ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expsubap Unicode version
Theorem expsubap 10541
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 9260 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
2 expaddzap 10537 . . 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 9234 . . . . 5 |- ( M e. ZZ -> M e. CC )
5 zcn 9234 . . . . 5 |- ( N e. ZZ -> N e. CC )
6 negsub 8182 . . . . 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 5884 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( A ^ ( M - N ) ) )
10 expnegzap 10527 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
11103expa 1203 . . . . 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 5884 . . 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 10518 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ M e. ZZ ) -> ( A ^ M ) e. CC )
15143expa 1203 . . . . 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 10518 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
18173expa 1203 . . . . 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 10525 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
21203expa 1203 . . . . 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 8726 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) / ( A ^ N ) ) = ( ( A ^ M ) x. ( 1 / ( A ^ N ) ) ) )
2413, 23eqtr4d 2213 . 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 2218 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 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5868   CCcc 7787   0cc0 7789   1c1 7790   + caddc 7792   x. cmul 7794   - cmin 8105   -ucneg 8106   # cap 8515   / cdiv 8605   ZZcz 9229   ^cexp 10492
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419  df-exp 10493
This theorem is referenced by:  expm1ap  10543  ltexp2a  10545  leexp2a  10546  iexpcyc  10597  expsubapd  10637
  Copyright terms: Public domain W3C validator