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Theorem issubassa 19598
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 1242 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ AssAlg)
2 assaring 19594 . . . . . 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 19594 . . . . . . 7 (𝑆 ∈ AssAlg → 𝑆 ∈ Ring)
65adantl 473 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ Ring)
74, 6syl5eqelr 2849 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊s 𝐴) ∈ Ring)
83, 7jca 507 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring))
9 simpl3 1246 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝑉)
10 simpl2 1244 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 1𝐴)
119, 10jca 507 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝑉1𝐴))
12 issubassa.v . . . . 5 𝑉 = (Base‘𝑊)
13 issubassa.o . . . . 5 1 = (1r𝑊)
1412, 13issubrg 19049 . . . 4 (𝐴 ∈ (SubRing‘𝑊) ↔ ((𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring) ∧ (𝐴𝑉1𝐴)))
158, 11, 14sylanbrc 578 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ (SubRing‘𝑊))
16 assalmod 19593 . . . . 5 (𝑆 ∈ AssAlg → 𝑆 ∈ LMod)
1716adantl 473 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ LMod)
18 assalmod 19593 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
19 issubassa.l . . . . . 6 𝐿 = (LSubSp‘𝑊)
204, 12, 19islss3 19231 . . . . 5 (𝑊 ∈ LMod → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
211, 18, 203syl 18 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
229, 17, 21mpbir2and 704 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝐿)
2315, 22jca 507 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿))
2412subrgss 19050 . . . . . 6 (𝐴 ∈ (SubRing‘𝑊) → 𝐴𝑉)
2524ad2antrl 719 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴𝑉)
264, 12ressbas2 16205 . . . . 5 (𝐴𝑉𝐴 = (Base‘𝑆))
2725, 26syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴 = (Base‘𝑆))
28 eqid 2765 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
294, 28resssca 16305 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑆))
3029ad2antrl 719 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
31 eqidd 2766 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
32 eqid 2765 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
334, 32ressvsca 16306 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
3433ad2antrl 719 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
35 eqid 2765 . . . . . 6 (.r𝑊) = (.r𝑊)
364, 35ressmulr 16280 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r𝑆))
3736ad2antrl 719 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (.r𝑊) = (.r𝑆))
38 simpr 477 . . . . 5 ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿) → 𝐴𝐿)
394, 19lsslmod 19232 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐴𝐿) → 𝑆 ∈ LMod)
4018, 38, 39syl2an 589 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ LMod)
414subrgring 19052 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
4241ad2antrl 719 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ Ring)
4328assasca 19595 . . . . 5 (𝑊 ∈ AssAlg → (Scalar‘𝑊) ∈ CRing)
4443adantr 472 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) ∈ CRing)
45 simpll 783 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑊 ∈ AssAlg)
46 simpr1 1248 . . . . 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 1250 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
4947, 48sseldd 3762 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝑉)
50 simpr3 1252 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
5147, 50sseldd 3762 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝑉)
52 eqid 2765 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5312, 28, 52, 32, 35assaass 19591 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → ((𝑥( ·𝑠𝑊)𝑦)(.r𝑊)𝑧) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5445, 46, 49, 51, 53syl13anc 1491 . . . 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 19592 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → (𝑦(.r𝑊)(𝑥( ·𝑠𝑊)𝑧)) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5645, 46, 49, 51, 55syl13anc 1491 . . . 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 19597 . . 3 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
58573ad2antl1 1236 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
5923, 58impbida 835 1 ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wss 3732  cfv 6068  (class class class)co 6842  Basecbs 16132  s cress 16133  .rcmulr 16217  Scalarcsca 16219   ·𝑠 cvsca 16220  1rcur 18768  Ringcrg 18814  CRingccrg 18815  SubRingcsubrg 19045  LModclmod 19132  LSubSpclss 19201  AssAlgcasa 19583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-sca 16232  df-vsca 16233  df-0g 16370  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-grp 17694  df-minusg 17695  df-sbg 17696  df-subg 17857  df-mgp 18757  df-ur 18769  df-ring 18816  df-subrg 19047  df-lmod 19134  df-lss 19202  df-assa 19586
This theorem is referenced by:  mplassa  19728  ply1assa  19842
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