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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 10875 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 10607 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 10635 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 10597 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 10598 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 10650 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 10976 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 233 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2847 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7158 0cc0 10539 1c1 10540 ici 10541 + caddc 10542 · cmul 10544 − cmin 10872 -cneg 10873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 |
This theorem is referenced by: recextlem1 11272 inelr 11630 cju 11636 irec 13567 i2 13568 crre 14475 remim 14478 remullem 14489 sqrtneglem 14628 absi 14648 sinhval 15509 coshval 15510 cosadd 15520 absefib 15553 efieq1re 15554 demoivreALT 15556 ncvspi 23762 cphipval2 23846 itgmulc2 24436 tanarg 25204 atandm2 25457 efiasin 25468 asinsinlem 25471 asinsin 25472 asin1 25474 efiatan 25492 atanlogsublem 25495 efiatan2 25497 2efiatan 25498 tanatan 25499 atantan 25503 atans2 25511 dvatan 25515 log2cnv 25524 nvpi 28446 ipasslem10 28618 polid2i 28936 lnophmlem2 29796 1nei 30474 iexpire 32969 itgmulc2nc 34962 dvasin 34980 |
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