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Theorem issubassa 19145
 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 1057 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑊 ∈ AssAlg)
2 assaring 19141 . . . . . 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 19141 . . . . . . 7 (𝑆 ∈ AssAlg → 𝑆 ∈ Ring)
65adantl 481 . . . . . 6 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ Ring)
74, 6syl5eqelr 2693 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊s 𝐴) ∈ Ring)
83, 7jca 553 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring))
9 simpl3 1059 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝑉)
10 simpl2 1058 . . . . 5 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 1𝐴)
119, 10jca 553 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝑉1𝐴))
12 issubassa.v . . . . 5 𝑉 = (Base‘𝑊)
13 issubassa.o . . . . 5 1 = (1r𝑊)
1412, 13issubrg 18603 . . . 4 (𝐴 ∈ (SubRing‘𝑊) ↔ ((𝑊 ∈ Ring ∧ (𝑊s 𝐴) ∈ Ring) ∧ (𝐴𝑉1𝐴)))
158, 11, 14sylanbrc 695 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴 ∈ (SubRing‘𝑊))
16 assalmod 19140 . . . . 5 (𝑆 ∈ AssAlg → 𝑆 ∈ LMod)
1716adantl 481 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝑆 ∈ LMod)
18 assalmod 19140 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
19 issubassa.l . . . . . 6 𝐿 = (LSubSp‘𝑊)
204, 12, 19islss3 18780 . . . . 5 (𝑊 ∈ LMod → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
211, 18, 203syl 18 . . . 4 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴𝐿 ↔ (𝐴𝑉𝑆 ∈ LMod)))
229, 17, 21mpbir2and 959 . . 3 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → 𝐴𝐿)
2315, 22jca 553 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ 𝑆 ∈ AssAlg) → (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿))
2412subrgss 18604 . . . . . 6 (𝐴 ∈ (SubRing‘𝑊) → 𝐴𝑉)
2524ad2antrl 760 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴𝑉)
264, 12ressbas2 15758 . . . . 5 (𝐴𝑉𝐴 = (Base‘𝑆))
2725, 26syl 17 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝐴 = (Base‘𝑆))
28 eqid 2610 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
294, 28resssca 15854 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (Scalar‘𝑊) = (Scalar‘𝑆))
3029ad2antrl 760 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
31 eqidd 2611 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
32 eqid 2610 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
334, 32ressvsca 15855 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
3433ad2antrl 760 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → ( ·𝑠𝑊) = ( ·𝑠𝑆))
35 eqid 2610 . . . . . 6 (.r𝑊) = (.r𝑊)
364, 35ressmulr 15829 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r𝑆))
3736ad2antrl 760 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (.r𝑊) = (.r𝑆))
38 simpr 476 . . . . 5 ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿) → 𝐴𝐿)
394, 19lsslmod 18781 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐴𝐿) → 𝑆 ∈ LMod)
4018, 38, 39syl2an 493 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ LMod)
414subrgring 18606 . . . . 5 (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
4241ad2antrl 760 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ Ring)
4328assasca 19142 . . . . 5 (𝑊 ∈ AssAlg → (Scalar‘𝑊) ∈ CRing)
4443adantr 480 . . . 4 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → (Scalar‘𝑊) ∈ CRing)
45 simpll 786 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑊 ∈ AssAlg)
46 simpr1 1060 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
4725adantr 480 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝐴𝑉)
48 simpr2 1061 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝐴)
4947, 48sseldd 3569 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑦𝑉)
50 simpr3 1062 . . . . . 6 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝐴)
5147, 50sseldd 3569 . . . . 5 (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝐴𝑧𝐴)) → 𝑧𝑉)
52 eqid 2610 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5312, 28, 52, 32, 35assaass 19138 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → ((𝑥( ·𝑠𝑊)𝑦)(.r𝑊)𝑧) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5445, 46, 49, 51, 53syl13anc 1320 . . . 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 19139 . . . . 5 ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑉𝑧𝑉)) → (𝑦(.r𝑊)(𝑥( ·𝑠𝑊)𝑧)) = (𝑥( ·𝑠𝑊)(𝑦(.r𝑊)𝑧)))
5645, 46, 49, 51, 55syl13anc 1320 . . . 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 19144 . . 3 ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
58573ad2antl1 1216 . 2 (((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)) → 𝑆 ∈ AssAlg)
5923, 58impbida 873 1 ((𝑊 ∈ AssAlg ∧ 1𝐴𝐴𝑉) → (𝑆 ∈ AssAlg ↔ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴𝐿)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  .rcmulr 15769  Scalarcsca 15771   ·𝑠 cvsca 15772  1rcur 18324  Ringcrg 18370  CRingccrg 18371  SubRingcsubrg 18599  LModclmod 18686  LSubSpclss 18753  AssAlgcasa 19130 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-assa 19133 This theorem is referenced by:  mplassa  19275  ply1assa  19390
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