Theorem dvdsnprmd 12127

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 11801	. . . 4 |- ( A || N -> ( A e. ZZ /\ N e. ZZ ) )
3		divides 11798	. . . 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 9283	. . . . . . . . 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 4009	. . . . . . . . . . . 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 9259	. . . . . . . . . . . . . . . . . . 19 |- ( A e. ZZ -> A e. RR )
16	15	3ad2ant1 1018	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> A e. RR )
17		zre 9259	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ZZ -> k e. RR )
18	17	3ad2ant3 1020	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> k e. RR )
19		0lt1 8086	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 1
20		0red 7960	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 0 e. RR )
21		1red 7974	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 1 e. RR )
22		lttr 8033	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
23	20, 21, 15, 22	syl3anc 1238	. . . . . . . . . . . . . . . . . . . . 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 1017	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> 0 < A )
27	16, 18, 26	3jca 1177	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
28	27	3exp 1202	. . . . . . . . . . . . . . . 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 8824	. . . . . . . . . . 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 8980	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
38	37	breq1i 4012	. . . . . . . . . 10 |- ( 2 <_ k <-> ( 1 + 1 ) <_ k )
39		1zzd 9282	. . . . . . . . . . . . . 14 |- ( k e. ZZ -> 1 e. ZZ )
40		zltp1le 9309	. . . . . . . . . . . . . 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 9536	. . . . . . . 8 |- ( k e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ k e. ZZ /\ 2 <_ k ) )
48	6, 7, 46, 47	syl3anbrc 1181	. . . . . . 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 9282	. . . . . . . . . . . . . . . . . 18 |- ( A e. ZZ -> 1 e. ZZ )
52		zltp1le 9309	. . . . . . . . . . . . . . . . . 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 4012	. . . . . . . . . . . . . . . 16 |- ( 2 <_ A <-> ( 1 + 1 ) <_ A )
56	54, 55	sylibr 134	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> 2 <_ A )
57	49, 50, 56	3jca 1177	. . . . . . . . . . . . . 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 9536	. . . . . . . . . 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 12125	. . . . . . 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 2240	. . . . . . . 8 |- ( ( k x. A ) = N -> ( ( k x. A ) e. Prime <-> N e. Prime ) )
69	68	notbid 667	. . . . . . 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 2594	. . 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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814   + caddc 7816   x. cmul 7818   < clt 7994   <_ cle 7995   2c2 8972   ZZcz 9255   ZZ>=cuz 9530   || cdvds 11796   Primecprime 12109

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-2o 6420  df-er 6537  df-en 6743  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-prm 12110

This theorem is referenced by: (None)