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Theorem divval 9680
 Description: Value of division: the (unique) element such that . This is meaningful only when is nonzero. (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 3927 . . 3
2 eqeq2 2445 . . . . 5
32riotabidv 6551 . . . 4
4 oveq1 6088 . . . . . 6
54eqeq1d 2444 . . . . 5
65riotabidv 6551 . . . 4
7 df-div 9678 . . . 4
8 riotaex 6553 . . . 4
93, 6, 7, 8ovmpt2 6209 . . 3
101, 9sylan2br 463 . 2
11103impb 1149 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725   wne 2599   cdif 3317  csn 3814  (class class class)co 6081  crio 6542  cc 8988  cc0 8990   cmul 8995   cdiv 9677 This theorem is referenced by:  divmul  9681  divcl  9684  cnflddiv  16731  divcn  18898  rexdiv  24172 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-div 9678
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