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Theorem expsubap 9524
Description: Exponent subtraction law for nonnegative 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 8382 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
2 expaddzap 9520 . . 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 397 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( ( A ^ M ) x. ( A ^ -u N ) ) )
4 zcn 8356 . . . . 5 |- ( M e. ZZ -> M e. CC )
5 zcn 8356 . . . . 5 |- ( N e. ZZ -> N e. CC )
6 negsub 7356 . . . . 5 |- ( ( M e. CC /\ N e. CC ) -> ( M + -u N ) = ( M - N ) )
74, 5, 6syl2an 283 . . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M + -u N ) = ( M - N ) )
87adantl 271 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( M + -u N ) = ( M - N ) )
98oveq2d 5548 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( A ^ ( M - N ) ) )
10 expnegzap 9510 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
11103expa 1138 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
1211adantrl 461 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
1312oveq2d 5548 . . 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 9501 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ M e. ZZ ) -> ( A ^ M ) e. CC )
15143expa 1138 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ M e. ZZ ) -> ( A ^ M ) e. CC )
1615adantrr 462 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ M ) e. CC )
17 expclzap 9501 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
18173expa 1138 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ N ) e. CC )
1918adantrl 461 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ N ) e. CC )
20 expap0i 9508 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
21203expa 1138 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ N e. ZZ ) -> ( A ^ N ) # 0 )
2221adantrl 461 . . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ N ) # 0 )
2316, 19, 22divrecapd 7880 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) / ( A ^ N ) ) = ( ( A ^ M ) x. ( 1 / ( A ^ N ) ) ) )
2413, 23eqtr4d 2116 . 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 2121 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 102   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   - cmin 7279   -ucneg 7280   # cap 7681   / cdiv 7760   ZZcz 8351   ^cexp 9475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094
This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476
This theorem is referenced by:  expm1ap  9526  ltexp2a  9528  leexp2a  9529  iexpcyc  9579  expsubapd  9616
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