Theorem efsub 14830

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

Step	Hyp	Ref	Expression
1		efcl 14813	. . . 4 |- ( A e. CC -> ( exp ` A ) e. CC )
2		efcl 14813	. . . 4 |- ( B e. CC -> ( exp ` B ) e. CC )
3		efne0 14827	. . . 4 |- ( B e. CC -> ( exp ` B ) =/= 0 )
4		divrec 10701	. . . 4 |- ( ( ( exp ` A ) e. CC /\ ( exp ` B ) e. CC /\ ( exp ` B ) =/= 0 ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) )
5	1, 2, 3, 4	syl3an 1368	. . 3 |- ( ( A e. CC /\ B e. CC /\ B e. CC ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) )
6	5	3anidm23 1385	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) / ( exp ` B ) ) = ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) )
7		efcan 14826	. . . . . . 7 |- ( B e. CC -> ( ( exp ` B ) x. ( exp ` -u B ) ) = 1 )
8	7	eqcomd 2628	. . . . . 6 |- ( B e. CC -> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) )
9		negcl 10281	. . . . . . . 8 |- ( B e. CC -> -u B e. CC )
10		efcl 14813	. . . . . . . 8 |- ( -u B e. CC -> ( exp ` -u B ) e. CC )
11	9, 10	syl 17	. . . . . . 7 |- ( B e. CC -> ( exp ` -u B ) e. CC )
12		ax-1cn 9994	. . . . . . . 8 |- 1 e. CC
13		divmul2 10689	. . . . . . . 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 ) ) ) )
14	12, 13	mp3an1 1411	. . . . . . 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 ) ) ) )
15	11, 2, 3, 14	syl12anc 1324	. . . . . 6 |- ( B e. CC -> ( ( 1 / ( exp ` B ) ) = ( exp ` -u B ) <-> 1 = ( ( exp ` B ) x. ( exp ` -u B ) ) ) )
16	8, 15	mpbird 247	. . . . 5 |- ( B e. CC -> ( 1 / ( exp ` B ) ) = ( exp ` -u B ) )
17	16	oveq2d 6666	. . . 4 |- ( B e. CC -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) )
18	17	adantl 482	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) )
19		efadd 14824	. . . 4 |- ( ( A e. CC /\ -u B e. CC ) -> ( exp ` ( A + -u B ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) )
20	9, 19	sylan2 491	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A + -u B ) ) = ( ( exp ` A ) x. ( exp ` -u B ) ) )
21	18, 20	eqtr4d 2659	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( exp ` A ) x. ( 1 / ( exp ` B ) ) ) = ( exp ` ( A + -u B ) ) )
22		negsub 10329	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
23	22	fveq2d 6195	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A + -u B ) ) = ( exp ` ( A - B ) ) )
24	6, 21, 23	3eqtrrd 2661	1 |- ( ( A e. CC /\ B e. CC ) -> ( exp ` ( A - B ) ) = ( ( exp ` A ) / ( exp ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   expce 14792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798

This theorem is referenced by:  efeq1  24275  efif1olem4  24291  relogdiv  24339  eflogeq  24348  efiarg  24353  logneg2  24361  logdiv2  24363  logcnlem4  24391  efopn  24404  ang180lem1  24539  efiatan  24639  2efiatan  24645  atantan  24650  birthdaylem2  24679  gamcvg2lem  24785  efchtdvds  24885  bposlem9  25017  iprodgam  31628