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

Theorem xnegeqd 41573
Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
xnegeqd.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
xnegeqd (𝜑 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeqd
StepHypRef Expression
1 xnegeqd.1 . 2 (𝜑𝐴 = 𝐵)
2 xnegeq 12593 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2syl 17 1 (𝜑 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  -𝑒cxne 12497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-ov 7154  df-neg 10865  df-xneg 12500
This theorem is referenced by:  supminfxr  41602  supminfxr2  41607  supminfxrrnmpt  41609  monoord2xrv  41622  liminfvalxr  41926  liminfvalxrmpt  41929  liminfval4  41932  liminfval3  41933  limsupval4  41937  liminfvaluz2  41938  limsupvaluz4  41943  climliminflimsupd  41944  xlimpnfxnegmnf  41957  liminfpnfuz  41959  xlimpnfxnegmnf2  42001  smfliminflem  42967
  Copyright terms: Public domain W3C validator