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Mirrors > Home > MPE Home > Th. List > reccl | Structured version Visualization version GIF version |
Description: Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
reccl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11174 | . 2 ⊢ 1 ∈ ℂ | |
2 | divcl 11885 | . 2 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
3 | 1, 2 | mp3an1 1447 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ≠ wne 2939 (class class class)co 7412 ℂcc 11114 0cc0 11116 1c1 11117 / 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: divrec 11895 divrec2 11896 divass 11897 divdir 11904 divneg 11913 recrec 11918 rec11 11919 divdiv32 11929 conjmul 11938 recclzi 11946 reccld 11990 expclzlem 14056 exprec 14076 expdiv 14086 rlimdiv 15599 geoisumr 15831 cndrng 21178 divcnOLD 24617 divccn 24624 divccnOLD 24626 divccncf 24659 dvexp3 25743 quotlem 26063 quotcl2 26065 quotdgr 26066 aareccl 26089 logtayllem 26418 logtayl 26419 cxpeq 26516 logrec 26519 dchrisum0lem2a 27271 dchrisum0lem2 27272 mulogsum 27286 pntlemr 27356 axcontlem2 28505 nvmul0or 30185 hvmul0or 30560 h1datomi 31116 nmopleid 31674 |
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