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 12603 | . 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 1536 -𝑒cxne 12507 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-neg 10876 df-xneg 12510 |
This theorem is referenced by: supminfxr 41746 supminfxr2 41751 supminfxrrnmpt 41753 monoord2xrv 41766 liminfvalxr 42070 liminfvalxrmpt 42073 liminfval4 42076 liminfval3 42077 limsupval4 42081 liminfvaluz2 42082 limsupvaluz4 42087 climliminflimsupd 42088 xlimpnfxnegmnf 42101 liminfpnfuz 42103 xlimpnfxnegmnf2 42145 smfliminflem 43111 |
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