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Mirrors > Home > MPE Home > Th. List > div0i | Structured version Visualization version GIF version |
Description: Division into zero is zero. (Contributed by NM, 12-Aug-1999.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
reccl.2 | ⊢ 𝐴 ≠ 0 |
Ref | Expression |
---|---|
div0i | ⊢ (0 / 𝐴) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | reccl.2 | . 2 ⊢ 𝐴 ≠ 0 | |
3 | div0 11066 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0) | |
4 | 1, 2, 3 | mp2an 682 | 1 ⊢ (0 / 𝐴) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ≠ wne 2969 (class class class)co 6924 ℂcc 10272 0cc0 10274 / cdiv 11035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 |
This theorem is referenced by: zeo 11820 arisum 15005 arisum2 15006 tan0 15292 nn0o 15523 xrhmeo 23164 pcoass 23242 dcubic 25035 atantayl2 25127 lgsquad2lem2 25573 2lgsoddprmlem3a 25598 dip0r 28161 lnopeq0i 29455 sinccvglem 32171 ftc1anclem6 34124 jm2.23 38536 fourierdlem62 41326 fourierdlem103 41367 fourierdlem104 41368 sqwvfoura 41386 sqwvfourb 41387 fourierswlem 41388 fouriersw 41389 0evenALTV 42638 |
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