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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 2741 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞)) | |
| 2 | eqeq1 2741 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞)) | |
| 3 | negeq 11500 | . . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
| 4 | 2, 3 | ifbieq2d 4552 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵)) |
| 5 | 1, 4 | ifbieq2d 4552 | . 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))) |
| 6 | df-xneg 13154 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
| 7 | df-xneg 13154 | . 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)) | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ifcif 4525 +∞cpnf 11292 -∞cmnf 11293 -cneg 11493 -𝑒cxne 13151 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-neg 11495 df-xneg 13154 |
| This theorem is referenced by: xnegcl 13255 xnegneg 13256 xneg11 13257 xltnegi 13258 xnegid 13280 xnegdi 13290 xsubge0 13303 xlesubadd 13305 xmulneg1 13311 xmulneg2 13312 xmulmnf1 13318 xmulm1 13323 xrsdsval 21428 xrsdsreclblem 21430 xblss2ps 24411 xblss2 24412 xrhmeo 24977 xaddeq0 32757 xrsmulgzz 33011 xrge0npcan 33025 carsgclctunlem2 34321 xnegeqd 45448 xnegeqi 45451 supminfxr2 45480 supminfxrrnmpt 45482 liminflbuz2 45830 |
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