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Mirrors > Home > MPE Home > Th. List > divassi | Structured version Visualization version GIF version |
Description: An associative law for division. (Contributed by NM, 15-Feb-1995.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
divclz.2 | ⊢ 𝐵 ∈ ℂ |
divmulz.3 | ⊢ 𝐶 ∈ ℂ |
divass.4 | ⊢ 𝐶 ≠ 0 |
Ref | Expression |
---|---|
divassi | ⊢ ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divass.4 | . 2 ⊢ 𝐶 ≠ 0 | |
2 | divclz.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
3 | divclz.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | divmulz.3 | . . 3 ⊢ 𝐶 ∈ ℂ | |
5 | 2, 3, 4 | divasszi 10965 | . 2 ⊢ (𝐶 ≠ 0 → ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶))) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ ((𝐴 · 𝐵) / 𝐶) = (𝐴 · (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1630 ∈ wcel 2137 ≠ wne 2930 (class class class)co 6811 ℂcc 10124 0cc0 10126 · cmul 10131 / cdiv 10874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-8 2139 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pow 4990 ax-pr 5053 ax-un 7112 ax-resscn 10183 ax-1cn 10184 ax-icn 10185 ax-addcl 10186 ax-addrcl 10187 ax-mulcl 10188 ax-mulrcl 10189 ax-mulcom 10190 ax-addass 10191 ax-mulass 10192 ax-distr 10193 ax-i2m1 10194 ax-1ne0 10195 ax-1rid 10196 ax-rnegex 10197 ax-rrecex 10198 ax-cnre 10199 ax-pre-lttri 10200 ax-pre-lttrn 10201 ax-pre-ltadd 10202 ax-pre-mulgt0 10203 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-nel 3034 df-ral 3053 df-rex 3054 df-reu 3055 df-rmo 3056 df-rab 3057 df-v 3340 df-sbc 3575 df-csb 3673 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-pw 4302 df-sn 4320 df-pr 4322 df-op 4326 df-uni 4587 df-br 4803 df-opab 4863 df-mpt 4880 df-id 5172 df-po 5185 df-so 5186 df-xp 5270 df-rel 5271 df-cnv 5272 df-co 5273 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 df-iota 6010 df-fun 6049 df-fn 6050 df-f 6051 df-f1 6052 df-fo 6053 df-f1o 6054 df-fv 6055 df-riota 6772 df-ov 6814 df-oprab 6815 df-mpt2 6816 df-er 7909 df-en 8120 df-dom 8121 df-sdom 8122 df-pnf 10266 df-mnf 10267 df-xr 10268 df-ltxr 10269 df-le 10270 df-sub 10458 df-neg 10459 df-div 10875 |
This theorem is referenced by: cos2bnd 15115 6lcm4e12 15529 sincos6thpi 24464 cxpsqrt 24646 1cubrlem 24765 efiatan 24836 log2cnv 24868 log2ublem1 24870 birthday 24878 bclbnd 25202 bposlem8 25213 ex-lcm 27624 dpmul4 29929 ballotth 30906 hgt750lem 31036 quad3 31869 areaquad 38302 41prothprmlem1 42042 |
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