Theorem prmexpb 12588

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 12548	. . . . . . . 8 |- ( P e. Prime -> P e. ZZ )
2	1	adantr 276	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime ) -> P e. ZZ )
3	2	3ad2ant1 1021	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. ZZ )
4		simp2l 1026	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. NN )
5		iddvdsexp 12241	. . . . . 6 |- ( ( P e. ZZ /\ M e. NN ) -> P || ( P ^ M ) )
6	3, 4, 5	syl2anc 411	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P || ( P ^ M ) )
7		breq2 4063	. . . . . . 7 |- ( ( P ^ M ) = ( Q ^ N ) -> ( P || ( P ^ M ) <-> P || ( Q ^ N ) ) )
8	7	3ad2ant3 1023	. . . . . 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 1024	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. Prime )
10		simp1r 1025	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> Q e. Prime )
11		simp2r 1027	. . . . . . 7 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. NN )
12		prmdvdsexpb 12586	. . . . . . 7 |- ( ( P e. Prime /\ Q e. Prime /\ N e. NN ) -> ( P || ( Q ^ N ) <-> P = Q ) )
13	9, 10, 11, 12	syl3anc 1250	. . . . . 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 188	. . . . 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 147	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P = Q )
16	3	zred 9530	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> P e. RR )
17	4	nnzd 9529	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M e. ZZ )
18	11	nnzd 9529	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> N e. ZZ )
19		prmgt1 12569	. . . . . . 7 |- ( P e. Prime -> 1 < P )
20	19	ad2antrr 488	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> 1 < P )
21	20	3adant3 1020	. . . . 5 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> 1 < P )
22		simp3 1002	. . . . . 6 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P ^ M ) = ( Q ^ N ) )
23	15	oveq1d 5982	. . . . . 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 2243	. . . . 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 10899	. . . 4 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> M = N )
26	15, 25	jca 306	. . 3 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) /\ ( P ^ M ) = ( Q ^ N ) ) -> ( P = Q /\ M = N ) )
27	26	3expia 1208	. 2 |- ( ( ( P e. Prime /\ Q e. Prime ) /\ ( M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( Q ^ N ) -> ( P = Q /\ M = N ) ) )
28		oveq12 5976	. 2 |- ( ( P = Q /\ M = N ) -> ( P ^ M ) = ( Q ^ N ) )
29	27, 28	impbid1 142	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 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   1c1 7961   < clt 8142   NNcn 9071   ZZcz 9407   ^cexp 10720   || cdvds 12213   Primecprime 12544

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545

This theorem is referenced by: (None)