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Theorem prmexpb 14120
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
StepHypRef Expression
1 prmz 14083 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ZZ )
21adantr 465 . . . . . . 7  |-  ( ( P  e.  Prime  /\  Q  e.  Prime )  ->  P  e.  ZZ )
323ad2ant1 1017 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  P  e.  ZZ )
4 simp2l 1022 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  M  e.  NN )
5 iddvdsexp 13871 . . . . . 6  |-  ( ( P  e.  ZZ  /\  M  e.  NN )  ->  P  ||  ( P ^ M ) )
63, 4, 5syl2anc 661 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  P  ||  ( P ^ M ) )
7 breq2 4451 . . . . . . 7  |-  ( ( P ^ M )  =  ( Q ^ N )  ->  ( P  ||  ( P ^ M )  <->  P  ||  ( Q ^ N ) ) )
873ad2ant3 1019 . . . . . 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 1020 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  P  e.  Prime )
10 simp1r 1021 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  Q  e.  Prime )
11 simp2r 1023 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  N  e.  NN )
12 prmdvdsexpb 14118 . . . . . . 7  |-  ( ( P  e.  Prime  /\  Q  e.  Prime  /\  N  e.  NN )  ->  ( P 
||  ( Q ^ N )  <->  P  =  Q ) )
139, 10, 11, 12syl3anc 1228 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P  ||  ( Q ^ N
)  <->  P  =  Q
) )
148, 13bitrd 253 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P  ||  ( P ^ M
)  <->  P  =  Q
) )
156, 14mpbid 210 . . . 4  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  P  =  Q )
163zred 10967 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  P  e.  RR )
174nnzd 10966 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  M  e.  ZZ )
1811nnzd 10966 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  N  e.  ZZ )
19 prmuz2 14097 . . . . . . . 8  |-  ( P  e.  Prime  ->  P  e.  ( ZZ>= `  2 )
)
20 eluz2b1 11154 . . . . . . . . 9  |-  ( P  e.  ( ZZ>= `  2
)  <->  ( P  e.  ZZ  /\  1  < 
P ) )
2120simprbi 464 . . . . . . . 8  |-  ( P  e.  ( ZZ>= `  2
)  ->  1  <  P )
2219, 21syl 16 . . . . . . 7  |-  ( P  e.  Prime  ->  1  < 
P )
2322ad2antrr 725 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  1  <  P )
24233adant3 1016 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  1  <  P )
25 simp3 998 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P ^ M )  =  ( Q ^ N ) )
2615oveq1d 6300 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P ^ N )  =  ( Q ^ N ) )
2725, 26eqtr4d 2511 . . . . 5  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P ^ M )  =  ( P ^ N ) )
2816, 17, 18, 24, 27expcand 12310 . . . 4  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  M  =  N )
2915, 28jca 532 . . 3  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN )  /\  ( P ^ M )  =  ( Q ^ N ) )  ->  ( P  =  Q  /\  M  =  N ) )
30293expia 1198 . 2  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  ( ( P ^ M )  =  ( Q ^ N
)  ->  ( P  =  Q  /\  M  =  N ) ) )
31 oveq12 6294 . 2  |-  ( ( P  =  Q  /\  M  =  N )  ->  ( P ^ M
)  =  ( Q ^ N ) )
3230, 31impbid1 203 1  |-  ( ( ( P  e.  Prime  /\  Q  e.  Prime )  /\  ( M  e.  NN  /\  N  e.  NN ) )  ->  ( ( P ^ M )  =  ( Q ^ N
)  <->  ( P  =  Q  /\  M  =  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   1c1 9494    < clt 9629   NNcn 10537   2c2 10586   ZZcz 10865   ZZ>=cuz 11083   ^cexp 12135    || cdivides 13850   Primecprime 14079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570  ax-pre-sup 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-2nd 6786  df-recs 7043  df-rdg 7077  df-1o 7131  df-2o 7132  df-oadd 7135  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11084  df-rp 11222  df-fl 11898  df-mod 11966  df-seq 12077  df-exp 12136  df-cj 12898  df-re 12899  df-im 12900  df-sqrt 13034  df-abs 13035  df-dvds 13851  df-gcd 14007  df-prm 14080
This theorem is referenced by:  fsumvma  23313
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