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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 10862 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 10594 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 10622 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 10585 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 10637 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 10963 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 232 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2845 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 (class class class)co 7145 0cc0 10526 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 − cmin 10859 -cneg 10860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 |
This theorem is referenced by: recextlem1 11259 inelr 11617 cju 11623 irec 13554 i2 13555 crre 14463 remim 14466 remullem 14477 sqrtneglem 14616 absi 14636 sinhval 15497 coshval 15498 cosadd 15508 absefib 15541 efieq1re 15542 demoivreALT 15544 ncvspi 23689 cphipval2 23773 itgmulc2 24363 tanarg 25129 atandm2 25382 efiasin 25393 asinsinlem 25396 asinsin 25397 asin1 25399 efiatan 25417 atanlogsublem 25420 efiatan2 25422 2efiatan 25423 tanatan 25424 atantan 25428 atans2 25436 dvatan 25440 log2cnv 25450 nvpi 28372 ipasslem10 28544 polid2i 28862 lnophmlem2 29722 1nei 30399 iexpire 32865 itgmulc2nc 34842 dvasin 34860 |
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