ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expsubap Unicode version
Theorem expsubap 10978
Description: Exponent subtraction law for integer exponentiation. (Contributed by Jim Kingdon, 11-Jun-2020.)
Assertion
Ref Expression
expsubap |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )
Proof of Theorem expsubap
StepHypRef Expression
1 znegcl 9630 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
2 expaddzap 10974 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ -u N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( ( A ^ M ) x. ( A ^ -u N ) ) )
31, 2sylanr2 405 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( ( A ^ M ) x. ( A ^ -u N ) ) )
4 zcn 9604 . . . . 5 |- ( M e. ZZ -> M e. CC )
5 zcn 9604 . . . . 5 |- ( N e. ZZ -> N e. CC )
6 negsub 8540 . . . . 5 |- ( ( M e. CC /\ N e. CC ) -> ( M + -u N ) = ( M - N ) )
74, 5, 6syl2an 289 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + -u N ) = ( M - N ) )
87adantl 277 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M + -u N ) = ( M - N ) )
98oveq2d 6076 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( A ^ ( M - N ) ) )
10 expnegzap 10964 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
11103expa 1230 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
1211adantrl 478 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
1312oveq2d 6076 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) x. ( A ^ -u N ) ) = ( ( A ^ M ) x. ( 1 / ( A ^ N ) ) ) )
14 expclzap 10955 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ M e. ZZ ) -> ( A ^ M ) e. CC )
15143expa 1230 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( A ^ M ) e. CC )
1615adantrr 479 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ M ) e. CC )
17 expclzap 10955 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
18173expa 1230 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ N ) e. CC )
1918adantrl 478 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ N ) e. CC )
20 expap0i 10962 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
21203expa 1230 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ N ) # 0 )
2221adantrl 478 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ N ) # 0 )
2316, 19, 22divrecapd 9089 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) / ( A ^ N ) ) = ( ( A ^ M ) x. ( 1 / ( A ^ N ) ) ) )
2413, 23eqtr4d 2270 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) x. ( A ^ -u N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )
253, 9, 243eqtr3d 2275 1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M - N ) ) = ( ( A ^ M ) / ( A ^ N ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6060   CCcc 8143   0cc0 8145   1c1 8146   + caddc 8148   x. cmul 8150   - cmin 8463   -ucneg 8464   # cap 8875   / cdiv 8968   ZZcz 9599   ^cexp 10929
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-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263
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 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-frec 6637  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-n0 9519  df-z 9600  df-uz 9877  df-seqfrec 10839  df-exp 10930
This theorem is referenced by:  expm1ap  10980  ltexp2a  10982  leexp2a  10983  iexpcyc  11035  expsubapd  11076
  Copyright terms: Public domain W3C validator