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Theorem xdivval 28053
 Description: Value of division: the (unique) element such that . This is meaningful only when is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Assertion
Ref Expression
xdivval /𝑒
Distinct variable groups:   ,   ,

Proof of Theorem xdivval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4096 . . 3
2 simpl 455 . . . . . 6
32eqeq2d 2416 . . . . 5
43riotabidva 6255 . . . 4
5 simpl 455 . . . . . . 7
65oveq1d 6292 . . . . . 6
76eqeq1d 2404 . . . . 5
87riotabidva 6255 . . . 4
9 df-xdiv 28052 . . . 4 /𝑒
10 riotaex 6243 . . . 4
114, 8, 9, 10ovmpt2 6418 . . 3 /𝑒
121, 11sylan2br 474 . 2 /𝑒
13123impb 1193 1 /𝑒
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   w3a 974   wceq 1405   wcel 1842   wne 2598   cdif 3410  csn 3971  crio 6238  (class class class)co 6277  cr 9520  cc0 9521  cxr 9656  cxmu 11369   /𝑒 cxdiv 28051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-iota 5532  df-fun 5570  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-xdiv 28052 This theorem is referenced by:  xdivcld  28057  xdivmul  28059  rexdiv  28060  xdivpnfrp  28067
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