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Mirrors > Home > MPE Home > Th. List > xmulm1 | Structured version Visualization version GIF version |
Description: Extended real version of mulm1 11654. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulm1 | โข (๐ด โ โ* โ (-1 ยทe ๐ด) = -๐๐ด) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 11213 | . . . . 5 โข 1 โ โ | |
2 | rexneg 13191 | . . . . 5 โข (1 โ โ โ -๐1 = -1) | |
3 | 1, 2 | ax-mp 5 | . . . 4 โข -๐1 = -1 |
4 | 3 | oveq1i 7412 | . . 3 โข (-๐1 ยทe ๐ด) = (-1 ยทe ๐ด) |
5 | 1xr 11272 | . . . 4 โข 1 โ โ* | |
6 | xmulneg1 13249 | . . . 4 โข ((1 โ โ* โง ๐ด โ โ*) โ (-๐1 ยทe ๐ด) = -๐(1 ยทe ๐ด)) | |
7 | 5, 6 | mpan 687 | . . 3 โข (๐ด โ โ* โ (-๐1 ยทe ๐ด) = -๐(1 ยทe ๐ด)) |
8 | 4, 7 | eqtr3id 2778 | . 2 โข (๐ด โ โ* โ (-1 ยทe ๐ด) = -๐(1 ยทe ๐ด)) |
9 | xmullid 13260 | . . 3 โข (๐ด โ โ* โ (1 ยทe ๐ด) = ๐ด) | |
10 | xnegeq 13187 | . . 3 โข ((1 ยทe ๐ด) = ๐ด โ -๐(1 ยทe ๐ด) = -๐๐ด) | |
11 | 9, 10 | syl 17 | . 2 โข (๐ด โ โ* โ -๐(1 ยทe ๐ด) = -๐๐ด) |
12 | 8, 11 | eqtrd 2764 | 1 โข (๐ด โ โ* โ (-1 ยทe ๐ด) = -๐๐ด) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 (class class class)co 7402 โcr 11106 1c1 11108 โ*cxr 11246 -cneg 11444 -๐cxne 13090 ยทe cxmu 13092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-xneg 13093 df-xmul 13095 |
This theorem is referenced by: (None) |
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