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Mathbox for Glauco Siliprandi |
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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 12286 | . 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 1653 -𝑒cxne 12189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-rex 3096 df-rab 3099 df-v 3388 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-iota 6065 df-fv 6110 df-ov 6882 df-neg 10560 df-xneg 12192 |
This theorem is referenced by: supminfxr 40432 supminfxr2 40437 supminfxrrnmpt 40439 monoord2xrv 40452 liminfvalxr 40754 liminfvalxrmpt 40757 liminfval4 40760 liminfval3 40761 limsupval4 40765 liminfvaluz2 40766 limsupvaluz4 40771 climliminflimsupd 40772 smfliminflem 41777 |
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