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Theorem issubassa 19518
Description: The subalgebras of an associative algebra are exactly the subrings (under the ring multiplication) that are simultaneously subspaces (under the scalar multiplication from the vector space). (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
issubassa.s 𝑆 = (𝑊s 𝐴)
issubassa.l 𝐿 = (LSubSp‘𝑊)
issubassa.v 𝑉 = (Base‘𝑊)
issubassa.o 1 = (1r𝑊)
Assertion
Ref Expression
issubassa ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))

Proof of Theorem issubassa
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1225 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ AssAlg)
2 assaring 19514 . . . . . 6 (𝑊 ∈ AssAlg → 𝑊 ∈ Ring)
31, 2syl 17 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ Ring)
4 issubassa.s . . . . . 6 𝑆 = (𝑊s 𝐴)
5 assaring 19514 . . . . . . 7 (𝑆 ∈ AssAlg → 𝑆 ∈ Ring)
65adantl 473 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ Ring)
74, 6syl5eqelr 2836 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊s 𝐴) ∈ Ring)
83, 7jca 555 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring))
9 simpl3 1229 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝑉)
10 simpl2 1227 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 1𝐴)
119, 10jca 555 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝑉1𝐴))
12 issubassa.v . . . . 5 𝑉 = (Base‘𝑊)
13 issubassa.o . . . . 5 1 = (1r𝑊)
1412, 13issubrg 18974 . . . 4 (𝐴 ∈ (SubRing‘𝑊) ↔ ((𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring) ∧ (𝐴𝑉1𝐴)))
158, 11, 14sylanbrc 701 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ (SubRing‘𝑊))
16 assalmod 19513 . . . . 5 (𝑆 ∈ AssAlg → 𝑆 ∈ LMod)
1716adantl 473 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ LMod)
18 assalmod 19513 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
19 issubassa.l . . . . . 6 𝐿 = (LSubSp‘𝑊)
204, 12, 19islss3 19153 . . . . 5 (𝑊 ∈ LMod → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
211, 18, 203syl 18 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
229, 17, 21mpbir2and 995 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝐿)
2315, 22jca 555 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿))
2412subrgss 18975 . . . . . 6 (𝐴 ∈ (SubRing‘𝑊) → 𝐴𝑉)
2524ad2antrl 766 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴𝑉)
264, 12ressbas2 16125 . . . . 5 (𝐴𝑉𝐴 = (Base‘𝑆))
2725, 26syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴 = (Base‘𝑆))
28 eqid 2752 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
294, 28resssca 16225 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑆))
3029ad2antrl 766 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
31 eqidd 2753 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
32 eqid 2752 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
334, 32ressvsca 16226 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
3433ad2antrl 766 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
35 eqid 2752 . . . . . 6 (.r𝑊) = (.r𝑊)
364, 35ressmulr 16200 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r𝑆))
3736ad2antrl 766 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (.r𝑊) = (.r𝑆))
38 simpr 479 . . . . 5 ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿) → 𝐴𝐿)
394, 19lsslmod 19154 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐴𝐿) → 𝑆 ∈ LMod)
4018, 38, 39syl2an 495 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ LMod)
414subrgring 18977 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
4241ad2antrl 766 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ Ring)
4328assasca 19515 . . . . 5 (𝑊 ∈ AssAlg → (Scalar‘𝑊) ∈ CRing)
4443adantr 472 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) ∈ CRing)
45 simpll 807 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑊 ∈ AssAlg)
46 simpr1 1231 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
4725adantr 472 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝐴𝑉)
48 simpr2 1233 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
4947, 48sseldd 3737 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝑉)
50 simpr3 1235 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
5147, 50sseldd 3737 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝑉)
52 eqid 2752 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5312, 28, 52, 32, 35assaass 19511 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → ((𝑥( ·𝑠𝑊)𝑦)(.r𝑊)𝑧) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5445, 46, 49, 51, 53syl13anc 1475 . . . 4 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → ((𝑥( ·𝑠𝑊)𝑦)(.r𝑊)𝑧) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5512, 28, 52, 32, 35assaassr 19512 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → (𝑦(.r𝑊)(𝑥( ·𝑠𝑊)𝑧)) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5645, 46, 49, 51, 55syl13anc 1475 . . . 4 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → (𝑦(.r𝑊)(𝑥( ·𝑠𝑊)𝑧)) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5727, 30, 31, 34, 37, 40, 42, 44, 54, 56isassad 19517 . . 3 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
58573ad2antl1 1198 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
5923, 58impbida 913 1 ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wss 3707  cfv 6041  (class class class)co 6805  Basecbs 16051  s cress 16052  .rcmulr 16136  Scalarcsca 16138   ·𝑠 cvsca 16139  1rcur 18693  Ringcrg 18739  CRingccrg 18740  SubRingcsubrg 18970  LModclmod 19057  LSubSpclss 19126  AssAlgcasa 19503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-mulr 16149  df-sca 16151  df-vsca 16152  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619  df-sbg 17620  df-subg 17784  df-mgp 18682  df-ur 18694  df-ring 18741  df-subrg 18972  df-lmod 19059  df-lss 19127  df-assa 19506
This theorem is referenced by:  mplassa  19648  ply1assa  19763
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