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Mirrors > Home > MPE Home > Th. List > divcli | Structured version Visualization version GIF version |
Description: Closure law for division. (Contributed by NM, 2-Feb-1995.) (Revised by Mario Carneiro, 17-Feb-2014.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divcl.3 | ⊢ 𝐵 ≠ 0 |
Ref | Expression |
---|---|
divcli | ⊢ (𝐴 / 𝐵) ∈ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcl.3 | . 2 ⊢ 𝐵 ≠ 0 | |
2 | divclz.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | 2, 3 | divclzi 12025 | . 2 ⊢ (𝐵 ≠ 0 → (𝐴 / 𝐵) ∈ ℂ) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ (𝐴 / 𝐵) ∈ ℂ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2103 ≠ wne 2942 (class class class)co 7445 ℂcc 11178 0cc0 11180 / cdiv 11943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 |
This theorem is referenced by: divcan1i 12034 halfpm6th 12510 sqdivi 14230 bpoly3 16100 bpoly4 16101 cos1bnd 16229 cospi 26524 sincosq1eq 26564 sincos6thpi 26567 sincos3rdpi 26568 cxpsqrt 26754 1cubr 26894 quart1cl 26906 quart1lem 26907 quart1 26908 dvatan 26987 log2cnv 26996 log2tlbnd 26997 bclbnd 27333 bposlem8 27344 bposlem9 27345 dp20h 32835 dpmul10 32851 dpmul100 32853 dp3mul10 32854 dpexpp1 32864 dpadd2 32866 quad3 35630 areacirc 37621 cxpi11d 42268 areaquad 43117 lhe4.4ex1a 44238 stoweidlem13 45868 stoweidlem26 45881 wallispilem4 45923 wallispi 45925 dirkerper 45951 fourierdlem103 46064 fourierswlem 46085 fouriersw 46086 |
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