Theorem prmexpb 11818

Description: Two positive prime powers are equal iff the primes and the powers are equal. (Contributed by Paul Chapman, 30-Nov-2012.)

Assertion

Ref	Expression
prmexpb	|- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) <-> ( P = Q /\ M = N ) ) )

Proof of Theorem prmexpb

Step	Hyp	Ref	Expression
1		prmz 11781	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
2	1	adantr 274	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime ) -> P e. ZZ )
3	2	3ad2ant1 1002	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. ZZ )
4		simp2l 1007	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. NN )
5		iddvdsexp 11506	. . . . . 6 |- ( ( P e. ZZ /\ M e. NN ) -> P || ( P ^ M ) )
6	3, 4, 5	syl2anc 408	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P || ( P ^ M ) )
7		breq2 3928	. . . . . . 7 |- ( ( P ^ M ) = ( Q ^ N ) -> ( P || ( P ^ M ) <-> P || ( Q ^ N ) ) )
8	7	3ad2ant3 1004	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P || ( P ^ M ) <-> P || ( Q ^ N ) ) )
9		simp1l 1005	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. Prime )
10		simp1r 1006	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> Q e. Prime )
11		simp2r 1008	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. NN )
12		prmdvdsexpb 11816	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )
13	9, 10, 11, 12	syl3anc 1216	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P || ( Q ^ N ) <-> P = Q ) )
14	8, 13	bitrd 187	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P || ( P ^ M ) <-> P = Q ) )
15	6, 14	mpbid 146	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P = Q )
16	3	zred 9166	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. RR )
17	4	nnzd 9165	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. ZZ )
18	11	nnzd 9165	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. ZZ )
19		prmgt1 11801	. . . . . . 7 |- ( P e. Prime -> 1 < P )
20	19	ad2antrr 479	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> 1 < P )
21	20	3adant3 1001	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> 1 < P )
22		simp3 983	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P ^ M ) = ( Q ^ N ) )
23	15	oveq1d 5782	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P ^ N ) = ( Q ^ N ) )
24	22, 23	eqtr4d 2173	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P ^ M ) = ( P ^ N ) )
25	16, 17, 18, 21, 24	expcand 10457	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M = N )
26	15, 25	jca 304	. . 3 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P = Q /\ M = N ) )
27	26	3expia 1183	. 2 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) -> ( P = Q /\ M = N ) ) )
28		oveq12 5776	. 2 |- ( ( P = Q /\ M = N ) -> ( P ^ M ) = ( Q ^ N ) )
29	27, 28	impbid1 141	1 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) <-> ( P = Q /\ M = N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   1c1 7614   < clt 7793   NNcn 8713   ZZcz 9047   ^cexp 10285   || cdvds 11482   Primecprime 11777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-1o 6306  df-2o 6307  df-er 6422  df-en 6628  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625  df-prm 11778

This theorem is referenced by: (None)