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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 13439 |
. . . 4
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3 | ringsrg 13292 |
. . . 4
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4 | 2, 3 | syl 14 |
. . 3
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5 | reldvdsrsrg 13335 |
. . . 4
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6 | subrgdvds.3 |
. . . . 5
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7 | 6 | releqi 4721 |
. . . 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 13445 |
. . . . . . 7
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11 | eqid 2187 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 11 | subrgss 13437 |
. . . . . . 7
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13 | 10, 12 | eqsstrrd 3204 |
. . . . . 6
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14 | 13 | sseld 3166 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | subrgrcl 13441 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | eqid 2187 |
. . . . . . . . . . 11
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17 | 1, 16 | ressmulrg 12617 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | 15, 17 | mpdan 421 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 18 | oveqd 5905 |
. . . . . . . 8
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20 | 19 | eqeq1d 2196 |
. . . . . . 7
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21 | 20 | rexbidv 2488 |
. . . . . 6
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22 | ssrexv 3232 |
. . . . . . 7
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23 | 13, 22 | syl 14 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 21, 23 | sylbird 170 |
. . . . 5
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25 | 14, 24 | anim12d 335 |
. . . 4
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26 | eqidd 2188 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
27 | 6 | a1i 9 |
. . . . 5
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28 | eqidd 2188 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 26, 27, 4, 28 | dvdsrd 13337 |
. . . 4
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30 | eqidd 2188 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | subrgdvds.2 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 31 | a1i 9 |
. . . . 5
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33 | ringsrg 13292 |
. . . . . 6
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34 | 15, 33 | syl 14 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | eqidd 2188 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
36 | 30, 32, 34, 35 | dvdsrd 13337 |
. . . 4
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37 | 25, 29, 36 | 3imtr4d 203 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | df-br 4016 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
39 | df-br 4016 |
. . 3
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40 | 37, 38, 39 | 3imtr3g 204 |
. 2
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41 | 9, 40 | relssdv 4730 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltirr 7936 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-inn 8933 df-2 8991 df-3 8992 df-ndx 12478 df-slot 12479 df-base 12481 df-sets 12482 df-iress 12483 df-plusg 12563 df-mulr 12564 df-0g 12724 df-mgm 12793 df-sgrp 12826 df-mnd 12839 df-grp 12901 df-minusg 12902 df-subg 13061 df-cmn 13122 df-abl 13123 df-mgp 13171 df-ur 13207 df-srg 13211 df-ring 13245 df-dvdsr 13332 df-subrg 13434 |
This theorem is referenced by: subrguss 13451 |
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