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Theorem efsub 13692
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 13676 . . . 4  |-  ( A  e.  CC  ->  ( exp `  A )  e.  CC )
2 efcl 13676 . . . 4  |-  ( B  e.  CC  ->  ( exp `  B )  e.  CC )
3 efne0 13689 . . . 4  |-  ( B  e.  CC  ->  ( exp `  B )  =/=  0 )
4 divrec 10219 . . . 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 1270 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  e.  CC )  ->  (
( exp `  A
)  /  ( exp `  B ) )  =  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) ) )
653anidm23 1287 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  /  ( exp `  B ) )  =  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) ) )
7 efcan 13688 . . . . . . 7  |-  ( B  e.  CC  ->  (
( exp `  B
)  x.  ( exp `  -u B ) )  =  1 )
87eqcomd 2475 . . . . . 6  |-  ( B  e.  CC  ->  1  =  ( ( exp `  B )  x.  ( exp `  -u B ) ) )
9 negcl 9816 . . . . . . . 8  |-  ( B  e.  CC  ->  -u B  e.  CC )
10 efcl 13676 . . . . . . . 8  |-  ( -u B  e.  CC  ->  ( exp `  -u B
)  e.  CC )
119, 10syl 16 . . . . . . 7  |-  ( B  e.  CC  ->  ( exp `  -u B )  e.  CC )
12 ax-1cn 9546 . . . . . . . 8  |-  1  e.  CC
13 divmul2 10207 . . . . . . . 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 1311 . . . . . . 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 1226 . . . . . 6  |-  ( B  e.  CC  ->  (
( 1  /  ( exp `  B ) )  =  ( exp `  -u B
)  <->  1  =  ( ( exp `  B
)  x.  ( exp `  -u B ) ) ) )
168, 15mpbird 232 . . . . 5  |-  ( B  e.  CC  ->  (
1  /  ( exp `  B ) )  =  ( exp `  -u B
) )
1716oveq2d 6298 . . . 4  |-  ( B  e.  CC  ->  (
( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( ( exp `  A
)  x.  ( exp `  -u B ) ) )
1817adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( ( exp `  A
)  x.  ( exp `  -u B ) ) )
19 efadd 13687 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( ( exp `  A )  x.  ( exp `  -u B
) ) )
209, 19sylan2 474 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( ( exp `  A )  x.  ( exp `  -u B
) ) )
2118, 20eqtr4d 2511 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( exp `  A
)  x.  ( 1  /  ( exp `  B
) ) )  =  ( exp `  ( A  +  -u B ) ) )
22 negsub 9863 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
2322fveq2d 5868 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  +  -u B ) )  =  ( exp `  ( A  -  B
) ) )
246, 21, 233eqtrrd 2513 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( exp `  ( A  -  B )
)  =  ( ( exp `  A )  /  ( exp `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5586  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   expce 13655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-pm 7420  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-oi 7931  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-ico 11531  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661
This theorem is referenced by:  efeq1  22649  efif1olem4  22665  relogdiv  22705  eflogeq  22714  efiarg  22720  logneg2  22728  logdiv2  22730  logcnlem4  22754  efopn  22767  ang180lem1  22869  efiatan  22971  2efiatan  22977  atantan  22982  birthdaylem2  23010  efchtdvds  23161  bposlem9  23295  gamcvg2lem  28241  iprodgam  28702
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