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( 1 < k <-> A < ( k x. A ) ) )
36	13, 35	mpbird 167	. . . . . . . . 9 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> 1 < k )
37		df-2 9313	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
38	37	breq1i 4121	. . . . . . . . . 10 |- ( 2 <_ k <-> ( 1 + 1 ) <_ k )
39		1zzd 9621	. . . . . . . . . . . . . 14 |- ( k e. ZZ -> 1 e. ZZ )
40		zltp1le 9649	. . . . . . . . . . . . . 14 |- ( ( 1 e. ZZ /\ k e. ZZ ) -> ( 1 < k <-> ( 1 + 1 ) <_ k ) )
41	39, 40	mpancom 422	. . . . . . . . . . . . 13 |- ( k e. ZZ -> ( 1 < k <-> ( 1 + 1 ) <_ k ) )
42	41	bicomd 141	. . . . . . . . . . . 12 |- ( k e. ZZ -> ( ( 1 + 1 ) <_ k <-> 1 < k ) )
43	42	adantl 277	. . . . . . . . . . 11 |- ( ( ph /\ k e. ZZ ) -> ( ( 1 + 1 ) <_ k <-> 1 < k ) )
44	43	adantr 276	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( ( 1 + 1 ) <_ k <-> 1 < k ) )
45	38, 44	bitrid 192	. . . . . . . . 9 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( 2 <_ k <-> 1 < k ) )
46	36, 45	mpbird 167	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> 2 <_ k )
47		eluz2 9877	. . . . . . . 8 |- ( k e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ k e. ZZ /\ 2 <_ k ) )
48	6, 7, 46, 47	syl3anbrc 1208	. . . . . . 7 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> k e. ( ZZ>= ` 2 ) )
49	5	a1i 9	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> 2 e. ZZ )
50		simpl 109	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> A e. ZZ )
51		1zzd 9621	. . . . . . . . . . . . . . . . . 18 |- ( A e. ZZ -> 1 e. ZZ )
52		zltp1le 9649	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 < A <-> ( 1 + 1 ) <_ A ) )
53	51, 52	mpancom 422	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> ( 1 < A <-> ( 1 + 1 ) <_ A ) )
54	53	biimpa 296	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ 1 < A ) -> ( 1 + 1 ) <_ A )
55	37	breq1i 4121	. . . . . . . . . . . . . . . 16 |- ( 2 <_ A <-> ( 1 + 1 ) <_ A )
56	54, 55	sylibr 134	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> 2 <_ A )
57	49, 50, 56	3jca 1204	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ 1 < A ) -> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) )
58	57	ex 115	. . . . . . . . . . . . 13 |- ( A e. ZZ -> ( 1 < A -> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) ) )
59	58	adantr 276	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ N e. ZZ ) -> ( 1 < A -> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) ) )
60	1, 2, 59	3syl 17	. . . . . . . . . . 11 |- ( ph -> ( 1 < A -> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) ) )
61	14, 60	mpd 13	. . . . . . . . . 10 |- ( ph -> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) )
62		eluz2 9877	. . . . . . . . . 10 |- ( A e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ A e. ZZ /\ 2 <_ A ) )
63	61, 62	sylibr 134	. . . . . . . . 9 |- ( ph -> A e. ( ZZ>= ` 2 ) )
64	63	adantr 276	. . . . . . . 8 |- ( ( ph /\ k e. ZZ ) -> A e. ( ZZ>= ` 2 ) )
65	64	adantr 276	. . . . . . 7 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> A e. ( ZZ>= ` 2 ) )
66		nprm 12845	. . . . . . 7 |- ( ( k e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( k x. A ) e. Prime )
67	48, 65, 66	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> -. ( k x. A ) e. Prime )
68		eleq1 2297	. . . . . . . 8 |- ( ( k x. A ) = N -> ( ( k x. A ) e. Prime <-> N e. Prime ) )
69	68	notbid 673	. . . . . . 7 |- ( ( k x. A ) = N -> ( -. ( k x. A ) e. Prime <-> -. N e. Prime ) )
70	69	adantl 277	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> ( -. ( k x. A ) e. Prime <-> -. N e. Prime ) )
71	67, 70	mpbid 147	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> -. N e. Prime )
72	71	ex 115	. . . 4 |- ( ( ph /\ k e. ZZ ) -> ( ( k x. A ) = N -> -. N e. Prime ) )
73	72	rexlimdva 2662	. . 3 |- ( ph -> ( E. k e. ZZ ( k x. A ) = N -> -. N e. Prime ) )
74	4, 73	sylbid 150	. 2 |- ( ph -> ( A || N -> -. N e. Prime ) )
75	1, 74	mpd 13	1 |- ( ph -> -. N e. Prime )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   2c2 9305   ZZcz 9594   ZZ>=cuz 9871   || cdvds 12498   Primecprime 12829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-prm 12830

This theorem is referenced by: (None)