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Mirrors > Home > ILE Home > Th. List > xnegeq | 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 2171 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞)) | |
2 | eqeq1 2171 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞)) | |
3 | negeq 8082 | . . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
4 | 2, 3 | ifbieq2d 3539 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵)) |
5 | 1, 4 | ifbieq2d 3539 | . 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))) |
6 | df-xneg 9699 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
7 | df-xneg 9699 | . 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)) | |
8 | 5, 6, 7 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ifcif 3515 +∞cpnf 7921 -∞cmnf 7922 -cneg 8061 -𝑒cxne 9696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-if 3516 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-neg 8063 df-xneg 9699 |
This theorem is referenced by: xnegcl 9759 xnegneg 9760 xneg11 9761 xltnegi 9762 xnegid 9786 xnegdi 9795 xsubge0 9808 xposdif 9809 xlesubadd 9810 xrnegiso 11189 infxrnegsupex 11190 xrminmax 11192 xrminrecl 11200 xrminadd 11202 xblss2ps 12951 xblss2 12952 |
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