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Mirrors > Home > ILE Home > Th. List > unitmulcl | Unicode version |
Description: The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
unitmulcl.1 |
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unitmulcl.2 |
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Ref | Expression |
---|---|
unitmulcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. . 3
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2 | eqidd 2178 |
. . . . . 6
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3 | unitmulcl.1 |
. . . . . . 7
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4 | 3 | a1i 9 |
. . . . . 6
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5 | ringsrg 13229 |
. . . . . . 7
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6 | 1, 5 | syl 14 |
. . . . . 6
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7 | simp3 999 |
. . . . . 6
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8 | 2, 4, 6, 7 | unitcld 13282 |
. . . . 5
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9 | simp2 998 |
. . . . . . 7
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10 | eqidd 2178 |
. . . . . . . 8
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11 | eqidd 2178 |
. . . . . . . 8
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12 | eqidd 2178 |
. . . . . . . 8
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13 | eqidd 2178 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 4, 10, 11, 12, 13, 6 | isunitd 13280 |
. . . . . . 7
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15 | 9, 14 | mpbid 147 |
. . . . . 6
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16 | 15 | simpld 112 |
. . . . 5
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17 | eqid 2177 |
. . . . . 6
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18 | eqid 2177 |
. . . . . 6
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19 | unitmulcl.2 |
. . . . . 6
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20 | 17, 18, 19 | dvdsrmul1 13276 |
. . . . 5
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21 | 1, 8, 16, 20 | syl3anc 1238 |
. . . 4
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22 | eqid 2177 |
. . . . . 6
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23 | 17, 19, 22 | ringlidm 13211 |
. . . . 5
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24 | 1, 8, 23 | syl2anc 411 |
. . . 4
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25 | 21, 24 | breqtrd 4031 |
. . 3
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26 | 4, 10, 11, 12, 13, 6 | isunitd 13280 |
. . . . 5
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27 | 7, 26 | mpbid 147 |
. . . 4
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28 | 27 | simpld 112 |
. . 3
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29 | 17, 18 | dvdsrtr 13275 |
. . 3
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30 | 1, 25, 28, 29 | syl3anc 1238 |
. 2
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31 | eqid 2177 |
. . . . 5
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32 | 31 | opprring 13254 |
. . . 4
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33 | 1, 32 | syl 14 |
. . 3
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34 | 2, 4, 6, 9 | unitcld 13282 |
. . . . . 6
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35 | 31, 17 | opprbasg 13252 |
. . . . . . 7
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36 | 1, 35 | syl 14 |
. . . . . 6
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37 | 34, 36 | eleqtrd 2256 |
. . . . 5
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38 | 27 | simprd 114 |
. . . . 5
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39 | eqid 2177 |
. . . . . 6
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40 | eqid 2177 |
. . . . . 6
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41 | eqid 2177 |
. . . . . 6
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42 | 39, 40, 41 | dvdsrmul1 13276 |
. . . . 5
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43 | 33, 37, 38, 42 | syl3anc 1238 |
. . . 4
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44 | 17, 19, 31, 41 | opprmulg 13248 |
. . . . 5
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45 | 44 | 3com23 1209 |
. . . 4
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46 | 17, 22 | srgidcl 13164 |
. . . . . . 7
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47 | 6, 46 | syl 14 |
. . . . . 6
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48 | 17, 19, 31, 41 | opprmulg 13248 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 1, 47, 9, 48 | syl3anc 1238 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
50 | 17, 19, 22 | ringridm 13212 |
. . . . . 6
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51 | 1, 34, 50 | syl2anc 411 |
. . . . 5
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52 | 49, 51 | eqtrd 2210 |
. . . 4
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53 | 43, 45, 52 | 3brtr3d 4036 |
. . 3
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54 | 15 | simprd 114 |
. . 3
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55 | 39, 40 | dvdsrtr 13275 |
. . 3
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56 | 33, 53, 54, 55 | syl3anc 1238 |
. 2
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57 | 4, 10, 11, 12, 13, 6 | isunitd 13280 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
58 | 30, 56, 57 | mpbir2and 944 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-pre-ltirr 7925 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-tpos 6248 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-base 12470 df-sets 12471 df-plusg 12551 df-mulr 12552 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-grp 12885 df-minusg 12886 df-cmn 13095 df-abl 13096 df-mgp 13136 df-ur 13148 df-srg 13152 df-ring 13186 df-oppr 13245 df-dvdsr 13263 df-unit 13264 |
This theorem is referenced by: unitmulclb 13288 unitgrp 13290 unitdvcl 13310 rdivmuldivd 13318 lringuplu 13342 subrgugrp 13366 |
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