 Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xnegeqi Structured version   Visualization version   GIF version

Theorem xnegeqi 41277
 Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
xnegeqi.1 𝐴 = 𝐵
Assertion
Ref Expression
xnegeqi -𝑒𝐴 = -𝑒𝐵

Proof of Theorem xnegeqi
StepHypRef Expression
1 xnegeqi.1 . 2 𝐴 = 𝐵
2 xnegeq 12454 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2ax-mp 5 1 -𝑒𝐴 = -𝑒𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1525  -𝑒cxne 12358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-iota 6196  df-fv 6240  df-ov 7026  df-neg 10726  df-xneg 12361 This theorem is referenced by:  supminfxr2  41308  liminfvalxr  41627  liminf0  41637  liminfpnfuz  41660
 Copyright terms: Public domain W3C validator