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Mirrors > Home > MPE Home > Th. List > nsgsubg | Structured version Unicode version |
Description: A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
nsgsubg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2451 |
. . 3
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2 | eqid 2451 |
. . 3
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3 | 1, 2 | isnsg 15812 |
. 2
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4 | 3 | simplbi 460 |
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-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 |
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-rab 2804 df-v 3070 df-sbc 3285 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fv 5524 df-ov 6193 df-subg 15780 df-nsg 15781 |
This theorem is referenced by: nsgconj 15816 isnsg3 15817 eqgcpbl 15837 divsgrp 15838 divseccl 15839 divsadd 15840 divs0 15841 divsinv 15842 divssub 15843 ghmnsgima 15872 ghmnsgpreima 15873 conjnsg 15884 divsghm 15885 sylow3lem4 16233 clsnsg 19796 divstgpopn 19806 divstgphaus 19809 |
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