Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11312 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ≠ 0) | |
6 | 1, 2, 3, 4, 5 | syl22anc 836 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3018 (class class class)co 7158 ℂcc 10537 0cc0 10539 / cdiv 11299 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 |
This theorem is referenced by: ntrivcvgtail 15258 tanval3 15489 lcmgcdlem 15952 pcdiv 16191 pcqdiv 16196 sylow1lem1 18725 fincygsubgodd 19236 i1fmulc 24306 itg1mulc 24307 dvcnvlem 24575 plydivlem4 24887 tanarg 25204 logcnlem4 25230 angcld 25385 angrteqvd 25386 cosangneg2d 25387 angrtmuld 25388 ang180lem1 25389 ang180lem2 25390 ang180lem3 25391 ang180lem4 25392 ang180lem5 25393 lawcoslem1 25395 lawcos 25396 isosctrlem2 25399 isosctrlem3 25400 angpieqvdlem2 25409 mcubic 25427 cubic2 25428 cubic 25429 quartlem4 25440 tanatan 25499 dmgmdivn0 25607 lgamgulmlem2 25609 gamcvg2lem 25638 qqhval2lem 31224 iprodgam 32976 pellexlem6 39438 bccm1k 40681 ioodvbdlimc1lem2 42224 ioodvbdlimc2lem 42226 wallispilem4 42360 stirlinglem1 42366 stirlinglem3 42368 stirlinglem4 42369 stirlinglem7 42372 stirlinglem13 42378 stirlinglem14 42379 stirlinglem15 42380 |
Copyright terms: Public domain | W3C validator |