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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 11293 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 / cdiv 11285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 |
This theorem is referenced by: recgt0 11474 expmulz 13463 rlimdiv 14990 rlimno1 14998 isumdivc 15107 fsumdivc 15129 geolim 15214 georeclim 15216 clim2div 15233 prodfdiv 15240 dvmptdivc 24489 dvmptdiv 24498 dvexp3 24502 logtayl 25170 dvcncxp1 25251 cxpeq 25265 logbrec 25287 ang180lem1 25314 ang180lem2 25315 ang180lem3 25316 isosctrlem2 25324 dvatan 25440 efrlim 25474 amgm 25495 lgamgulmlem2 25534 lgamgulmlem3 25535 igamf 25555 igamcl 25556 lgam1 25568 dchrinvcl 25756 dchrabs 25763 2lgslem3c 25901 dchrmusumlem 26025 vmalogdivsum2 26041 pntrlog2bndlem2 26081 pntrlog2bndlem6 26086 nmlno0lem 28497 nmlnop0iALT 29699 branmfn 29809 leopmul 29838 logdivsqrle 31820 dvtan 34823 dvasin 34859 areacirclem1 34863 areacirclem4 34866 exp11d 39067 pell14qrdich 39344 mpaaeu 39628 areaquad 39701 hashnzfzclim 40531 binomcxplemnotnn0 40565 oddfl 41419 climrec 41760 climdivf 41769 reclimc 41810 divlimc 41813 ioodvbdlimc1lem2 42093 ioodvbdlimc2lem 42095 stoweidlem7 42169 stoweidlem37 42199 wallispilem4 42230 wallispi 42232 wallispi2lem1 42233 stirlinglem1 42236 stirlinglem3 42238 stirlinglem4 42239 stirlinglem5 42240 stirlinglem7 42242 stirlinglem10 42245 stirlinglem11 42246 stirlinglem12 42247 stirlinglem15 42250 dirkertrigeq 42263 fourierdlem30 42299 fourierdlem83 42351 fourierdlem95 42363 eenglngeehlnmlem1 44652 eenglngeehlnmlem2 44653 seccl 44777 csccl 44778 young2d 44834 |
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