Theorem xnegeq 12038

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

Step	Hyp	Ref	Expression
1		eqeq1 2626	. . 3 |- ( A = B -> ( A = +oo <-> B = +oo ) )
2		eqeq1 2626	. . . 4 |- ( A = B -> ( A = -oo <-> B = -oo ) )
3		negeq 10273	. . . 4 |- ( A = B -> -u A = -u B )
4	2, 3	ifbieq2d 4111	. . 3 |- ( A = B -> if ( A = -oo , +oo , -u A ) = if ( B = -oo , +oo , -u B ) )
5	1, 4	ifbieq2d 4111	. 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 11946	. 2 |- -e A = if ( A = +oo , -oo , if ( A = -oo , +oo , -u A ) )
7		df-xneg 11946	. 2 |- -e B = if ( B = +oo , -oo , if ( B = -oo , +oo , -u B ) )
8	5, 6, 7	3eqtr4g 2681	1 |- ( A = B -> -e A = -e B )

Syntax hints:   -> wi 4   = wceq 1483   ifcif 4086   +oocpnf 10071   -oocmnf 10072   -ucneg 10267   -ecxne 11943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-neg 10269  df-xneg 11946

This theorem is referenced by:  xnegcl  12044  xnegneg  12045  xneg11  12046  xltnegi  12047  xnegid  12069  xnegdi  12078  xsubge0  12091  xlesubadd  12093  xmulneg1  12099  xmulneg2  12100  xmulmnf1  12106  xmulm1  12111  xrsdsval  19790  xrsdsreclblem  19792  xblss2ps  22206  xblss2  22207  xrhmeo  22745  xaddeq0  29518  xrsmulgzz  29678  xrge0npcan  29694  carsgclctunlem2  30381  xnegeqd  39664  xnegeqi  39667  supminfxr2  39699  supminfxrrnmpt  39701