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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 12508 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
2 | renepnf 10689 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
3 | ifnefalse 4479 | . . . 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 10690 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
6 | ifnefalse 4479 | . . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ ℝ → if(𝐴 = -∞, +∞, -𝐴) = -𝐴) |
8 | 4, 7 | eqtrd 2856 | . 2 ⊢ (𝐴 ∈ ℝ → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = -𝐴) |
9 | 1, 8 | syl5eq 2868 | 1 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ifcif 4467 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 -cneg 10871 -𝑒cxne 12505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xneg 12508 |
This theorem is referenced by: xneg0 12606 xnegcl 12607 xnegneg 12608 xltnegi 12610 rexsub 12627 xnegid 12632 xnegdi 12642 xpncan 12645 xnpcan 12646 xmulneg1 12663 xmulm1 12675 xadddi 12689 xlt2addrd 30482 xrsmulgzz 30665 rexnegd 41432 xnegrecl 41732 |
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