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Mirrors > Home > MPE Home > Th. List > div23d | Structured version Visualization version GIF version |
Description: A commutative/associative law for division. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divassd.4 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
div23d | ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divmuld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | divassd.4 | . 2 ⊢ (𝜑 → 𝐶 ≠ 0) | |
5 | div23 11317 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl112anc 1370 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 0cc0 10537 · cmul 10542 / cdiv 11297 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 |
This theorem is referenced by: bcpasc 13682 abslem2 14699 geolim 15226 bpolydiflem 15408 efaddlem 15446 eftlub 15462 bitsinv1lem 15790 pjthlem1 24040 itg2monolem3 24353 dvmulbr 24536 dvrecg 24570 dvmptdiv 24571 dvtaylp 24958 itgulm 24996 tanregt0 25123 logtayl2 25245 cxpeq 25338 heron 25416 dcubic2 25422 cubic2 25426 dquartlem1 25429 dquartlem2 25430 dquart 25431 quart1lem 25433 quart1 25434 dvatan 25513 atantayl 25515 jensenlem2 25565 lgamgulmlem2 25607 lgamgulmlem3 25608 ftalem2 25651 bclbnd 25856 bposlem9 25868 lgseisenlem4 25954 lgsquadlem1 25956 lgsquadlem2 25957 dchrvmasumlem1 26071 mulog2sumlem2 26111 2vmadivsumlem 26116 selberg3lem1 26133 selberg4lem1 26136 selberg4 26137 selberg3r 26145 pntrlog2bndlem4 26156 pntrlog2bndlem5 26157 pntibndlem2 26167 pntlemo 26183 brbtwn2 26691 colinearalg 26696 axsegconlem10 26712 axpaschlem 26726 axcontlem8 26757 pjhthlem1 29168 sinccvglem 32915 knoppndvlem14 33864 bj-bary1lem 34594 dvtan 34957 binomcxplemnotnn0 40708 dvnprodlem2 42252 itgsinexp 42260 stirlinglem3 42381 stirlinglem4 42382 dirkertrigeqlem3 42405 fourierdlem95 42506 eenglngeehlnmlem1 44744 eenglngeehlnmlem2 44745 |
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