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Mirrors > Home > ILE Home > Th. List > ablsub2inv | Unicode version |
Description: Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.) |
Ref | Expression |
---|---|
ablsub2inv.b |
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ablsub2inv.m |
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ablsub2inv.n |
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ablsub2inv.g |
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ablsub2inv.x |
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ablsub2inv.y |
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Ref | Expression |
---|---|
ablsub2inv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablsub2inv.b |
. . 3
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2 | eqid 2177 |
. . 3
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3 | ablsub2inv.m |
. . 3
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4 | ablsub2inv.n |
. . 3
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5 | ablsub2inv.g |
. . . 4
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6 | ablgrp 12917 |
. . . 4
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7 | 5, 6 | syl 14 |
. . 3
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8 | ablsub2inv.x |
. . . 4
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9 | 1, 4 | grpinvcl 12808 |
. . . 4
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10 | 7, 8, 9 | syl2anc 411 |
. . 3
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11 | ablsub2inv.y |
. . 3
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12 | 1, 2, 3, 4, 7, 10, 11 | grpsubinv 12829 |
. 2
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13 | 1, 2 | ablcom 12930 |
. . . . . 6
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14 | 5, 10, 11, 13 | syl3anc 1238 |
. . . . 5
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15 | 1, 4 | grpinvinv 12823 |
. . . . . . 7
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16 | 7, 11, 15 | syl2anc 411 |
. . . . . 6
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17 | 16 | oveq1d 5884 |
. . . . 5
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18 | 14, 17 | eqtr4d 2213 |
. . . 4
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19 | 1, 4 | grpinvcl 12808 |
. . . . . 6
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20 | 7, 11, 19 | syl2anc 411 |
. . . . 5
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21 | 1, 2, 4 | grpinvadd 12834 |
. . . . 5
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22 | 7, 8, 20, 21 | syl3anc 1238 |
. . . 4
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23 | 18, 22 | eqtr4d 2213 |
. . 3
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24 | 1, 2, 4, 3 | grpsubval 12806 |
. . . . 5
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25 | 8, 11, 24 | syl2anc 411 |
. . . 4
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26 | 25 | fveq2d 5515 |
. . 3
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27 | 23, 26 | eqtr4d 2213 |
. 2
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28 | 1, 3, 4 | grpinvsub 12838 |
. . 3
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29 | 7, 8, 11, 28 | syl3anc 1238 |
. 2
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30 | 12, 27, 29 | 3eqtrd 2214 |
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-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7890 ax-resscn 7891 ax-1re 7893 ax-addrcl 7896 |
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-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 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-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 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-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-inn 8906 df-2 8964 df-ndx 12445 df-slot 12446 df-base 12448 df-plusg 12528 df-0g 12652 df-mgm 12664 df-sgrp 12697 df-mnd 12707 df-grp 12767 df-minusg 12768 df-sbg 12769 df-cmn 12914 df-abl 12915 |
This theorem is referenced by: (None) |
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