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Mirrors > Home > ILE Home > Th. List > subgmulgcl | GIF version |
Description: Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
subgmulgcl.t | โข ยท = (.gโ๐บ) |
Ref | Expression |
---|---|
subgmulgcl | โข ((๐ โ (SubGrpโ๐บ) โง ๐ โ โค โง ๐ โ ๐) โ (๐ ยท ๐) โ ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . 2 โข (Baseโ๐บ) = (Baseโ๐บ) | |
2 | subgmulgcl.t | . 2 โข ยท = (.gโ๐บ) | |
3 | eqid 2177 | . 2 โข (+gโ๐บ) = (+gโ๐บ) | |
4 | subgrcl 12970 | . 2 โข (๐ โ (SubGrpโ๐บ) โ ๐บ โ Grp) | |
5 | 1 | subgss 12965 | . 2 โข (๐ โ (SubGrpโ๐บ) โ ๐ โ (Baseโ๐บ)) |
6 | 3 | subgcl 12975 | . 2 โข ((๐ โ (SubGrpโ๐บ) โง ๐ฅ โ ๐ โง ๐ฆ โ ๐) โ (๐ฅ(+gโ๐บ)๐ฆ) โ ๐) |
7 | eqid 2177 | . 2 โข (0gโ๐บ) = (0gโ๐บ) | |
8 | 7 | subg0cl 12973 | . 2 โข (๐ โ (SubGrpโ๐บ) โ (0gโ๐บ) โ ๐) |
9 | eqid 2177 | . 2 โข (invgโ๐บ) = (invgโ๐บ) | |
10 | 9 | subginvcl 12974 | . 2 โข ((๐ โ (SubGrpโ๐บ) โง ๐ฅ โ ๐) โ ((invgโ๐บ)โ๐ฅ) โ ๐) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mulgsubcl 12929 | 1 โข ((๐ โ (SubGrpโ๐บ) โง ๐ โ โค โง ๐ โ ๐) โ (๐ ยท ๐) โ ๐) |
Colors of variables: wff set class |
Syntax hints: โ wi 4 โง w3a 978 = wceq 1353 โ wcel 2148 โcfv 5215 (class class class)co 5872 โคcz 9249 Basecbs 12454 +gcplusg 12528 0gc0g 12693 Grpcgrp 12809 invgcminusg 12810 .gcmg 12915 SubGrpcsubg 12958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-2 8974 df-n0 9173 df-z 9250 df-uz 9525 df-seqfrec 10441 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-mulg 12916 df-subg 12961 |
This theorem is referenced by: subgmulg 12979 zsssubrg 13348 |
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