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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rexnegd | Structured version Visualization version GIF version |
Description: Minus a real number. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
rexnegd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
rexnegd | ⊢ (𝜑 → -𝑒𝐴 = -𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexnegd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | rexneg 13249 | . 2 ⊢ (𝐴 ∈ ℝ → -𝑒𝐴 = -𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → -𝑒𝐴 = -𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ℝcr 11150 -cneg 11489 -𝑒cxne 13147 |
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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-resscn 11208 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xneg 13150 |
This theorem is referenced by: supminfxr 45448 supminfxr2 45453 liminfval4 45777 liminfvaluz2 45783 limsupvaluz4 45788 climliminflimsupd 45789 liminfltlem 45792 smfliminflem 46818 |
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