Theorem expcan 11103

Description: Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Assertion

Ref	Expression
expcan	|- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ M ) = ( A ^ N ) <-> M = N ) )

Proof of Theorem expcan

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> A e. RR )
2		simpl2 1028	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> M e. ZZ )
3		simpl3 1029	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> N e. ZZ )
4		simpr 110	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> 1 < A )
5	1, 2, 3, 4	expcanlem 11102	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ M ) <_ ( A ^ N ) -> M <_ N ) )
6	1, 3, 2, 4	expcanlem 11102	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ N ) <_ ( A ^ M ) -> N <_ M ) )
7	5, 6	anim12d 335	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( ( A ^ M ) <_ ( A ^ N ) /\ ( A ^ N ) <_ ( A ^ M ) ) -> ( M <_ N /\ N <_ M ) ) )
8		0red 8291	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> 0 e. RR )
9		1red 8305	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> 1 e. RR )
10		0lt1 8416	. . . . . . . 8 |- 0 < 1
11	10	a1i 9	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> 0 < 1 )
12	8, 9, 1, 11, 4	lttrd 8415	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> 0 < A )
13	1, 12	gt0ap0d 8920	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> A # 0 )
14	1, 13, 2	reexpclzapd 11085	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( A ^ M ) e. RR )
15	1, 13, 3	reexpclzapd 11085	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( A ^ N ) e. RR )
16	14, 15	letri3d 8405	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ M ) = ( A ^ N ) <-> ( ( A ^ M ) <_ ( A ^ N ) /\ ( A ^ N ) <_ ( A ^ M ) ) ) )
17	2	zred 9718	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> M e. RR )
18	3	zred 9718	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> N e. RR )
19	17, 18	letri3d 8405	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
20	7, 16, 19	3imtr4d 203	. 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ M ) = ( A ^ N ) -> M = N ) )
21		oveq2 6066	. 2 |- ( M = N -> ( A ^ M ) = ( A ^ N ) )
22	20, 21	impbid1 142	1 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( ( A ^ M ) = ( A ^ N ) <-> M = N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   1c1 8144   < clt 8324   <_ cle 8325   ZZcz 9594   ^cexp 10924

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-rp 10005  df-seqfrec 10834  df-exp 10925

This theorem is referenced by:  expcand  11104