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Mirrors > Home > MPE Home > Th. List > divne0d | Structured version Visualization version GIF version |
Description: The ratio of nonzero numbers is nonzero. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divne0d.3 | ⊢ (𝜑 → 𝐴 ≠ 0) |
divne0d.4 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
divne0d | ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divne0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | divne0d.4 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
5 | divne0 11932 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ≠ 0) | |
6 | 1, 2, 3, 4, 5 | syl22anc 839 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 0cc0 11153 / cdiv 11918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 |
This theorem is referenced by: ntrivcvgtail 15933 tanval3 16167 lcmgcdlem 16640 pcdiv 16886 pcqdiv 16891 sylow1lem1 19631 fincygsubgodd 20147 i1fmulc 25753 itg1mulc 25754 dvcnvlem 26029 plydivlem4 26353 tanarg 26676 logcnlem4 26702 angcld 26863 angrteqvd 26864 cosangneg2d 26865 angrtmuld 26866 ang180lem1 26867 ang180lem2 26868 ang180lem3 26869 ang180lem4 26870 ang180lem5 26871 lawcoslem1 26873 lawcos 26874 isosctrlem2 26877 isosctrlem3 26878 angpieqvdlem2 26887 mcubic 26905 cubic2 26906 cubic 26907 quartlem4 26918 tanatan 26977 dmgmdivn0 27086 lgamgulmlem2 27088 gamcvg2lem 27117 nrt2irr 30502 constrrtlc1 33738 qqhval2lem 33944 iprodgam 35722 recbothd 41974 aks4d1p1p7 42056 aks6d1c2p2 42101 unitscyglem2 42178 pellexlem6 42822 bccm1k 44338 ioodvbdlimc1lem2 45888 ioodvbdlimc2lem 45890 wallispilem4 46024 stirlinglem1 46030 stirlinglem3 46032 stirlinglem4 46033 stirlinglem7 46036 stirlinglem13 46042 stirlinglem14 46043 stirlinglem15 46044 |
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