Theorem prmexpb 12083

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 12043	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
2	1	adantr 274	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime ) -> P e. ZZ )
3	2	3ad2ant1 1008	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. ZZ )
4		simp2l 1013	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. NN )
5		iddvdsexp 11755	. . . . . 6 |- ( ( P e. ZZ /\ M e. NN ) -> P || ( P ^ M ) )
6	3, 4, 5	syl2anc 409	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P || ( P ^ M ) )
7		breq2 3986	. . . . . . 7 |- ( ( P ^ M ) = ( Q ^ N ) -> ( P || ( P ^ M ) <-> P || ( Q ^ N ) ) )
8	7	3ad2ant3 1010	. . . . . 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 1011	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. Prime )
10		simp1r 1012	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> Q e. Prime )
11		simp2r 1014	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. NN )
12		prmdvdsexpb 12081	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )
13	9, 10, 11, 12	syl3anc 1228	. . . . . 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 9313	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. RR )
17	4	nnzd 9312	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. ZZ )
18	11	nnzd 9312	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. ZZ )
19		prmgt1 12064	. . . . . . 7 |- ( P e. Prime -> 1 < P )
20	19	ad2antrr 480	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> 1 < P )
21	20	3adant3 1007	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> 1 < P )
22		simp3 989	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P ^ M ) = ( Q ^ N ) )
23	15	oveq1d 5857	. . . . . 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 2201	. . . . 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 10630	. . . 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 1195	. 2 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) -> ( P = Q /\ M = N ) ) )
28		oveq12 5851	. 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 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   1c1 7754   < clt 7933   NNcn 8857   ZZcz 9191   ^cexp 10454   || cdvds 11727   Primecprime 12039

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-sup 6949  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-gcd 11876  df-prm 12040

This theorem is referenced by: (None)