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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 11297 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (1 / 𝐴) ∈ ℂ) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (1 / 𝐴) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 3014 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 / cdiv 11289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 |
This theorem is referenced by: recgt0 11478 expmulz 13467 rlimdiv 14994 rlimno1 15002 isumdivc 15111 fsumdivc 15133 geolim 15218 georeclim 15220 clim2div 15237 prodfdiv 15244 dvmptdivc 24554 dvmptdiv 24563 dvexp3 24567 logtayl 25235 dvcncxp1 25316 cxpeq 25330 logbrec 25352 ang180lem1 25379 ang180lem2 25380 ang180lem3 25381 isosctrlem2 25389 dvatan 25505 efrlim 25539 amgm 25560 lgamgulmlem2 25599 lgamgulmlem3 25600 igamf 25620 igamcl 25621 lgam1 25633 dchrinvcl 25821 dchrabs 25828 2lgslem3c 25966 dchrmusumlem 26090 vmalogdivsum2 26106 pntrlog2bndlem2 26146 pntrlog2bndlem6 26151 nmlno0lem 28562 nmlnop0iALT 29764 branmfn 29874 leopmul 29903 logdivsqrle 31914 dvtan 34934 dvasin 34970 areacirclem1 34974 areacirclem4 34977 exp11d 39180 pell14qrdich 39457 mpaaeu 39741 areaquad 39814 hashnzfzclim 40645 binomcxplemnotnn0 40679 oddfl 41533 climrec 41874 climdivf 41883 reclimc 41924 divlimc 41927 ioodvbdlimc1lem2 42207 ioodvbdlimc2lem 42209 stoweidlem7 42283 stoweidlem37 42313 wallispilem4 42344 wallispi 42346 wallispi2lem1 42347 stirlinglem1 42350 stirlinglem3 42352 stirlinglem4 42353 stirlinglem5 42354 stirlinglem7 42356 stirlinglem10 42359 stirlinglem11 42360 stirlinglem12 42361 stirlinglem15 42364 dirkertrigeq 42377 fourierdlem30 42413 fourierdlem83 42465 fourierdlem95 42477 eenglngeehlnmlem1 44715 eenglngeehlnmlem2 44716 seccl 44840 csccl 44841 young2d 44897 |
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