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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 11907 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0 / 𝐴) = 0) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (0 / 𝐴) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ∈ wcel 2105 ≠ wne 2939 (class class class)co 7412 ℂcc 11112 0cc0 11114 / cdiv 11876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 |
This theorem is referenced by: zeo 12653 arisum 15811 arisum2 15812 tan0 16099 nn0o 16331 xrhmeo 24692 pcoass 24772 dcubic 26588 atantayl2 26680 lgsquad2lem2 27125 2lgsoddprmlem3a 27150 dip0r 30238 lnopeq0i 31528 sinccvglem 34956 ftc1anclem6 36870 jm2.23 42038 sqrtcval 42695 fourierdlem62 45183 fourierdlem103 45224 fourierdlem104 45225 sqwvfoura 45243 sqwvfourb 45244 fourierswlem 45245 fouriersw 45246 0evenALTV 46655 |
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