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Mirrors > Home > MPE Home > Th. List > irec | Structured version Visualization version GIF version |
Description: The reciprocal of i. (Contributed by NM, 11-Oct-1999.) |
Ref | Expression |
---|---|
irec | ⊢ (1 / i) = -i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 9995 | . . . 4 ⊢ i ∈ ℂ | |
2 | 1, 1 | mulneg2i 10477 | . . 3 ⊢ (i · -i) = -(i · i) |
3 | ixi 10656 | . . . 4 ⊢ (i · i) = -1 | |
4 | ax-1cn 9994 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 1, 1 | mulcli 10045 | . . . . 5 ⊢ (i · i) ∈ ℂ |
6 | 4, 5 | negcon2i 10364 | . . . 4 ⊢ (1 = -(i · i) ↔ (i · i) = -1) |
7 | 3, 6 | mpbir 221 | . . 3 ⊢ 1 = -(i · i) |
8 | 2, 7 | eqtr4i 2647 | . 2 ⊢ (i · -i) = 1 |
9 | negicn 10282 | . . 3 ⊢ -i ∈ ℂ | |
10 | ine0 10465 | . . 3 ⊢ i ≠ 0 | |
11 | 4, 1, 9, 10 | divmuli 10779 | . 2 ⊢ ((1 / i) = -i ↔ (i · -i) = 1) |
12 | 8, 11 | mpbir 221 | 1 ⊢ (1 / i) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 (class class class)co 6650 1c1 9937 ici 9938 · cmul 9941 -cneg 10267 / cdiv 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 |
This theorem is referenced by: imre 13848 crim 13855 cnpart 13980 sinhval 14884 dvsincos 23744 dvatan 24662 atantayl2 24665 sinh-conventional 42480 |
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