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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 11175 | . . . 4 ⊢ i ∈ ℂ | |
2 | 1, 1 | mulneg2i 11668 | . . 3 ⊢ (i · -i) = -(i · i) |
3 | ixi 11850 | . . . 4 ⊢ (i · i) = -1 | |
4 | ax-1cn 11174 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 1, 1 | mulcli 11228 | . . . . 5 ⊢ (i · i) ∈ ℂ |
6 | 4, 5 | negcon2i 11550 | . . . 4 ⊢ (1 = -(i · i) ↔ (i · i) = -1) |
7 | 3, 6 | mpbir 230 | . . 3 ⊢ 1 = -(i · i) |
8 | 2, 7 | eqtr4i 2762 | . 2 ⊢ (i · -i) = 1 |
9 | negicn 11468 | . . 3 ⊢ -i ∈ ℂ | |
10 | ine0 11656 | . . 3 ⊢ i ≠ 0 | |
11 | 4, 1, 9, 10 | divmuli 11975 | . 2 ⊢ ((1 / i) = -i ↔ (i · -i) = 1) |
12 | 8, 11 | mpbir 230 | 1 ⊢ (1 / i) = -i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 (class class class)co 7412 1c1 11117 ici 11118 · cmul 11121 -cneg 11452 / cdiv 11878 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 |
This theorem is referenced by: imre 15062 crim 15069 cnpart 15194 sinhval 16104 dvsincos 25746 dvatan 26691 atantayl2 26694 sinh-conventional 47884 |
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