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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 2819 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | issubg 18271 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
4 | 3 | simp3bi 1142 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
5 | 1, 4 | eqeltrid 2915 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 ↾s cress 16476 Grpcgrp 18095 SubGrpcsubg 18265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7151 df-subg 18268 |
This theorem is referenced by: subg0 18277 subginv 18278 subg0cl 18279 subginvcl 18280 subgcl 18281 issubg2 18286 issubgrpd 18288 subsubg 18294 resghm 18366 resghm2b 18368 subgga 18422 gasubg 18424 odsubdvds 18688 pgp0 18713 subgpgp 18714 sylow2blem2 18738 slwhash 18741 fislw 18742 subglsm 18791 pj1ghm 18821 subgabl 18948 cntrabl 18955 cycsubgcyg 19013 subgdmdprd 19148 subgdprd 19149 ablfacrplem 19179 pgpfaclem1 19195 pgpfaclem3 19197 ablfaclem3 19201 issubrg2 19547 subdrgint 19574 islss3 19723 mplgrp 20222 zringcyg 20630 cnmsgngrp 20715 psgnghm 20716 scmatghm 21134 m2cpmrngiso 21358 subgtgp 22705 subgngp 23236 reefgim 25030 amgmlemALT 44895 |
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