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Mirrors > Home > MPE Home > Th. List > abladdsub4 | Structured version Visualization version Unicode version |
Description: Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
ablsubadd.b |
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ablsubadd.p |
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ablsubadd.m |
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Ref | Expression |
---|---|
abladdsub4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablgrp 17447 |
. . . 4
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2 | 1 | 3ad2ant1 1030 |
. . 3
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3 | simp2l 1035 |
. . . 4
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4 | simp2r 1036 |
. . . 4
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5 | ablsubadd.b |
. . . . 5
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6 | ablsubadd.p |
. . . . 5
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7 | 5, 6 | grpcl 16691 |
. . . 4
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8 | 2, 3, 4, 7 | syl3anc 1269 |
. . 3
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9 | simp3l 1037 |
. . . 4
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10 | simp3r 1038 |
. . . 4
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11 | 5, 6 | grpcl 16691 |
. . . 4
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12 | 2, 9, 10, 11 | syl3anc 1269 |
. . 3
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13 | 5, 6 | grpcl 16691 |
. . . 4
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14 | 2, 9, 4, 13 | syl3anc 1269 |
. . 3
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15 | ablsubadd.m |
. . . 4
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16 | 5, 15 | grpsubrcan 16747 |
. . 3
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17 | 2, 8, 12, 14, 16 | syl13anc 1271 |
. 2
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18 | simp1 1009 |
. . . . 5
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19 | 5, 6, 15 | ablsub4 17467 |
. . . . 5
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20 | 18, 3, 4, 9, 4, 19 | syl122anc 1278 |
. . . 4
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21 | eqid 2453 |
. . . . . . 7
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22 | 5, 21, 15 | grpsubid 16750 |
. . . . . 6
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23 | 2, 4, 22 | syl2anc 667 |
. . . . 5
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24 | 23 | oveq2d 6311 |
. . . 4
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25 | 5, 15 | grpsubcl 16746 |
. . . . . 6
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26 | 2, 3, 9, 25 | syl3anc 1269 |
. . . . 5
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27 | 5, 6, 21 | grprid 16709 |
. . . . 5
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28 | 2, 26, 27 | syl2anc 667 |
. . . 4
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29 | 20, 24, 28 | 3eqtrd 2491 |
. . 3
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30 | 5, 6, 15 | ablsub4 17467 |
. . . . 5
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31 | 18, 9, 10, 9, 4, 30 | syl122anc 1278 |
. . . 4
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32 | 5, 21, 15 | grpsubid 16750 |
. . . . . 6
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33 | 2, 9, 32 | syl2anc 667 |
. . . . 5
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34 | 33 | oveq1d 6310 |
. . . 4
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35 | 5, 15 | grpsubcl 16746 |
. . . . . 6
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36 | 2, 10, 4, 35 | syl3anc 1269 |
. . . . 5
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37 | 5, 6, 21 | grplid 16708 |
. . . . 5
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38 | 2, 36, 37 | syl2anc 667 |
. . . 4
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39 | 31, 34, 38 | 3eqtrd 2491 |
. . 3
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40 | 29, 39 | eqeq12d 2468 |
. 2
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41 | 17, 40 | bitr3d 259 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-rep 4518 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-id 4752 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-1st 6798 df-2nd 6799 df-0g 15352 df-mgm 16500 df-sgrp 16539 df-mnd 16549 df-grp 16685 df-minusg 16686 df-sbg 16687 df-cmn 17444 df-abl 17445 |
This theorem is referenced by: lmodvaddsub4 18152 |
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