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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 2736 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞)) | |
2 | eqeq1 2736 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞)) | |
3 | negeq 11456 | . . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
4 | 2, 3 | ifbieq2d 4554 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵)) |
5 | 1, 4 | ifbieq2d 4554 | . 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))) |
6 | df-xneg 13096 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
7 | df-xneg 13096 | . 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)) | |
8 | 5, 6, 7 | 3eqtr4g 2797 | 1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ifcif 4528 +∞cpnf 11249 -∞cmnf 11250 -cneg 11449 -𝑒cxne 13093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7414 df-neg 11451 df-xneg 13096 |
This theorem is referenced by: xnegcl 13196 xnegneg 13197 xneg11 13198 xltnegi 13199 xnegid 13221 xnegdi 13231 xsubge0 13244 xlesubadd 13246 xmulneg1 13252 xmulneg2 13253 xmulmnf1 13259 xmulm1 13264 xrsdsval 21189 xrsdsreclblem 21191 xblss2ps 24127 xblss2 24128 xrhmeo 24686 xaddeq0 32221 xrsmulgzz 32434 xrge0npcan 32450 carsgclctunlem2 33604 xnegeqd 44446 xnegeqi 44449 supminfxr2 44478 supminfxrrnmpt 44480 liminflbuz2 44830 |
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