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Theorem divval 10271
 Description: Value of division: if and are complex numbers with nonzero, then is the (unique) complex number such that . (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.)
Assertion
Ref Expression
divval
Distinct variable groups:   ,   ,

Proof of Theorem divval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4128 . . 3
2 eqeq2 2444 . . . . 5
32riotabidv 6269 . . . 4
4 oveq1 6312 . . . . . 6
54eqeq1d 2431 . . . . 5
65riotabidv 6269 . . . 4
7 df-div 10269 . . . 4
8 riotaex 6271 . . . 4
93, 6, 7, 8ovmpt2 6446 . . 3
101, 9sylan2br 478 . 2
11103impb 1201 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   w3a 982   wceq 1437   wcel 1870   wne 2625   cdif 3439  csn 4002  crio 6266  (class class class)co 6305  cc 9536  cc0 9538   cmul 9543   cdiv 10268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-div 10269 This theorem is referenced by:  divmul  10272  divcl  10275  cnflddiv  18933  divcn  21796  rexdiv  28233
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