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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 11509 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) | |
6 | 1, 2, 3, 4, 5 | syl112anc 1376 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵) / 𝐶) = ((𝐴 / 𝐶) · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 (class class class)co 7213 ℂcc 10727 0cc0 10729 · cmul 10734 / cdiv 11489 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 |
This theorem is referenced by: bcpasc 13887 abslem2 14903 geolim 15434 bpolydiflem 15616 efaddlem 15654 eftlub 15670 bitsinv1lem 16000 pjthlem1 24334 itg2monolem3 24650 dvmulbr 24836 dvrecg 24870 dvmptdiv 24871 dvtaylp 25262 itgulm 25300 tanregt0 25428 logtayl2 25550 cxpeq 25643 heron 25721 dcubic2 25727 cubic2 25731 dquartlem1 25734 dquartlem2 25735 dquart 25736 quart1lem 25738 quart1 25739 dvatan 25818 atantayl 25820 jensenlem2 25870 lgamgulmlem2 25912 lgamgulmlem3 25913 ftalem2 25956 bclbnd 26161 bposlem9 26173 lgseisenlem4 26259 lgsquadlem1 26261 lgsquadlem2 26262 dchrvmasumlem1 26376 mulog2sumlem2 26416 2vmadivsumlem 26421 selberg3lem1 26438 selberg4lem1 26441 selberg4 26442 selberg3r 26450 pntrlog2bndlem4 26461 pntrlog2bndlem5 26462 pntibndlem2 26472 pntlemo 26488 brbtwn2 26996 colinearalg 27001 axsegconlem10 27017 axpaschlem 27031 axcontlem8 27062 pjhthlem1 29472 sinccvglem 33343 knoppndvlem14 34442 bj-bary1lem 35215 dvtan 35564 lcmineqlem10 39780 aks4d1p1p7 39815 binomcxplemnotnn0 41647 dvnprodlem2 43163 itgsinexp 43171 stirlinglem3 43292 stirlinglem4 43293 dirkertrigeqlem3 43316 fourierdlem95 43417 eenglngeehlnmlem1 45756 eenglngeehlnmlem2 45757 |
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