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Theorem xnegeqd 43758
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 13132 . 2 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
31, 2syl 17 1 (𝜑 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  -𝑒cxne 13035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-neg 11393  df-xneg 13038
This theorem is referenced by:  supminfxr  43785  supminfxr2  43790  supminfxrrnmpt  43792  monoord2xrv  43805  liminfvalxr  44110  liminfvalxrmpt  44113  liminfval4  44116  liminfval3  44117  limsupval4  44121  liminfvaluz2  44122  limsupvaluz4  44127  climliminflimsupd  44128  xlimpnfxnegmnf  44141  liminfpnfuz  44143  xlimpnfxnegmnf2  44185  smfliminflem  45157
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