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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 10876 | . 2 ⊢ -1 = (0 − 1) | |
2 | ax-i2m1 10608 | . . 3 ⊢ ((i · i) + 1) = 0 | |
3 | 0cn 10636 | . . . 4 ⊢ 0 ∈ ℂ | |
4 | ax-1cn 10598 | . . . 4 ⊢ 1 ∈ ℂ | |
5 | ax-icn 10599 | . . . . 5 ⊢ i ∈ ℂ | |
6 | 5, 5 | mulcli 10651 | . . . 4 ⊢ (i · i) ∈ ℂ |
7 | 3, 4, 6 | subadd2i 10977 | . . 3 ⊢ ((0 − 1) = (i · i) ↔ ((i · i) + 1) = 0) |
8 | 2, 7 | mpbir 233 | . 2 ⊢ (0 − 1) = (i · i) |
9 | 1, 8 | eqtr2i 2848 | 1 ⊢ (i · i) = -1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 (class class class)co 7159 0cc0 10540 1c1 10541 ici 10542 + caddc 10543 · cmul 10545 − cmin 10873 -cneg 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 |
This theorem is referenced by: recextlem1 11273 inelr 11631 cju 11637 irec 13567 i2 13568 crre 14476 remim 14479 remullem 14490 sqrtneglem 14629 absi 14649 sinhval 15510 coshval 15511 cosadd 15521 absefib 15554 efieq1re 15555 demoivreALT 15557 ncvspi 23763 cphipval2 23847 itgmulc2 24437 tanarg 25205 atandm2 25458 efiasin 25469 asinsinlem 25472 asinsin 25473 asin1 25475 efiatan 25493 atanlogsublem 25496 efiatan2 25498 2efiatan 25499 tanatan 25500 atantan 25504 atans2 25512 dvatan 25516 log2cnv 25525 nvpi 28447 ipasslem10 28619 polid2i 28937 lnophmlem2 29797 1nei 30475 iexpire 32971 itgmulc2nc 34964 dvasin 34982 |
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