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Mirrors > Home > MPE Home > Th. List > nsgconj | Structured version Unicode version |
Description: The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.) |
Ref | Expression |
---|---|
isnsg3.1 |
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isnsg3.2 |
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isnsg3.3 |
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Ref | Expression |
---|---|
nsgconj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 15817 |
. . . . 5
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2 | 1 | 3ad2ant1 1009 |
. . . 4
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3 | subgrcl 15790 |
. . . 4
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4 | 2, 3 | syl 16 |
. . 3
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5 | simp2 989 |
. . 3
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6 | isnsg3.1 |
. . . . . 6
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7 | 6 | subgss 15786 |
. . . . 5
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8 | 2, 7 | syl 16 |
. . . 4
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9 | simp3 990 |
. . . 4
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10 | 8, 9 | sseldd 3457 |
. . 3
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11 | isnsg3.2 |
. . . 4
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12 | isnsg3.3 |
. . . 4
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13 | 6, 11, 12 | grpaddsubass 15719 |
. . 3
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14 | 4, 5, 10, 5, 13 | syl13anc 1221 |
. 2
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15 | 6, 11, 12 | grpnpcan 15721 |
. . . . 5
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16 | 4, 10, 5, 15 | syl3anc 1219 |
. . . 4
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17 | 16, 9 | eqeltrd 2539 |
. . 3
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18 | simp1 988 |
. . . 4
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19 | 6, 12 | grpsubcl 15710 |
. . . . 5
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20 | 4, 10, 5, 19 | syl3anc 1219 |
. . . 4
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21 | 6, 11 | nsgbi 15816 |
. . . 4
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22 | 18, 20, 5, 21 | syl3anc 1219 |
. . 3
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23 | 17, 22 | mpbid 210 |
. 2
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24 | 14, 23 | eqeltrd 2539 |
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 1952 ax-ext 2430 ax-rep 4503 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 |
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 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-op 3984 df-uni 4192 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-id 4736 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-1st 6679 df-2nd 6680 df-0g 14484 df-mnd 15519 df-grp 15649 df-minusg 15650 df-sbg 15651 df-subg 15782 df-nsg 15783 |
This theorem is referenced by: isnsg3 15819 ghmnsgima 15874 ghmnsgpreima 15875 clsnsg 19798 |
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