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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 2825 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞)) | |
2 | eqeq1 2825 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞)) | |
3 | negeq 10878 | . . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵) | |
4 | 2, 3 | ifbieq2d 4492 | . . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵)) |
5 | 1, 4 | ifbieq2d 4492 | . 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))) |
6 | df-xneg 12508 | . 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) | |
7 | df-xneg 12508 | . 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)) | |
8 | 5, 6, 7 | 3eqtr4g 2881 | 1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ifcif 4467 +∞cpnf 10672 -∞cmnf 10673 -cneg 10871 -𝑒cxne 12505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-neg 10873 df-xneg 12508 |
This theorem is referenced by: xnegcl 12607 xnegneg 12608 xneg11 12609 xltnegi 12610 xnegid 12632 xnegdi 12642 xsubge0 12655 xlesubadd 12657 xmulneg1 12663 xmulneg2 12664 xmulmnf1 12670 xmulm1 12675 xrsdsval 20589 xrsdsreclblem 20591 xblss2ps 23011 xblss2 23012 xrhmeo 23550 xaddeq0 30477 xrsmulgzz 30665 xrge0npcan 30681 carsgclctunlem2 31577 xnegeqd 41731 xnegeqi 41734 supminfxr2 41765 supminfxrrnmpt 41767 liminflbuz2 42116 |
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