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Mirrors > Home > ILE Home > Th. List > ixi | 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 8063 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 7849 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 7882 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 7837 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 7839 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 7895 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 8177 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 145 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2186 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 (class class class)co 5836 0cc0 7744 1c1 7745 ici 7746 + caddc 7747 · cmul 7749 − cmin 8060 -cneg 8061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: inelr 8473 mulreim 8493 recextlem1 8539 cju 8847 irec 10544 i2 10545 crre 10785 remim 10788 remullem 10799 absi 10987 cosadd 11664 absefib 11697 efieq1re 11698 demoivreALT 11700 |
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