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Mirrors > Home > MPE Home > Th. List > subgsubcl | Structured version Visualization version GIF version |
Description: A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
subgsubcl.p | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
subgsubcl | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | subgss 17908 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
3 | 2 | 3ad2ant1 1164 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 ⊆ (Base‘𝐺)) |
4 | simp2 1168 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
5 | 3, 4 | sseldd 3799 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘𝐺)) |
6 | simp3 1169 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ 𝑆) | |
7 | 3, 6 | sseldd 3799 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘𝐺)) |
8 | eqid 2799 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
9 | eqid 2799 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | subgsubcl.p | . . . 4 ⊢ − = (-g‘𝐺) | |
11 | 1, 8, 9, 10 | grpsubval 17781 | . . 3 ⊢ ((𝑋 ∈ (Base‘𝐺) ∧ 𝑌 ∈ (Base‘𝐺)) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
12 | 5, 7, 11 | syl2anc 580 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) = (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌))) |
13 | 9 | subginvcl 17916 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) ∈ 𝑆) |
14 | 13 | 3adant2 1162 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → ((invg‘𝐺)‘𝑌) ∈ 𝑆) |
15 | 8 | subgcl 17917 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝑆) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ 𝑆) |
16 | 14, 15 | syld3an3 1529 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐺)((invg‘𝐺)‘𝑌)) ∈ 𝑆) |
17 | 12, 16 | eqeltrd 2878 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 − 𝑌) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ⊆ wss 3769 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 +gcplusg 16267 invgcminusg 17739 -gcsg 17740 SubGrpcsubg 17901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-minusg 17742 df-sbg 17743 df-subg 17904 |
This theorem is referenced by: issubg4 17926 ssnmz 17949 subgdisj1 18417 dprdfeq0 18737 pgpfac1lem3 18792 lidlsubcl 19539 chfacfisfcpmat 20988 subgntr 22238 opnsubg 22239 clssubg 22240 clmsubcl 23213 |
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