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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 9446 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | renepnf 7731 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | ifnefalse 3449 | . . . 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 7732 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
6 | ifnefalse 3449 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) |
8 | 4, 7 | eqtrd 2145 | . 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴) |
9 | 1, 8 | syl5eq 2157 | 1 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 ≠ wne 2280 ifcif 3438 ℝcr 7540 +∞cpnf 7715 -∞cmnf 7716 -cneg 7851 -𝑒cxne 9443 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-uni 3701 df-pnf 7720 df-mnf 7721 df-xneg 9446 |
This theorem is referenced by: xneg0 9501 xnegcl 9502 xnegneg 9503 xltnegi 9505 rexsub 9523 xnegid 9529 xnegdi 9538 xpncan 9541 xnpcan 9542 xposdif 9552 xrmaxaddlem 10915 xrminrecl 10928 xrminrpcl 10929 |
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