Theorem dvdsprime 12827

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 12822	. . 3 |- ( P e. Prime <-> ( P e. ( ZZ>= ` 2 ) /\ A. m e. NN ( m || P -> ( m = 1 \/ m = P ) ) ) )
2		breq1 4114	. . . . . 6 |- ( m = M -> ( m || P <-> M || P ) )
3		eqeq1 2241	. . . . . . . 8 |- ( m = M -> ( m = 1 <-> M = 1 ) )
4		eqeq1 2241	. . . . . . . 8 |- ( m = M -> ( m = P <-> M = P ) )
5	3, 4	orbi12d 801	. . . . . . 7 |- ( m = M -> ( ( m = 1 \/ m = P ) <-> ( M = 1 \/ M = P ) ) )
6		orcom 736	. . . . . . 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 2922	. . . 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 12816	. . . . . 6 |- ( P e. Prime -> P e. ZZ )
13		iddvds 12498	. . . . . 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 4114	. . . 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 12499	. . . . . 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 4114	. . . 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 725	. 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 716   = wceq 1398   e. wcel 2205   A.wral 2522   class class class wbr 4111   ` cfv 5354   1c1 8133   NNcn 9242   2c2 9293   ZZcz 9582   ZZ>=cuz 9859   || cdvds 12481   Primecprime 12812

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-dvds 12482  df-prm 12813

This theorem is referenced by:  prm2orodd  12831  pythagtriplem4  12974  2lgs  16026