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Mirrors > Home > ILE Home > Th. List > rexneg | 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 9699 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | renepnf 7937 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | ifnefalse 3526 | . . . 4 ⊢ (𝐴 ≠ +∞ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴)) | |
4 | 2, 3 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐴 = -∞, +∞, -𝐴)) |
5 | renemnf 7938 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
6 | ifnefalse 3526 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) |
8 | 4, 7 | eqtrd 2197 | . 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴) |
9 | 1, 8 | syl5eq 2209 | 1 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ifcif 3515 ℝcr 7743 +∞cpnf 7921 -∞cmnf 7922 -cneg 8061 -𝑒cxne 9696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-pnf 7926 df-mnf 7927 df-xneg 9699 |
This theorem is referenced by: xneg0 9758 xnegcl 9759 xnegneg 9760 xltnegi 9762 rexsub 9780 xnegid 9786 xnegdi 9795 xpncan 9798 xnpcan 9799 xposdif 9809 xrmaxaddlem 11187 xrminrecl 11200 xrminrpcl 11201 |
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