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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 7717 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 7511 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 7541 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 7499 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 7501 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 7554 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 7831 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 145 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2110 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 (class class class)co 5666 0cc0 7411 1c1 7412 ici 7413 + caddc 7414 · cmul 7416 − cmin 7714 -cneg 7715 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-setind 4366 ax-resscn 7498 ax-1cn 7499 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-addcom 7506 ax-addass 7508 ax-distr 7510 ax-i2m1 7511 ax-0id 7514 ax-rnegex 7515 ax-cnre 7517 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-sub 7716 df-neg 7717 |
This theorem is referenced by: inelr 8122 mulreim 8142 recextlem1 8181 cju 8482 irec 10115 i2 10116 crre 10352 remim 10355 remullem 10366 absi 10553 cosadd 11089 absefib 11121 efieq1re 11122 demoivreALT 11124 |
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