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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 11497 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 11226 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 11256 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 11216 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 11217 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 11271 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 11598 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 230 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2755 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7424 0cc0 11158 1c1 11159 ici 11160 + caddc 11161 · cmul 11163 − cmin 11494 -cneg 11495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-neg 11497 |
This theorem is referenced by: recextlem1 11894 inelr 12254 cju 12260 irec 14219 i2 14220 crre 15119 remim 15122 remullem 15133 sqrtneglem 15271 absi 15291 sinhval 16156 coshval 16157 cosadd 16167 absefib 16200 efieq1re 16201 demoivreALT 16203 ncvspi 25175 cphipval2 25260 itgmulc2 25854 tanarg 26646 atandm2 26905 efiasin 26916 asinsinlem 26919 asinsin 26920 asin1 26922 efiatan 26940 atanlogsublem 26943 efiatan2 26945 2efiatan 26946 tanatan 26947 atantan 26951 atans2 26959 dvatan 26963 log2cnv 26972 nvpi 30600 ipasslem10 30772 polid2i 31090 lnophmlem2 31950 1nei 32650 iexpire 35557 itgmulc2nc 37389 dvasin 37405 sqrtcval 43308 |
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