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Mirrors > Home > ILE Home > Th. List > subrgmcl | GIF version |
Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subrgmcl.p | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
subrgmcl | β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β (π Β· π) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . . . 5 β’ (π βΎs π΄) = (π βΎs π΄) | |
2 | 1 | subrgring 13283 | . . . 4 β’ (π΄ β (SubRingβπ ) β (π βΎs π΄) β Ring) |
3 | 2 | 3ad2ant1 1018 | . . 3 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β (π βΎs π΄) β Ring) |
4 | simp2 998 | . . . 4 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β π β π΄) | |
5 | 1 | subrgbas 13289 | . . . . 5 β’ (π΄ β (SubRingβπ ) β π΄ = (Baseβ(π βΎs π΄))) |
6 | 5 | 3ad2ant1 1018 | . . . 4 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β π΄ = (Baseβ(π βΎs π΄))) |
7 | 4, 6 | eleqtrd 2256 | . . 3 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β π β (Baseβ(π βΎs π΄))) |
8 | simp3 999 | . . . 4 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β π β π΄) | |
9 | 8, 6 | eleqtrd 2256 | . . 3 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β π β (Baseβ(π βΎs π΄))) |
10 | eqid 2177 | . . . 4 β’ (Baseβ(π βΎs π΄)) = (Baseβ(π βΎs π΄)) | |
11 | eqid 2177 | . . . 4 β’ (.rβ(π βΎs π΄)) = (.rβ(π βΎs π΄)) | |
12 | 10, 11 | ringcl 13127 | . . 3 β’ (((π βΎs π΄) β Ring β§ π β (Baseβ(π βΎs π΄)) β§ π β (Baseβ(π βΎs π΄))) β (π(.rβ(π βΎs π΄))π) β (Baseβ(π βΎs π΄))) |
13 | 3, 7, 9, 12 | syl3anc 1238 | . 2 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β (π(.rβ(π βΎs π΄))π) β (Baseβ(π βΎs π΄))) |
14 | subrgrcl 13285 | . . . . 5 β’ (π΄ β (SubRingβπ ) β π β Ring) | |
15 | subrgmcl.p | . . . . . 6 β’ Β· = (.rβπ ) | |
16 | 1, 15 | ressmulrg 12595 | . . . . 5 β’ ((π΄ β (SubRingβπ ) β§ π β Ring) β Β· = (.rβ(π βΎs π΄))) |
17 | 14, 16 | mpdan 421 | . . . 4 β’ (π΄ β (SubRingβπ ) β Β· = (.rβ(π βΎs π΄))) |
18 | 17 | 3ad2ant1 1018 | . . 3 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β Β· = (.rβ(π βΎs π΄))) |
19 | 18 | oveqd 5889 | . 2 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β (π Β· π) = (π(.rβ(π βΎs π΄))π)) |
20 | 13, 19, 6 | 3eltr4d 2261 | 1 β’ ((π΄ β (SubRingβπ ) β§ π β π΄ β§ π β π΄) β (π Β· π) β π΄) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ w3a 978 = wceq 1353 β wcel 2148 βcfv 5215 (class class class)co 5872 Basecbs 12454 βΎs cress 12455 .rcmulr 12529 Ringcrg 13110 SubRingcsubrg 13276 |
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-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 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-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-pre-ltirr 7920 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 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-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-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 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-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-pnf 7990 df-mnf 7991 df-ltxr 7993 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-iress 12462 df-plusg 12541 df-mulr 12542 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-subg 12961 df-mgp 13062 df-ring 13112 df-subrg 13278 |
This theorem is referenced by: issubrg2 13300 subrgintm 13302 |
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