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Theorem xnegeq 12588
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 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeq
StepHypRef Expression
1 eqeq1 2822 . . 3 (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2 eqeq1 2822 . . . 4 (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3 negeq 10866 . . . 4 (𝐴 = 𝐵 → -𝐴 = -𝐵)
42, 3ifbieq2d 4488 . . 3 (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
51, 4ifbieq2d 4488 . 2 (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6 df-xneg 12495 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7 df-xneg 12495 . 2 -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
85, 6, 73eqtr4g 2878 1 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  ifcif 4463  +∞cpnf 10660  -∞cmnf 10661  -cneg 10859  -𝑒cxne 12492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-neg 10861  df-xneg 12495
This theorem is referenced by:  xnegcl  12594  xnegneg  12595  xneg11  12596  xltnegi  12597  xnegid  12619  xnegdi  12629  xsubge0  12642  xlesubadd  12644  xmulneg1  12650  xmulneg2  12651  xmulmnf1  12657  xmulm1  12662  xrsdsval  20517  xrsdsreclblem  20519  xblss2ps  22938  xblss2  22939  xrhmeo  23477  xaddeq0  30403  xrsmulgzz  30592  xrge0npcan  30608  carsgclctunlem2  31476  xnegeqd  41587  xnegeqi  41590  supminfxr2  41621  supminfxrrnmpt  41623  liminflbuz2  41972
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