| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > xnegeqd | Structured version Visualization version GIF version | ||
| Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
| Ref | Expression |
|---|---|
| xnegeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| xnegeqd | ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xnegeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | xnegeq 13249 | . 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → -𝑒𝐴 = -𝑒𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 -𝑒cxne 13151 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-xneg 13154 |
| This theorem is referenced by: supminfxr 45475 supminfxr2 45480 supminfxrrnmpt 45482 monoord2xrv 45494 liminfvalxr 45798 liminfvalxrmpt 45801 liminfval4 45804 liminfval3 45805 limsupval4 45809 liminfvaluz2 45810 limsupvaluz4 45815 climliminflimsupd 45816 xlimpnfxnegmnf 45829 liminfpnfuz 45831 xlimpnfxnegmnf2 45873 smfliminflem 46845 |
| Copyright terms: Public domain | W3C validator |