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Mirrors > Home > MPE Home > Th. List > subrgmcl | Structured version Visualization version GIF version |
Description: A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subrgmcl.p | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
subrgmcl | ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (𝑅 ↾s 𝐴) = (𝑅 ↾s 𝐴) | |
2 | 1 | subrgring 19650 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑅 ↾s 𝐴) ∈ Ring) |
3 | 2 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑅 ↾s 𝐴) ∈ Ring) |
4 | simp2 1138 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
5 | 1 | subrgbas 19656 | . . . . 5 ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
6 | 5 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝐴 = (Base‘(𝑅 ↾s 𝐴))) |
7 | 4, 6 | eleqtrd 2835 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴))) |
8 | simp3 1139 | . . . 4 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴) | |
9 | 8, 6 | eleqtrd 2835 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ (Base‘(𝑅 ↾s 𝐴))) |
10 | eqid 2738 | . . . 4 ⊢ (Base‘(𝑅 ↾s 𝐴)) = (Base‘(𝑅 ↾s 𝐴)) | |
11 | eqid 2738 | . . . 4 ⊢ (.r‘(𝑅 ↾s 𝐴)) = (.r‘(𝑅 ↾s 𝐴)) | |
12 | 10, 11 | ringcl 19426 | . . 3 ⊢ (((𝑅 ↾s 𝐴) ∈ Ring ∧ 𝑋 ∈ (Base‘(𝑅 ↾s 𝐴)) ∧ 𝑌 ∈ (Base‘(𝑅 ↾s 𝐴))) → (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
13 | 3, 7, 9, 12 | syl3anc 1372 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌) ∈ (Base‘(𝑅 ↾s 𝐴))) |
14 | subrgmcl.p | . . . . 5 ⊢ · = (.r‘𝑅) | |
15 | 1, 14 | ressmulr 16721 | . . . 4 ⊢ (𝐴 ∈ (SubRing‘𝑅) → · = (.r‘(𝑅 ↾s 𝐴))) |
16 | 15 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → · = (.r‘(𝑅 ↾s 𝐴))) |
17 | 16 | oveqd 7181 | . 2 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) = (𝑋(.r‘(𝑅 ↾s 𝐴))𝑌)) |
18 | 13, 17, 6 | 3eltr4d 2848 | 1 ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 Basecbs 16579 ↾s cress 16580 .rcmulr 16662 Ringcrg 19409 SubRingcsubrg 19643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-subg 18387 df-mgp 19352 df-ring 19411 df-subrg 19645 |
This theorem is referenced by: issubrg2 19667 subrgint 19669 abvres 19722 cnsubrg 20270 issubassa2 20699 rnasclmulcl 20701 resspsrmul 20789 mplbas2 20846 mpfmulcl 20913 pf1mulcl 21117 chfacfisfcpmat 21599 clmmcl 23830 cphdivcl 23927 cphabscl 23930 cphsqrtcl2 23931 cphsqrtcl3 23932 plypf1 24953 dvply2g 25025 taylply2 25107 idlinsubrg 31172 fedgmullem2 31275 cnsrexpcl 40546 cnsrplycl 40548 rngunsnply 40554 |
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