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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xnegeqi | 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 |
---|---|
xnegeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
xnegeqi | ⊢ -𝑒𝐴 = -𝑒𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | xnegeq 12283 | . 2 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -𝑒𝐴 = -𝑒𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 -𝑒cxne 12186 |
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 2354 ax-ext 2775 |
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 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-rex 3093 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-iota 6062 df-fv 6107 df-ov 6879 df-neg 10557 df-xneg 12189 |
This theorem is referenced by: supminfxr2 40430 liminfvalxr 40747 liminf0 40757 |
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