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| Mirrors > Home > MPE Home > Th. List > dvrcl | Structured version Visualization version GIF version | ||
| Description: Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.) |
| Ref | Expression |
|---|---|
| dvrcl.b | β’ π΅ = (Baseβπ ) |
| dvrcl.o | β’ π = (Unitβπ ) |
| dvrcl.d | β’ / = (/rβπ ) |
| Ref | Expression |
|---|---|
| dvrcl | β’ ((π β Ring β§ π β π΅ β§ π β π) β (π / π) β π΅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvrcl.b | . . . 4 β’ π΅ = (Baseβπ ) | |
| 2 | eqid 2724 | . . . 4 β’ (.rβπ ) = (.rβπ ) | |
| 3 | dvrcl.o | . . . 4 β’ π = (Unitβπ ) | |
| 4 | eqid 2724 | . . . 4 β’ (invrβπ ) = (invrβπ ) | |
| 5 | dvrcl.d | . . . 4 β’ / = (/rβπ ) | |
| 6 | 1, 2, 3, 4, 5 | dvrval 20301 | . . 3 β’ ((π β π΅ β§ π β π) β (π / π) = (π(.rβπ )((invrβπ )βπ))) |
| 7 | 6 | 3adant1 1127 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π / π) = (π(.rβπ )((invrβπ )βπ))) |
| 8 | 3, 4, 1 | ringinvcl 20290 | . . . 4 β’ ((π β Ring β§ π β π) β ((invrβπ )βπ) β π΅) |
| 9 | 8 | 3adant2 1128 | . . 3 β’ ((π β Ring β§ π β π΅ β§ π β π) β ((invrβπ )βπ) β π΅) |
| 10 | 1, 2 | ringcl 20151 | . . 3 β’ ((π β Ring β§ π β π΅ β§ ((invrβπ )βπ) β π΅) β (π(.rβπ )((invrβπ )βπ)) β π΅) |
| 11 | 9, 10 | syld3an3 1406 | . 2 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π(.rβπ )((invrβπ )βπ)) β π΅) |
| 12 | 7, 11 | eqeltrd 2825 | 1 β’ ((π β Ring β§ π β π΅ β§ π β π) β (π / π) β π΅) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17149 .rcmulr 17203 Ringcrg 20134 Unitcui 20253 invrcinvr 20285 /rcdvr 20298 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 |
| This theorem is referenced by: rdivmuldivd 20311 irredrmul 20325 lringuplu 20440 isdrng2 20597 cnflddiv 21280 qsssubdrg 21309 cvsdivcl 25004 sum2dchr 27147 sdrgdvcl 32889 isarchiofld 32927 qqhf 33485 |
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