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 12588 | . 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 1528 -𝑒cxne 12492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-neg 10861 df-xneg 12495 |
This theorem is referenced by: supminfxr 41616 supminfxr2 41621 supminfxrrnmpt 41623 monoord2xrv 41636 liminfvalxr 41940 liminfvalxrmpt 41943 liminfval4 41946 liminfval3 41947 limsupval4 41951 liminfvaluz2 41952 limsupvaluz4 41957 climliminflimsupd 41958 xlimpnfxnegmnf 41971 liminfpnfuz 41973 xlimpnfxnegmnf2 42015 smfliminflem 42981 |
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