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Mirrors > Home > ILE Home > Th. List > grpsubadd0sub | Unicode version |
Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.) |
Ref | Expression |
---|---|
grpsubid.b |
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grpsubid.o |
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grpsubid.m |
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grpsubadd0sub.p |
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Ref | Expression |
---|---|
grpsubadd0sub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubid.b |
. . . 4
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2 | grpsubadd0sub.p |
. . . 4
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3 | eqid 2177 |
. . . 4
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4 | grpsubid.m |
. . . 4
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5 | 1, 2, 3, 4 | grpsubval 12796 |
. . 3
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6 | 5 | 3adant1 1015 |
. 2
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7 | grpsubid.o |
. . . . 5
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8 | 1, 4, 3, 7 | grpinvval2 12829 |
. . . 4
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9 | 8 | 3adant2 1016 |
. . 3
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10 | 9 | oveq2d 5884 |
. 2
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11 | 6, 10 | eqtrd 2210 |
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 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
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 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-grp 12757 df-minusg 12758 df-sbg 12759 |
This theorem is referenced by: (None) |
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