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| Mirrors > Home > MPE Home > Th. List > subgcl | Structured version Visualization version GIF version | ||
| Description: A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| Ref | Expression |
|---|---|
| subgcl.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| subgcl | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
| 2 | 1 | subggrp 19147 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
| 3 | 2 | 3ad2ant1 1134 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝐺 ↾s 𝑆) ∈ Grp) |
| 4 | simp2 1138 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ 𝑆) | |
| 5 | 1 | subgbas 19148 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 6 | 5 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 7 | 4, 6 | eleqtrd 2843 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑋 ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 8 | simp3 1139 | . . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ 𝑆) | |
| 9 | 8, 6 | eleqtrd 2843 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → 𝑌 ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 10 | eqid 2737 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
| 11 | eqid 2737 | . . . 4 ⊢ (+g‘(𝐺 ↾s 𝑆)) = (+g‘(𝐺 ↾s 𝑆)) | |
| 12 | 10, 11 | grpcl 18959 | . . 3 ⊢ (((𝐺 ↾s 𝑆) ∈ Grp ∧ 𝑋 ∈ (Base‘(𝐺 ↾s 𝑆)) ∧ 𝑌 ∈ (Base‘(𝐺 ↾s 𝑆))) → (𝑋(+g‘(𝐺 ↾s 𝑆))𝑌) ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 13 | 3, 7, 9, 12 | syl3anc 1373 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘(𝐺 ↾s 𝑆))𝑌) ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 14 | subgcl.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
| 15 | 1, 14 | ressplusg 17334 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → + = (+g‘(𝐺 ↾s 𝑆))) |
| 16 | 15 | 3ad2ant1 1134 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → + = (+g‘(𝐺 ↾s 𝑆))) |
| 17 | 16 | oveqd 7448 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑋(+g‘(𝐺 ↾s 𝑆))𝑌)) |
| 18 | 13, 17, 6 | 3eltr4d 2856 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 Grpcgrp 18951 SubGrpcsubg 19138 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-subg 19141 |
| This theorem is referenced by: subgsubcl 19155 subgmulgcl 19157 issubg2 19159 subgint 19168 ssnmz 19184 eqger 19196 eqgcpbl 19200 resghm 19250 ghmpreima 19256 subgga 19318 gasubg 19320 sylow2blem2 19639 lsmmod 19693 pj1ghm 19721 dprdfadd 20040 pgpfac1lem3 20097 subrngacl 20556 subrgacl 20583 islss4 20960 dflidl2rng 21228 mpllsslem 22020 subgntr 24115 taylply2 26409 taylply2OLD 26410 efgh 26583 subgcld 33046 grplsmid 33432 nsgmgclem 33439 nsgqusf1olem1 33441 nsgqusf1olem3 33443 algextdeglem8 33765 |
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