Theorem dvdsprime 12717

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 12712	. . 3 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) ) )
2		breq1 4092	. . . . . 6 |- ( m = M -> ( m || P <-> M || P ) )
3		eqeq1 2237	. . . . . . . 8 |- ( m = M -> ( m = 1 <-> M = 1 ) )
4		eqeq1 2237	. . . . . . . 8 |- ( m = M -> ( m = P <-> M = P ) )
5	3, 4	orbi12d 800	. . . . . . 7 |- ( m = M -> ( ( m = 1 \/ m = P ) <-> ( M = 1 \/ M = P ) ) )
6		orcom 735	. . . . . . 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 2908	. . . 4 |- ( ( A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) /\ M e. NN ) -> ( M || P -> ( M = P \/ M = 1 ) ) )
10	9	adantll 476	. . 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 12706	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
13		iddvds 12388	. . . . . 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 4092	. . . 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 12389	. . . . . 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 4092	. . . 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 724	. 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   e. wcel 2201   A.wral 2509   class class class wbr 4089   ` cfv 5328   1c1 8038   NNcn 9148   2c2 9199   ZZcz 9484   ZZ>=cuz 9760   || cdvds 12371   Primecprime 12702

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-1o 6587  df-2o 6588  df-er 6707  df-en 6915  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-seqfrec 10716  df-exp 10807  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-dvds 12372  df-prm 12703

This theorem is referenced by:  prm2orodd  12721  pythagtriplem4  12864  2lgs  15862