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Theorem xnegeq 12599
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 2825 . . 3 (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2 eqeq1 2825 . . . 4 (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3 negeq 10877 . . . 4 (𝐴 = 𝐵 → -𝐴 = -𝐵)
42, 3ifbieq2d 4491 . . 3 (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
51, 4ifbieq2d 4491 . 2 (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6 df-xneg 12506 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7 df-xneg 12506 . 2 -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
85, 6, 73eqtr4g 2881 1 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ifcif 4466  +∞cpnf 10671  -∞cmnf 10672  -cneg 10870  -𝑒cxne 12503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362  df-ov 7158  df-neg 10872  df-xneg 12506
This theorem is referenced by:  xnegcl  12605  xnegneg  12606  xneg11  12607  xltnegi  12608  xnegid  12630  xnegdi  12640  xsubge0  12653  xlesubadd  12655  xmulneg1  12661  xmulneg2  12662  xmulmnf1  12668  xmulm1  12673  xrsdsval  20588  xrsdsreclblem  20590  xblss2ps  23010  xblss2  23011  xrhmeo  23549  xaddeq0  30476  xrsmulgzz  30665  xrge0npcan  30681  carsgclctunlem2  31577  xnegeqd  41709  xnegeqi  41712  supminfxr2  41743  supminfxrrnmpt  41745  liminflbuz2  42094
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