Theorem dvdsprime 12902

Description: If M divides a prime, then M is either the prime or one. (Contributed by Scott Fenton, 8-Apr-2014.)

Assertion

Ref	Expression
dvdsprime	|- ( ( P e. Prime /\ M e. NN ) -> ( M || P <-> ( M = P \/ M = 1 ) ) )

Proof of Theorem dvdsprime

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isprm2 12897	. . 3 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) ) )
2		breq1 4133	. . . . . 6 |- ( m = M -> ( m || P <-> M || P ) )
3		eqeq1 2245	. . . . . . . 8 |- ( m = M -> ( m = 1 <-> M = 1 ) )
4		eqeq1 2245	. . . . . . . 8 |- ( m = M -> ( m = P <-> M = P ) )
5	3, 4	orbi12d 805	. . . . . . 7 |- ( m = M -> ( ( m = 1 \/ m = P ) <-> ( M = 1 \/ M = P ) ) )
6		orcom 740	. . . . . . 7 |- ( ( M = 1 \/ M = P ) <-> ( M = P \/ M = 1 ) )
7	5, 6	bitrdi 196	. . . . . 6 |- ( m = M -> ( ( m = 1 \/ m = P ) <-> ( M = P \/ M = 1 ) ) )
8	2, 7	imbi12d 234	. . . . 5 |- ( m = M -> ( ( m || P -> ( m = 1 \/ m = P ) ) <-> ( M || P -> ( M = P \/ M = 1 ) ) ) )
9	8	rspccva 2928	. . . 4 |- ( ( A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) /\ M e. NN ) -> ( M || P -> ( M = P \/ M = 1 ) ) )
10	9	adantll 480	. . 3 |- ( ( ( P e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) ) /\ M e. NN ) -> ( M || P -> ( M = P \/ M = 1 ) ) )
11	1, 10	sylanb 284	. 2 |- ( ( P e. Prime /\ M e. NN ) -> ( M || P -> ( M = P \/ M = 1 ) ) )
12		prmz 12891	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
13		iddvds 12573	. . . . . 6 |- ( P e. ZZ -> P || P )
14	12, 13	syl 14	. . . . 5 |- ( P e. Prime -> P || P )
15	14	adantr 276	. . . 4 |- ( ( P e. Prime /\ M e. NN ) -> P || P )
16		breq1 4133	. . . 4 |- ( M = P -> ( M || P <-> P || P ) )
17	15, 16	syl5ibrcom 157	. . 3 |- ( ( P e. Prime /\ M e. NN ) -> ( M = P -> M || P ) )
18		1dvds 12574	. . . . . 6 |- ( P e. ZZ -> 1 || P )
19	12, 18	syl 14	. . . . 5 |- ( P e. Prime -> 1 || P )
20	19	adantr 276	. . . 4 |- ( ( P e. Prime /\ M e. NN ) -> 1 || P )
21		breq1 4133	. . . 4 |- ( M = 1 -> ( M || P <-> 1 || P ) )
22	20, 21	syl5ibrcom 157	. . 3 |- ( ( P e. Prime /\ M e. NN ) -> ( M = 1 -> M || P ) )
23	17, 22	jaod 729	. 2 |- ( ( P e. Prime /\ M e. NN ) -> ( ( M = P \/ M = 1 ) -> M || P ) )
24	11, 23	impbid 129	1 |- ( ( P e. Prime /\ M e. NN ) -> ( M || P <-> ( M = P \/ M = 1 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4130   ` cfv 5377   1c1 8180   NNcn 9305   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:  prm2orodd  12906  pythagtriplem4  13049  2lgs  16235