Theorem dvdsnprmd 12633

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 12289	. . . 4 |- ( A || N -> ( A e. ZZ /\ N e. ZZ ) )
3		divides 12286	. . . 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 9462	. . . . . . . . 9 |- 2 e. ZZ
6	5	a1i 9	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> 2 e. ZZ )
7		simplr 528	. . . . . . . 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 4086	. . . . . . . . . . . 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 9438	. . . . . . . . . . . . . . . . . . 19 |- ( A e. ZZ -> A e. RR )
16	15	3ad2ant1 1042	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> A e. RR )
17		zre 9438	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ZZ -> k e. RR )
18	17	3ad2ant3 1044	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> k e. RR )
19		0lt1 8261	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 1
20		0red 8135	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 0 e. RR )
21		1red 8149	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 1 e. RR )
22		lttr 8208	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
23	20, 21, 15, 22	syl3anc 1271	. . . . . . . . . . . . . . . . . . . . 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 1041	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> 0 < A )
27	16, 18, 26	3jca 1201	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
28	27	3exp 1226	. . . . . . . . . . . . . . . 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 9000	. . . . . . . . . . 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 9157	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
38	37	breq1i 4089	. . . . . . . . . 10 |- ( 2 <_ k <-> ( 1 + 1 ) <_ k )
39		1zzd 9461	. . . . . . . . . . . . . 14 |- ( k e. ZZ -> 1 e. ZZ )
40		zltp1le 9489	. . . . . . . . . . . . . 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 9716	. . . . . . . 8 |- ( k e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ k e. ZZ /\ 2 <_ k ) )
48	6, 7, 46, 47	syl3anbrc 1205	. . . . . . 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 9461	. . . . . . . . . . . . . . . . . 18 |- ( A e. ZZ -> 1 e. ZZ )
52		zltp1le 9489	. . . . . . . . . . . . . . . . . 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 4089	. . . . . . . . . . . . . . . 16 |- ( 2 <_ A <-> ( 1 + 1 ) <_ A )
56	54, 55	sylibr 134	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> 2 <_ A )
57	49, 50, 56	3jca 1201	. . . . . . . . . . . . . 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 9716	. . . . . . . . . 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 12631	. . . . . . 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 2292	. . . . . . . 8 |- ( ( k x. A ) = N -> ( ( k x. A ) e. Prime <-> N e. Prime ) )
69	68	notbid 671	. . . . . . 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 2648	. . 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 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082   ` cfv 5314  (class class class)co 5994   RRcr 7986   0cc0 7987   1c1 7988   + caddc 7990   x. cmul 7992   < clt 8169   <_ cle 8170   2c2 9149   ZZcz 9434   ZZ>=cuz 9710   || cdvds 12284   Primecprime 12615

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105  ax-arch 8106  ax-caucvg 8107

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-2o 6553  df-er 6670  df-en 6878  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-n0 9358  df-z 9435  df-uz 9711  df-q 9803  df-rp 9838  df-seqfrec 10657  df-exp 10748  df-cj 11339  df-re 11340  df-im 11341  df-rsqrt 11495  df-abs 11496  df-dvds 12285  df-prm 12616

This theorem is referenced by: (None)