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Theorem efsub 12477
Description: Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
Assertion
Ref Expression
efsub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B )
)  =  ( ( exp `  A )  /  ( exp `  B
) ) )

Proof of Theorem efsub
StepHypRef Expression
1 efcl 12461 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efcl 12461 . . . 4  |-  ( B  e.  CC  ->  ( exp `  B )  e.  CC )
3 efne0 12474 . . . 4  |-  ( B  e.  CC  ->  ( exp `  B )  =/=  0 )
4 divrec 9530 . . . 4  |-  ( ( ( exp `  A
)  e.  CC  /\  ( exp `  B )  e.  CC  /\  ( exp `  B )  =/=  0 )  ->  (
( exp `  A
)  /  ( exp `  B ) )  =  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) ) )
51, 2, 3, 4syl3an 1224 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( exp `  A
)  /  ( exp `  B ) )  =  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) ) )
653anidm23 1241 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  /  ( exp `  B ) )  =  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) ) )
7 efcan 12473 . . . . . . 7  |-  ( B  e.  CC  ->  (
( exp `  B
)  x.  ( exp `  -u B ) )  =  1 )
87eqcomd 2363 . . . . . 6  |-  ( B  e.  CC  ->  1  =  ( ( exp `  B )  x.  ( exp `  -u B ) ) )
9 negcl 9142 . . . . . . . 8  |-  ( B  e.  CC  ->  -u B  e.  CC )
10 efcl 12461 . . . . . . . 8  |-  ( -u B  e.  CC  ->  ( exp `  -u B
)  e.  CC )
119, 10syl 15 . . . . . . 7  |-  ( B  e.  CC  ->  ( exp `  -u B )  e.  CC )
12 ax-1cn 8885 . . . . . . . 8  |-  1  e.  CC
13 divmul2 9518 . . . . . . . 8  |-  ( ( 1  e.  CC  /\  ( exp `  -u B
)  e.  CC  /\  ( ( exp `  B
)  e.  CC  /\  ( exp `  B )  =/=  0 ) )  ->  ( ( 1  /  ( exp `  B
) )  =  ( exp `  -u B
)  <->  1  =  ( ( exp `  B
)  x.  ( exp `  -u B ) ) ) )
1412, 13mp3an1 1264 . . . . . . 7  |-  ( ( ( exp `  -u B
)  e.  CC  /\  ( ( exp `  B
)  e.  CC  /\  ( exp `  B )  =/=  0 ) )  ->  ( ( 1  /  ( exp `  B
) )  =  ( exp `  -u B
)  <->  1  =  ( ( exp `  B
)  x.  ( exp `  -u B ) ) ) )
1511, 2, 3, 14syl12anc 1180 . . . . . 6  |-  ( B  e.  CC  ->  (
( 1  /  ( exp `  B ) )  =  ( exp `  -u B
)  <->  1  =  ( ( exp `  B
)  x.  ( exp `  -u B ) ) ) )
168, 15mpbird 223 . . . . 5  |-  ( B  e.  CC  ->  (
1  /  ( exp `  B ) )  =  ( exp `  -u B
) )
1716oveq2d 5961 . . . 4  |-  ( B  e.  CC  ->  (
( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( ( exp `  A
)  x.  ( exp `  -u B ) ) )
1817adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( ( exp `  A
)  x.  ( exp `  -u B ) ) )
19 efadd 12472 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( ( exp `  A )  x.  ( exp `  -u B
) ) )
209, 19sylan2 460 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( ( exp `  A )  x.  ( exp `  -u B
) ) )
2118, 20eqtr4d 2393 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( exp `  ( A  +  -u B ) ) )
22 negsub 9185 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2322fveq2d 5612 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( exp `  ( A  -  B
) ) )
246, 21, 233eqtrrd 2395 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B )
)  =  ( ( exp `  A )  /  ( exp `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   ` cfv 5337  (class class class)co 5945   CCcc 8825   0cc0 8827   1c1 8828    + caddc 8830    x. cmul 8832    - cmin 9127   -ucneg 9128    / cdiv 9513   expce 12440
This theorem is referenced by:  efeq1  19998  efif1olem4  20014  relogdiv  20054  eflogeq  20063  efiarg  20069  logneg2  20077  logdiv2  20079  logcnlem4  20103  efopn  20116  ang180lem1  20218  efiatan  20319  2efiatan  20325  atantan  20330  birthdaylem2  20358  efchtdvds  20509  bposlem9  20643  gamcvg2lem  24092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-inf2 7432  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904  ax-pre-sup 8905  ax-addf 8906  ax-mulf 8907
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-pm 6863  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-sup 7284  df-oi 7315  df-card 7662  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514  df-nn 9837  df-2 9894  df-3 9895  df-n0 10058  df-z 10117  df-uz 10323  df-rp 10447  df-ico 10754  df-fz 10875  df-fzo 10963  df-fl 11017  df-seq 11139  df-exp 11198  df-fac 11382  df-bc 11409  df-hash 11431  df-shft 11658  df-cj 11680  df-re 11681  df-im 11682  df-sqr 11816  df-abs 11817  df-limsup 12041  df-clim 12058  df-rlim 12059  df-sum 12256  df-ef 12446
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