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Mirrors > Home > MPE Home > Th. List > reccli | Structured version Visualization version GIF version |
Description: Closure law for reciprocal. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
reccl.2 | ⊢ 𝐴 ≠ 0 |
Ref | Expression |
---|---|
reccli | ⊢ (1 / 𝐴) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reccl.2 | . 2 ⊢ 𝐴 ≠ 0 | |
2 | divclz.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | 2 | recclzi 11879 | . 2 ⊢ (𝐴 ≠ 0 → (1 / 𝐴) ∈ ℂ) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ (1 / 𝐴) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ≠ wne 2943 (class class class)co 7356 ℂcc 11048 0cc0 11050 1c1 11051 / cdiv 11811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 |
This theorem is referenced by: halfcn 12367 halfpm6th 12373 sqrecii 14086 bpoly2 15939 bpoly3 15940 bpoly4 15941 fsumcube 15942 sinhval 16035 cos01bnd 16067 cos1bnd 16068 flodddiv4 16294 dvmptim 25332 tan4thpi 25869 sincos6thpi 25870 sincos3rdpi 25871 1cubrlem 26189 1cubr 26190 cubic 26197 bposlem8 26637 pntibndlem2 26937 ftc1anclem6 36146 |
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