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Mirrors > Home > MPE Home > Th. List > rexneg | Structured version Visualization version GIF version |
Description: Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
rexneg | ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 11822 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | renepnf 9966 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | ifnefalse 4048 | . . . 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 9967 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
6 | ifnefalse 4048 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) |
8 | 4, 7 | eqtrd 2644 | . 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴) |
9 | 1, 8 | syl5eq 2656 | 1 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ifcif 4036 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 -cneg 10146 -𝑒cxne 11819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xneg 11822 |
This theorem is referenced by: xneg0 11917 xnegcl 11918 xnegneg 11919 xltnegi 11921 rexsub 11938 xnegid 11943 xnegdi 11950 xpncan 11953 xnpcan 11954 xmulneg1 11971 xmulm1 11983 xadddi 11997 xlt2addrd 28913 xrsmulgzz 29009 |
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