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Mirrors > Home > ILE Home > Th. List > crngpropd | Unicode version |
Description: If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
ringpropd.1 |
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ringpropd.2 |
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ringpropd.3 |
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ringpropd.4 |
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Ref | Expression |
---|---|
crngpropd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringpropd.1 |
. . . . . 6
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2 | eqid 2177 |
. . . . . . 7
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3 | eqid 2177 |
. . . . . . 7
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4 | 2, 3 | mgpbasg 12960 |
. . . . . 6
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5 | 1, 4 | sylan9eq 2230 |
. . . . 5
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6 | ringpropd.2 |
. . . . . . 7
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7 | 6 | adantr 276 |
. . . . . 6
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8 | ringpropd.3 |
. . . . . . . . 9
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9 | ringpropd.4 |
. . . . . . . . 9
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10 | 1, 6, 8, 9 | ringpropd 13040 |
. . . . . . . 8
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11 | 10 | biimpa 296 |
. . . . . . 7
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12 | eqid 2177 |
. . . . . . . 8
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13 | eqid 2177 |
. . . . . . . 8
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14 | 12, 13 | mgpbasg 12960 |
. . . . . . 7
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15 | 11, 14 | syl 14 |
. . . . . 6
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16 | 7, 15 | eqtrd 2210 |
. . . . 5
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17 | 9 | adantlr 477 |
. . . . . 6
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18 | eqid 2177 |
. . . . . . . . 9
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19 | 2, 18 | mgpplusgg 12958 |
. . . . . . . 8
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20 | 19 | adantl 277 |
. . . . . . 7
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21 | 20 | oveqdr 5897 |
. . . . . 6
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22 | eqid 2177 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 12, 22 | mgpplusgg 12958 |
. . . . . . . 8
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24 | 11, 23 | syl 14 |
. . . . . . 7
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25 | 24 | oveqdr 5897 |
. . . . . 6
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26 | 17, 21, 25 | 3eqtr3d 2218 |
. . . . 5
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27 | 5, 16, 26 | cmnpropd 12922 |
. . . 4
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28 | 27 | pm5.32da 452 |
. . 3
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29 | 10 | anbi1d 465 |
. . 3
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30 | 28, 29 | bitrd 188 |
. 2
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31 | 2 | iscrng 13009 |
. 2
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32 | 12 | iscrng 13009 |
. 2
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33 | 30, 31, 32 | 3bitr4g 223 |
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 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-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1cn 7892 ax-1re 7893 ax-icn 7894 ax-addcl 7895 ax-addrcl 7896 ax-mulcl 7897 ax-addcom 7899 ax-addass 7901 ax-i2m1 7904 ax-0lt1 7905 ax-0id 7907 ax-rnegex 7908 ax-pre-ltirr 7911 ax-pre-ltadd 7915 |
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-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7981 df-mnf 7982 df-ltxr 7984 df-inn 8906 df-2 8964 df-3 8965 df-ndx 12445 df-slot 12446 df-base 12448 df-sets 12449 df-plusg 12528 df-mulr 12529 df-0g 12652 df-mgm 12664 df-sgrp 12697 df-mnd 12707 df-grp 12767 df-cmn 12914 df-mgp 12955 df-ring 13004 df-cring 13005 |
This theorem is referenced by: (None) |
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