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Theorem rexneg 13186

Description: Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
rexneg	⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)

**Proof of Theorem rexneg**
Step	Hyp	Ref	Expression
1		df-xneg 13088	. 2 ⊢ -_𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		renepnf 11258	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3		ifnefalse 4539	. . . 4 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴))
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴))
5		renemnf 11259	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6		ifnefalse 4539	. . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
8	4, 7	eqtrd 2773	. 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴)
9	1, 8	eqtrid 2785	1 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ifcif 4527 ℝcr 11105 +∞cpnf 11241 -∞cmnf 11242 -cneg 11441 -_𝑒cxne 13085

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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163

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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xneg 13088

This theorem is referenced by: xneg0 13187 xnegcl 13188 xnegneg 13189 xltnegi 13191 rexsub 13208 xnegid 13213 xnegdi 13223 xpncan 13226 xnpcan 13227 xmulneg1 13244 xmulm1 13256 xadddi 13270 xlt2addrd 31949 xrsmulgzz 32157 rexnegd 43765 xnegrecl 44083

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