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Theorem issubassa 19246
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 1062 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ AssAlg)
2 assaring 19242 . . . . . 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 19242 . . . . . . 7 (𝑆 ∈ AssAlg → 𝑆 ∈ Ring)
65adantl 482 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ Ring)
74, 6syl5eqelr 2703 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊s 𝐴) ∈ Ring)
83, 7jca 554 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring))
9 simpl3 1064 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝑉)
10 simpl2 1063 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 1𝐴)
119, 10jca 554 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝑉1𝐴))
12 issubassa.v . . . . 5 𝑉 = (Base‘𝑊)
13 issubassa.o . . . . 5 1 = (1r𝑊)
1412, 13issubrg 18704 . . . 4 (𝐴 ∈ (SubRing‘𝑊) ↔ ((𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring) ∧ (𝐴𝑉1𝐴)))
158, 11, 14sylanbrc 697 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ (SubRing‘𝑊))
16 assalmod 19241 . . . . 5 (𝑆 ∈ AssAlg → 𝑆 ∈ LMod)
1716adantl 482 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ LMod)
18 assalmod 19241 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
19 issubassa.l . . . . . 6 𝐿 = (LSubSp‘𝑊)
204, 12, 19islss3 18881 . . . . 5 (𝑊 ∈ LMod → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
211, 18, 203syl 18 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
229, 17, 21mpbir2and 956 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝐿)
2315, 22jca 554 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿))
2412subrgss 18705 . . . . . 6 (𝐴 ∈ (SubRing‘𝑊) → 𝐴𝑉)
2524ad2antrl 763 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴𝑉)
264, 12ressbas2 15855 . . . . 5 (𝐴𝑉𝐴 = (Base‘𝑆))
2725, 26syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴 = (Base‘𝑆))
28 eqid 2621 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
294, 28resssca 15955 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑆))
3029ad2antrl 763 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
31 eqidd 2622 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
32 eqid 2621 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
334, 32ressvsca 15956 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
3433ad2antrl 763 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
35 eqid 2621 . . . . . 6 (.r𝑊) = (.r𝑊)
364, 35ressmulr 15930 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r𝑆))
3736ad2antrl 763 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (.r𝑊) = (.r𝑆))
38 simpr 477 . . . . 5 ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿) → 𝐴𝐿)
394, 19lsslmod 18882 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐴𝐿) → 𝑆 ∈ LMod)
4018, 38, 39syl2an 494 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ LMod)
414subrgring 18707 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
4241ad2antrl 763 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ Ring)
4328assasca 19243 . . . . 5 (𝑊 ∈ AssAlg → (Scalar‘𝑊) ∈ CRing)
4443adantr 481 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) ∈ CRing)
45 simpll 789 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑊 ∈ AssAlg)
46 simpr1 1065 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
4725adantr 481 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝐴𝑉)
48 simpr2 1066 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
4947, 48sseldd 3585 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝑉)
50 simpr3 1067 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
5147, 50sseldd 3585 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝑉)
52 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5312, 28, 52, 32, 35assaass 19239 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → ((𝑥( ·𝑠𝑊)𝑦)(.r𝑊)𝑧) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5445, 46, 49, 51, 53syl13anc 1325 . . . 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 19240 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → (𝑦(.r𝑊)(𝑥( ·𝑠𝑊)𝑧)) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5645, 46, 49, 51, 55syl13anc 1325 . . . 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 19245 . . 3 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
58573ad2antl1 1221 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
5923, 58impbida 876 1 ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wss 3556  cfv 5849  (class class class)co 6607  Basecbs 15784  s cress 15785  .rcmulr 15866  Scalarcsca 15868   ·𝑠 cvsca 15869  1rcur 18425  Ringcrg 18471  CRingccrg 18472  SubRingcsubrg 18700  LModclmod 18787  LSubSpclss 18854  AssAlgcasa 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-sca 15881  df-vsca 15882  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-subg 17515  df-mgp 18414  df-ur 18426  df-ring 18473  df-subrg 18702  df-lmod 18789  df-lss 18855  df-assa 19234
This theorem is referenced by:  mplassa  19376  ply1assa  19491
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