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Mirrors > Home > MPE Home > Th. List > divcl | Structured version Visualization version GIF version |
Description: Closure law for division. (Contributed by NM, 21-Jul-2001.) (Proof shortened by Mario Carneiro, 17-Feb-2014.) |
Ref | Expression |
---|---|
divcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divval 10889 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) | |
2 | receu 10874 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) | |
3 | riotacl 6767 | . . 3 ⊢ (∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴 → (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) |
5 | 1, 4 | eqeltrd 2850 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃!wreu 3063 ℩crio 6752 (class class class)co 6792 ℂcc 10136 0cc0 10138 · cmul 10143 / cdiv 10886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 |
This theorem is referenced by: reccl 10894 divcan2 10895 divcan1 10896 div23 10906 div12 10909 divmulasscom 10911 div11 10915 divsubdir 10923 divmuldiv 10927 divdivdiv 10928 divcan5 10929 divmuleq 10932 divcan6 10934 divdiv32 10935 dmdcan 10937 ddcan 10941 divsubdiv 10943 div2neg 10950 divclzi 10962 divcld 11003 nndivtr 11264 halfcl 11460 sqdiv 13131 cjdiv 14108 absdiv 14239 sinf 15056 efi4p 15069 dvrec 23934 efeq1 24492 efif1olem4 24508 axcontlem4 26064 dipcl 27903 spansncol 28763 subfaclim 31504 sinccvglem 31900 nndivsub 32789 ftc1anclem6 33818 lhe4.4ex1a 39051 |
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