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Mirrors > Home > MPE Home > Th. List > grpsubcl | Structured version Visualization version GIF version |
Description: Closure of group subtraction. (Contributed by NM, 31-Mar-2014.) |
Ref | Expression |
---|---|
grpsubcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubcl.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpsubcl | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpsubcl.m | . . 3 ⊢ − = (-g‘𝐺) | |
3 | 1, 2 | grpsubf 17541 | . 2 ⊢ (𝐺 ∈ Grp → − :(𝐵 × 𝐵)⟶𝐵) |
4 | fovrn 6846 | . 2 ⊢ (( − :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) | |
5 | 3, 4 | syl3an1 1399 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 × cxp 5141 ⟶wf 5922 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 Grpcgrp 17469 -gcsg 17471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 |
This theorem is referenced by: grpsubsub 17551 grpsubsub4 17555 grpnpncan 17557 grpnnncan2 17559 dfgrp3 17561 nsgconj 17674 nsgacs 17677 nsgid 17687 ghmnsgpreima 17732 ghmeqker 17734 ghmf1 17736 conjghm 17738 conjnmz 17741 conjnmzb 17742 sylow3lem2 18089 abladdsub4 18265 abladdsub 18266 ablpncan3 18268 ablsubsub4 18270 ablpnpcan 18271 ablnnncan 18274 ablnnncan1 18275 telgsumfzslem 18431 telgsumfzs 18432 telgsums 18436 lmodvsubcl 18956 lvecvscan2 19160 coe1subfv 19684 evl1subd 19754 ipsubdir 20035 ipsubdi 20036 ip2subdi 20037 dmatsubcl 20352 scmatsubcl 20371 mdetunilem9 20474 mdetuni0 20475 chmatcl 20681 chpmat1d 20689 chpdmatlem1 20691 chpscmat 20695 chpidmat 20700 chfacfisf 20707 cpmadugsumlemF 20729 cpmidgsum2 20732 tgpconncomp 21963 ghmcnp 21965 nrmmetd 22426 ngpds2 22457 ngpds3 22459 isngp4 22463 nmsub 22474 nm2dif 22476 nmtri2 22478 subgngp 22486 ngptgp 22487 nrgdsdi 22516 nrgdsdir 22517 nlmdsdi 22532 nlmdsdir 22533 nrginvrcnlem 22542 nmods 22595 tchcphlem1 23080 tchcph 23082 cphipval2 23086 4cphipval2 23087 cphipval 23088 ipcnlem2 23089 deg1sublt 23915 ply1divmo 23940 ply1divex 23941 r1pcl 23962 r1pid 23964 ply1remlem 23967 ig1peu 23976 dchr2sum 25043 lgsqrlem2 25117 lgsqrlem3 25118 lgsqrlem4 25119 ttgcontlem1 25810 ogrpsublt 29850 archiabllem1a 29873 archiabllem2a 29876 archiabllem2c 29877 ornglmulle 29933 orngrmulle 29934 lclkrlem2m 37125 idomrootle 38090 lidldomn1 42246 linply1 42506 |
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