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Theorem rexneg 13147

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 13048	. 2 ⊢ -_𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
2		renepnf 11198	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3		ifnefalse 4496	. . . 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 11199	. . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
6		ifnefalse 4496	. . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴)
8	4, 7	eqtrd 2764	. 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴)
9	1, 8	eqtrid 2776	1 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ifcif 4484 ℝcr 11043 +∞cpnf 11181 -∞cmnf 11182 -cneg 11382 -_𝑒cxne 13045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-resscn 11101

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xneg 13048

This theorem is referenced by: xneg0 13148 xnegcl 13149 xnegneg 13150 xltnegi 13152 rexsub 13169 xnegid 13174 xnegdi 13184 xpncan 13187 xnpcan 13188 xmulneg1 13205 xmulm1 13217 xadddi 13231 xlt2addrd 32655 xrsmulgzz 32920 rexnegd 45110 xnegrecl 45407

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