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Mirrors > Home > ILE Home > Th. List > grpaddsubass | Unicode version |
Description: Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
grpsubadd.b |
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grpsubadd.p |
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grpsubadd.m |
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Ref | Expression |
---|---|
grpaddsubass |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 109 |
. . 3
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2 | simpr1 1003 |
. . 3
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3 | simpr2 1004 |
. . 3
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4 | grpsubadd.b |
. . . . 5
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5 | eqid 2177 |
. . . . 5
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6 | 4, 5 | grpinvcl 12875 |
. . . 4
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7 | 6 | 3ad2antr3 1164 |
. . 3
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8 | grpsubadd.p |
. . . 4
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9 | 4, 8 | grpass 12840 |
. . 3
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10 | 1, 2, 3, 7, 9 | syl13anc 1240 |
. 2
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11 | 4, 8 | grpcl 12839 |
. . . 4
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12 | 11 | 3adant3r3 1214 |
. . 3
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13 | simpr3 1005 |
. . 3
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14 | grpsubadd.m |
. . . 4
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15 | 4, 8, 5, 14 | grpsubval 12873 |
. . 3
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16 | 12, 13, 15 | syl2anc 411 |
. 2
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17 | 4, 8, 5, 14 | grpsubval 12873 |
. . . 4
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18 | 3, 13, 17 | syl2anc 411 |
. . 3
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19 | 18 | oveq2d 5890 |
. 2
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20 | 10, 16, 19 | 3eqtr4d 2220 |
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 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-inn 8918 df-2 8976 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-sbg 12836 |
This theorem is referenced by: grppncan 12915 grpnpncan 12919 nsgconj 13019 abladdsub 13071 ablsubsub 13074 |
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