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Mirrors > Home > MPE Home > Th. List > div12 | Structured version Visualization version GIF version |
Description: A commutative/associative law for division. (Contributed by NM, 30-Apr-2005.) |
Ref | Expression |
---|---|
div12 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcl 11929 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐵 / 𝐶) ∈ ℂ) | |
2 | 1 | 3expb 1117 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐵 / 𝐶) ∈ ℂ) |
3 | mulcom 11244 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 / 𝐶) ∈ ℂ) → (𝐴 · (𝐵 / 𝐶)) = ((𝐵 / 𝐶) · 𝐴)) | |
4 | 2, 3 | sylan2 591 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0))) → (𝐴 · (𝐵 / 𝐶)) = ((𝐵 / 𝐶) · 𝐴)) |
5 | 4 | 3impb 1112 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 · (𝐵 / 𝐶)) = ((𝐵 / 𝐶) · 𝐴)) |
6 | div13 11944 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ 𝐴 ∈ ℂ) → ((𝐵 / 𝐶) · 𝐴) = ((𝐴 / 𝐶) · 𝐵)) | |
7 | 6 | 3comr 1122 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐵 / 𝐶) · 𝐴) = ((𝐴 / 𝐶) · 𝐵)) |
8 | divcl 11929 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐴 / 𝐶) ∈ ℂ) | |
9 | 8 | 3expb 1117 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / 𝐶) ∈ ℂ) |
10 | mulcom 11244 | . . . 4 ⊢ (((𝐴 / 𝐶) ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 / 𝐶) · 𝐵) = (𝐵 · (𝐴 / 𝐶))) | |
11 | 9, 10 | stoic3 1771 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) ∧ 𝐵 ∈ ℂ) → ((𝐴 / 𝐶) · 𝐵) = (𝐵 · (𝐴 / 𝐶))) |
12 | 11 | 3com23 1123 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) · 𝐵) = (𝐵 · (𝐴 / 𝐶))) |
13 | 5, 7, 12 | 3eqtrd 2770 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 · (𝐵 / 𝐶)) = (𝐵 · (𝐴 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 (class class class)co 7424 ℂcc 11156 0cc0 11158 · cmul 11163 / cdiv 11921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 |
This theorem is referenced by: div2neg 11988 div12d 12077 bpoly3 16060 efival 16154 cos01bnd 16188 cos01gt0 16193 sincosq4sgn 26529 bclbnd 27309 bposlem9 27321 dchrvmasum2lem 27525 dchrvmasumiflem1 27530 selbergr 27597 pntpbnd1a 27614 pntibndlem2 27620 dignn0flhalflem1 48003 |
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