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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 234 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2822 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 (class class class)co 7135 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 |
This theorem is referenced by: recextlem1 11259 inelr 11615 cju 11621 irec 13560 i2 13561 crre 14465 remim 14468 remullem 14479 sqrtneglem 14618 absi 14638 sinhval 15499 coshval 15500 cosadd 15510 absefib 15543 efieq1re 15544 demoivreALT 15546 ncvspi 23761 cphipval2 23845 itgmulc2 24437 tanarg 25210 atandm2 25463 efiasin 25474 asinsinlem 25477 asinsin 25478 asin1 25480 efiatan 25498 atanlogsublem 25501 efiatan2 25503 2efiatan 25504 tanatan 25505 atantan 25509 atans2 25517 dvatan 25521 log2cnv 25530 nvpi 28450 ipasslem10 28622 polid2i 28940 lnophmlem2 29800 1nei 30498 iexpire 33080 itgmulc2nc 35125 dvasin 35141 sqrtcval 40341 |
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