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Mirrors > Home > MPE Home > Th. List > xnegeq | Structured version Visualization version GIF version |
Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xnegeq | ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2739 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞)) | |
2 | eqeq1 2739 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞)) | |
3 | negeq 11498 | . . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
4 | 2, 3 | ifbieq2d 4557 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵)) |
5 | 1, 4 | ifbieq2d 4557 | . 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))) |
6 | df-xneg 13152 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
7 | df-xneg 13152 | . 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ifcif 4531 +∞cpnf 11290 -∞cmnf 11291 -cneg 11491 -𝑒cxne 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-neg 11493 df-xneg 13152 |
This theorem is referenced by: xnegcl 13252 xnegneg 13253 xneg11 13254 xltnegi 13255 xnegid 13277 xnegdi 13287 xsubge0 13300 xlesubadd 13302 xmulneg1 13308 xmulneg2 13309 xmulmnf1 13315 xmulm1 13320 xrsdsval 21446 xrsdsreclblem 21448 xblss2ps 24427 xblss2 24428 xrhmeo 24991 xaddeq0 32764 xrsmulgzz 32994 xrge0npcan 33008 carsgclctunlem2 34301 xnegeqd 45387 xnegeqi 45390 supminfxr2 45419 supminfxrrnmpt 45421 liminflbuz2 45771 |
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