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Mirrors > Home > ILE Home > Th. List > subrgdvds | Unicode version |
Description: If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
subrgdvds.1 |
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subrgdvds.2 |
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subrgdvds.3 |
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Ref | Expression |
---|---|
subrgdvds |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subrgdvds.1 |
. . . . 5
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2 | 1 | subrgring 13538 |
. . . 4
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3 | ringsrg 13366 |
. . . 4
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4 | 2, 3 | syl 14 |
. . 3
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5 | reldvdsrsrg 13409 |
. . . 4
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6 | subrgdvds.3 |
. . . . 5
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7 | 6 | releqi 4724 |
. . . 4
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8 | 5, 7 | sylibr 134 |
. . 3
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9 | 4, 8 | syl 14 |
. 2
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10 | 1 | subrgbas 13544 |
. . . . . . 7
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11 | eqid 2189 |
. . . . . . . 8
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12 | 11 | subrgss 13536 |
. . . . . . 7
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13 | 10, 12 | eqsstrrd 3207 |
. . . . . 6
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14 | 13 | sseld 3169 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | subrgrcl 13540 |
. . . . . . . . . 10
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16 | eqid 2189 |
. . . . . . . . . . 11
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17 | 1, 16 | ressmulrg 12628 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 15, 17 | mpdan 421 |
. . . . . . . . 9
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19 | 18 | oveqd 5908 |
. . . . . . . 8
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20 | 19 | eqeq1d 2198 |
. . . . . . 7
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21 | 20 | rexbidv 2491 |
. . . . . 6
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22 | ssrexv 3235 |
. . . . . . 7
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23 | 13, 22 | syl 14 |
. . . . . 6
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24 | 21, 23 | sylbird 170 |
. . . . 5
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25 | 14, 24 | anim12d 335 |
. . . 4
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26 | eqidd 2190 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 6 | a1i 9 |
. . . . 5
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28 | eqidd 2190 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 26, 27, 4, 28 | dvdsrd 13411 |
. . . 4
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30 | eqidd 2190 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | subrgdvds.2 |
. . . . . 6
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32 | 31 | a1i 9 |
. . . . 5
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33 | ringsrg 13366 |
. . . . . 6
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34 | 15, 33 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | eqidd 2190 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 30, 32, 34, 35 | dvdsrd 13411 |
. . . 4
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37 | 25, 29, 36 | 3imtr4d 203 |
. . 3
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38 | df-br 4019 |
. . 3
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39 | df-br 4019 |
. . 3
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40 | 37, 38, 39 | 3imtr3g 204 |
. 2
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41 | 9, 40 | relssdv 4733 |
1
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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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-pre-ltirr 7942 ax-pre-lttrn 7944 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8013 df-mnf 8014 df-ltxr 8016 df-inn 8939 df-2 8997 df-3 8998 df-ndx 12489 df-slot 12490 df-base 12492 df-sets 12493 df-iress 12494 df-plusg 12574 df-mulr 12575 df-0g 12735 df-mgm 12804 df-sgrp 12837 df-mnd 12850 df-grp 12920 df-minusg 12921 df-subg 13081 df-cmn 13192 df-abl 13193 df-mgp 13242 df-ur 13281 df-srg 13285 df-ring 13319 df-dvdsr 13406 df-subrg 13533 |
This theorem is referenced by: subrguss 13550 |
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