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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 11299 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) ≠ 0) | |
6 | 1, 2, 3, 4, 5 | syl22anc 837 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℂcc 10524 0cc0 10526 / cdiv 11286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 |
This theorem is referenced by: ntrivcvgtail 15248 tanval3 15479 lcmgcdlem 15940 pcdiv 16179 pcqdiv 16184 sylow1lem1 18715 fincygsubgodd 19227 i1fmulc 24307 itg1mulc 24308 dvcnvlem 24579 plydivlem4 24892 tanarg 25210 logcnlem4 25236 angcld 25391 angrteqvd 25392 cosangneg2d 25393 angrtmuld 25394 ang180lem1 25395 ang180lem2 25396 ang180lem3 25397 ang180lem4 25398 ang180lem5 25399 lawcoslem1 25401 lawcos 25402 isosctrlem2 25405 isosctrlem3 25406 angpieqvdlem2 25415 mcubic 25433 cubic2 25434 cubic 25435 quartlem4 25446 tanatan 25505 dmgmdivn0 25613 lgamgulmlem2 25615 gamcvg2lem 25644 qqhval2lem 31332 iprodgam 33087 recbothd 39280 pellexlem6 39775 bccm1k 41046 ioodvbdlimc1lem2 42574 ioodvbdlimc2lem 42576 wallispilem4 42710 stirlinglem1 42716 stirlinglem3 42718 stirlinglem4 42719 stirlinglem7 42722 stirlinglem13 42728 stirlinglem14 42729 stirlinglem15 42730 |
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