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Mirrors > Home > MPE Home > Th. List > reccld | Structured version Visualization version GIF version |
Description: Closure law for reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
reccld.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
reccld | ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | reccld.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | reccl 10571 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | syl2anc 691 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ≠ wne 2780 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 / cdiv 10563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 |
This theorem is referenced by: recgt0 10746 expmulz 12768 rlimdiv 14224 rlimno1 14232 isumdivc 14337 fsumdivc 14360 geolim 14440 georeclim 14442 clim2div 14460 prodfdiv 14467 dvmptdivc 23534 dvexp3 23545 logtayl 24206 dvcncxp1 24284 cxpeq 24298 logbrec 24320 ang180lem1 24339 ang180lem2 24340 ang180lem3 24341 isosctrlem2 24349 dvatan 24462 efrlim 24496 amgm 24517 lgamgulmlem2 24556 lgamgulmlem3 24557 igamf 24577 igamcl 24578 lgam1 24590 dchrinvcl 24778 dchrabs 24785 2lgslem3c 24923 dchrmusumlem 25011 vmalogdivsum2 25027 pntrlog2bndlem2 25067 pntrlog2bndlem6 25072 nmlno0lem 27032 nmlnop0iALT 28238 branmfn 28348 leopmul 28377 dvtan 32630 dvasin 32666 areacirclem1 32670 areacirclem4 32673 pell14qrdich 36451 mpaaeu 36739 areaquad 36821 hashnzfzclim 37543 binomcxplemnotnn0 37577 oddfl 38430 climrec 38670 climdivf 38679 reclimc 38720 divlimc 38723 dvmptdiv 38807 ioodvbdlimc1lem2 38822 ioodvbdlimc2lem 38824 stoweidlem7 38900 stoweidlem37 38930 wallispilem4 38961 wallispi 38963 wallispi2lem1 38964 stirlinglem1 38967 stirlinglem3 38969 stirlinglem4 38970 stirlinglem5 38971 stirlinglem7 38973 stirlinglem10 38976 stirlinglem11 38977 stirlinglem12 38978 stirlinglem15 38981 dirkertrigeq 38994 fourierdlem30 39030 fourierdlem83 39082 fourierdlem95 39094 seccl 42290 csccl 42291 young2d 42360 |
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