Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > subggrp | Structured version Visualization version GIF version |
Description: A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subggrp.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
Ref | Expression |
---|---|
subggrp | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subggrp.h | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
2 | eqid 2610 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | issubg 17417 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
4 | 3 | simp3bi 1071 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
5 | 1, 4 | syl5eqel 2692 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Grpcgrp 17245 SubGrpcsubg 17411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-subg 17414 |
This theorem is referenced by: subg0 17423 subginv 17424 subg0cl 17425 subginvcl 17426 subgcl 17427 issubg2 17432 issubgrpd 17434 subsubg 17440 resghm 17499 resghm2b 17501 subgga 17556 gasubg 17558 odsubdvds 17809 pgp0 17834 subgpgp 17835 sylow2blem2 17859 slwhash 17862 fislw 17863 subglsm 17909 pj1ghm 17939 subgabl 18064 cycsubgcyg 18125 subgdmdprd 18256 subgdprd 18257 ablfacrplem 18287 pgpfaclem1 18303 pgpfaclem3 18305 ablfaclem3 18309 issubrg2 18623 islss3 18780 mplgrp 19271 zringcyg 19658 cnmsgngrp 19744 psgnghm 19745 scmatghm 20158 m2cpmrngiso 20382 subgtgp 21719 subgngp 22249 reefgim 24008 amgmlemALT 42358 |
Copyright terms: Public domain | W3C validator |