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Theorem xnegeq 11415
Description: Equality of two extended numbers with  -e in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegeq  |-  ( A  =  B  ->  -e
A  =  -e
B )

Proof of Theorem xnegeq
StepHypRef Expression
1 eqeq1 2447 . . 3  |-  ( A  =  B  ->  ( A  = +oo  <->  B  = +oo ) )
2 eqeq1 2447 . . . 4  |-  ( A  =  B  ->  ( A  = -oo  <->  B  = -oo ) )
3 negeq 9817 . . . 4  |-  ( A  =  B  ->  -u A  =  -u B )
42, 3ifbieq2d 3951 . . 3  |-  ( A  =  B  ->  if ( A  = -oo , +oo ,  -u A
)  =  if ( B  = -oo , +oo ,  -u B ) )
51, 4ifbieq2d 3951 . 2  |-  ( A  =  B  ->  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )  =  if ( B  = +oo , -oo ,  if ( B  = -oo , +oo ,  -u B ) ) )
6 df-xneg 11327 . 2  |-  -e
A  =  if ( A  = +oo , -oo ,  if ( A  = -oo , +oo ,  -u A ) )
7 df-xneg 11327 . 2  |-  -e
B  =  if ( B  = +oo , -oo ,  if ( B  = -oo , +oo ,  -u B ) )
85, 6, 73eqtr4g 2509 1  |-  ( A  =  B  ->  -e
A  =  -e
B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   ifcif 3926   +oocpnf 9628   -oocmnf 9629   -ucneg 9811    -ecxne 11324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586  df-ov 6284  df-neg 9813  df-xneg 11327
This theorem is referenced by:  xnegcl  11421  xnegneg  11422  xneg11  11423  xltnegi  11424  xnegid  11444  xnegdi  11449  xsubge0  11462  xlesubadd  11464  xmulneg1  11470  xmulneg2  11471  xmulmnf1  11477  xmulm1  11482  xrsdsval  18336  xrsdsreclblem  18338  xblss2ps  20777  xblss2  20778  xrhmeo  21319  xaddeq0  27445  xrsmulgzz  27539  xrge0npcan  27557
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