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Theorem expsubap 10771
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 9440 . . 3 |- ( N e. ZZ -> -u N e. ZZ )
2 expaddzap 10767 . . 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 9414 . . . . 5 |- ( M e. ZZ -> M e. CC )
5 zcn 9414 . . . . 5 |- ( N e. ZZ -> N e. CC )
6 negsub 8357 . . . . 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 5985 . 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( A ^ ( M + -u N ) ) = ( A ^ ( M - N ) ) )
10 expnegzap 10757 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ -u N ) = ( 1 / ( A ^ N ) ) )
11103expa 1206 . . . . 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 5985 . . 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 10748 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ M e. ZZ ) -> ( A ^ M ) e. CC )
15143expa 1206 . . . . 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 10748 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) e. CC )
18173expa 1206 . . . . 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 10755 . . . . . 6 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( A ^ N ) # 0 )
21203expa 1206 . . . . 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 8903 . . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( M e. ZZ /\ N e. ZZ ) ) -> ( ( A ^ M ) / ( A ^ N ) ) = ( ( A ^ M ) x. ( 1 / ( A ^ N ) ) ) )
2413, 23eqtr4d 2243 . 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 2248 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 1373   e. wcel 2178   class class class wbr 4060  (class class class)co 5969   CCcc 7960   0cc0 7962   1c1 7963   + caddc 7965   x. cmul 7967   - cmin 8280   -ucneg 8281   # cap 8691   / cdiv 8782   ZZcz 9409   ^cexp 10722
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 4176  ax-sep 4179  ax-nul 4187  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-iinf 4655  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-mulrcl 8061  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-precex 8072  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-apti 8077  ax-pre-ltadd 8078  ax-pre-mulgt0 8079  ax-pre-mulext 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 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-if 3581  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-po 4362  df-iso 4363  df-iord 4432  df-on 4434  df-ilim 4435  df-suc 4437  df-iom 4658  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-1st 6251  df-2nd 6252  df-recs 6416  df-frec 6502  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-reap 8685  df-ap 8692  df-div 8783  df-inn 9074  df-n0 9333  df-z 9410  df-uz 9686  df-seqfrec 10632  df-exp 10723
This theorem is referenced by:  expm1ap  10773  ltexp2a  10775  leexp2a  10776  iexpcyc  10828  expsubapd  10868
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