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| Mirrors > Home > MPE Home > Th. List > ixi | Structured version Visualization version GIF version | ||
| Description: i times itself is minus 1. (Contributed by NM, 6-May-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
| Ref | Expression |
|---|---|
| ixi | ⊢ (i · i) = -1 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-neg 11495 | . 2 ⊢ -1 = (0 − 1) | |
| 2 | ax-i2m1 11223 | . . 3 ⊢ ((i · i) + 1) = 0 | |
| 3 | 0cn 11253 | . . . 4 ⊢ 0 ∈ ℂ | |
| 4 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 5 | ax-icn 11214 | . . . . 5 ⊢ i ∈ ℂ | |
| 6 | 5, 5 | mulcli 11268 | . . . 4 ⊢ (i · i) ∈ ℂ |
| 7 | 3, 4, 6 | subadd2i 11597 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
| 8 | 2, 7 | mpbir 231 | . 2 ⊢ (0 − 1) = (i · i) |
| 9 | 1, 8 | eqtr2i 2766 | 1 ⊢ (i · i) = -1 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 ici 11157 + caddc 11158 · cmul 11160 − cmin 11492 -cneg 11493 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: recextlem1 11893 inelr 12256 cju 12262 irec 14240 i2 14241 crre 15153 remim 15156 remullem 15167 sqrtneglem 15305 absi 15325 sinhval 16190 coshval 16191 cosadd 16201 absefib 16234 efieq1re 16235 demoivreALT 16237 ncvspi 25190 cphipval2 25275 itgmulc2 25869 tanarg 26661 atandm2 26920 efiasin 26931 asinsinlem 26934 asinsin 26935 asin1 26937 efiatan 26955 atanlogsublem 26958 efiatan2 26960 2efiatan 26961 tanatan 26962 atantan 26966 atans2 26974 dvatan 26978 log2cnv 26987 nvpi 30686 ipasslem10 30858 polid2i 31176 lnophmlem2 32036 1nei 32747 iexpire 35735 itgmulc2nc 37695 dvasin 37711 sqrtcval 43654 |
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