Theorem dvdsnprmd 12905

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 12561	. . . 4 |- ( A || N -> ( A e. ZZ /\ N e. ZZ ) )
3		divides 12558	. . . 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 9674	. . . . . . . . 9 |- 2 e. ZZ
6	5	a1i 9	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> 2 e. ZZ )
7		simplr 533	. . . . . . . 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 4134	. . . . . . . . . . . 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 9650	. . . . . . . . . . . . . . . . . . 19 |- ( A e. ZZ -> A e. RR )
16	15	3ad2ant1 1049	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> A e. RR )
17		zre 9650	. . . . . . . . . . . . . . . . . . 19 |- ( k e. ZZ -> k e. RR )
18	17	3ad2ant3 1051	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> k e. RR )
19		0lt1 8453	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 1
20		0red 8327	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 0 e. RR )
21		1red 8341	. . . . . . . . . . . . . . . . . . . . . 22 |- ( A e. ZZ -> 1 e. RR )
22		lttr 8399	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
23	20, 21, 15, 22	syl3anc 1278	. . . . . . . . . . . . . . . . . . . . 21 |- ( A e. ZZ -> ( ( 0 < 1 /\ 1 < A ) -> 0 < A ) )
24	19, 23	mpani 434	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. ZZ -> ( 1 < A -> 0 < A ) )
25	24	imp 124	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ZZ /\ 1 < A ) -> 0 < A )
26	25	3adant3 1048	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> 0 < A )
27	16, 18, 26	3jca 1208	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ 1 < A /\ k e. ZZ ) -> ( A e. RR /\ k e. RR /\ 0 < A ) )
28	27	3exp 1233	. . . . . . . . . . . . . . . 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 9196	. . . . . . . . . . 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 9364	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
38	37	breq1i 4137	. . . . . . . . . 10 |- ( 2 <_ k <-> ( 1 + 1 ) <_ k )
39		1zzd 9673	. . . . . . . . . . . . . 14 |- ( k e. ZZ -> 1 e. ZZ )
40		zltp1le 9701	. . . . . . . . . . . . . 14 |- ( ( 1 e. ZZ /\ k e. ZZ ) -> ( 1 < k <-> ( 1 + 1 ) <_ k ) )
41	39, 40	mpancom 426	. . . . . . . . . . . . 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 9929	. . . . . . . 8 |- ( k e. ( ZZ>= ` 2 ) <-> ( 2 e. ZZ /\ k e. ZZ /\ 2 <_ k ) )
48	6, 7, 46, 47	syl3anbrc 1212	. . . . . . 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 9673	. . . . . . . . . . . . . . . . . 18 |- ( A e. ZZ -> 1 e. ZZ )
52		zltp1le 9701	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. ZZ /\ A e. ZZ ) -> ( 1 < A <-> ( 1 + 1 ) <_ A ) )
53	51, 52	mpancom 426	. . . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . 16 |- ( 2 <_ A <-> ( 1 + 1 ) <_ A )
56	54, 55	sylibr 134	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ 1 < A ) -> 2 <_ A )
57	49, 50, 56	3jca 1208	. . . . . . . . . . . . . 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 9929	. . . . . . . . . 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 12903	. . . . . . 7 |- ( ( k e. ( ZZ>= ` 2 ) /\ A e. ( ZZ>= ` 2 ) ) -> -. ( k x. A ) e. Prime )
67	48, 65, 66	syl2anc 415	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ ( k x. A ) = N ) -> -. ( k x. A ) e. Prime )
68		eleq1 2301	. . . . . . . 8 |- ( ( k x. A ) = N -> ( ( k x. A ) e. Prime <-> N e. Prime ) )
69	68	notbid 677	. . . . . . 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 2668	. . 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 )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   2c2 9356   ZZcz 9646   ZZ>=cuz 9923   || cdvds 12556   Primecprime 12887

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8904  df-ap 8911  df-div 9004  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-n0 9566  df-z 9647  df-uz 9924  df-q 10022  df-rp 10057  df-seqfrec 10887  df-exp 10978  df-cj 11609  df-re 11610  df-im 11611  df-rsqrt 11766  df-abs 11767  df-dvds 12557  df-prm 12888

This theorem is used by: (None)