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Mirrors > Home > MPE Home > Th. List > grpsubadd | Structured version Unicode version |
Description: Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
grpsubadd.b |
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grpsubadd.p |
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grpsubadd.m |
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Ref | Expression |
---|---|
grpsubadd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubadd.b |
. . . . . . 7
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2 | grpsubadd.p |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | eqid 2454 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | grpsubadd.m |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 1, 2, 3, 4 | grpsubval 15704 |
. . . . . 6
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6 | 5 | 3adant3 1008 |
. . . . 5
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7 | 6 | adantl 466 |
. . . 4
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8 | 7 | eqeq1d 2456 |
. . 3
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9 | simpl 457 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | simpr1 994 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 1, 3 | grpinvcl 15706 |
. . . . . 6
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12 | 11 | 3ad2antr2 1154 |
. . . . 5
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13 | 1, 2 | grpcl 15674 |
. . . . 5
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14 | 9, 10, 12, 13 | syl3anc 1219 |
. . . 4
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15 | simpr3 996 |
. . . 4
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16 | simpr2 995 |
. . . 4
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17 | 1, 2 | grprcan 15694 |
. . . 4
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18 | 9, 14, 15, 16, 17 | syl13anc 1221 |
. . 3
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19 | 1, 2 | grpass 15675 |
. . . . . 6
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20 | 9, 10, 12, 16, 19 | syl13anc 1221 |
. . . . 5
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21 | eqid 2454 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 1, 2, 21, 3 | grplinv 15707 |
. . . . . . 7
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23 | 22 | 3ad2antr2 1154 |
. . . . . 6
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24 | 23 | oveq2d 6219 |
. . . . 5
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25 | 1, 2, 21 | grprid 15692 |
. . . . . 6
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26 | 25 | 3ad2antr1 1153 |
. . . . 5
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27 | 20, 24, 26 | 3eqtrd 2499 |
. . . 4
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28 | 27 | eqeq1d 2456 |
. . 3
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29 | 8, 18, 28 | 3bitr2d 281 |
. 2
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30 | eqcom 2463 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 29, 30 | syl6bb 261 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4514 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-iun 4284 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fo 5535 df-f1o 5536 df-fv 5537 df-riota 6164 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-1st 6690 df-2nd 6691 df-0g 14503 df-mnd 15538 df-grp 15668 df-minusg 15669 df-sbg 15670 |
This theorem is referenced by: grpsubsub4 15741 conjghm 15900 conjnmzb 15904 sylow3lem2 16252 ablsubadd 16426 pgpfac1lem2 16708 pgpfac1lem4 16711 lspexch 17343 coe1subfv 17853 ipsubdir 18206 ipsubdi 18207 zlmodzxzsub 30928 |
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