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Theorem issubassa2 20049
Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
Hypotheses
Ref Expression
issubassa2.a 𝐴 = (algSc‘𝑊)
issubassa2.l 𝐿 = (LSubSp‘𝑊)
Assertion
Ref Expression
issubassa2 ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))

Proof of Theorem issubassa2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issubassa2.a . . . . 5 𝐴 = (algSc‘𝑊)
2 eqid 2818 . . . . 5 (1r𝑊) = (1r𝑊)
3 eqid 2818 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
41, 2, 3rnascl 20048 . . . 4 (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
54ad2antrr 722 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
6 issubassa2.l . . . 4 𝐿 = (LSubSp‘𝑊)
7 assalmod 20020 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
87ad2antrr 722 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑊 ∈ LMod)
9 simpr 485 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑆𝐿)
102subrg1cl 19472 . . . . 5 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) ∈ 𝑆)
1110ad2antlr 723 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → (1r𝑊) ∈ 𝑆)
126, 3, 8, 9, 11lspsnel5a 19697 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ((LSpan‘𝑊)‘{(1r𝑊)}) ⊆ 𝑆)
135, 12eqsstrd 4002 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴𝑆)
14 subrgsubg 19470 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
1514ad2antlr 723 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16 simplll 771 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑊 ∈ AssAlg)
17 simprl 767 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18 eqid 2818 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
1918subrgss 19465 . . . . . . . . 9 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
2019ad2antlr 723 . . . . . . . 8 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ⊆ (Base‘𝑊))
2120sselda 3964 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝑊))
2221adantrl 712 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦 ∈ (Base‘𝑊))
23 eqid 2818 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2818 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25 eqid 2818 . . . . . . 7 (.r𝑊) = (.r𝑊)
26 eqid 2818 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
271, 23, 24, 18, 25, 26asclmul1 20042 . . . . . 6 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
2816, 17, 22, 27syl3anc 1363 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
29 simpllr 772 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30 simplr 765 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴𝑆)
311, 23, 24asclfn 20038 . . . . . . . . . 10 𝐴 Fn (Base‘(Scalar‘𝑊))
3231a1i 11 . . . . . . . . 9 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝐴 Fn (Base‘(Scalar‘𝑊)))
33 fnfvelrn 6840 . . . . . . . . 9 ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3432, 33sylan 580 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3530, 34sseldd 3965 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ 𝑆)
3635adantrr 713 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝐴𝑥) ∈ 𝑆)
37 simprr 769 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦𝑆)
3825subrgmcl 19476 . . . . . 6 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴𝑥) ∈ 𝑆𝑦𝑆) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
3929, 36, 37, 38syl3anc 1363 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
4028, 39eqeltrrd 2911 . . . 4 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4140ralrimivva 3188 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4223, 24, 18, 26, 6islss4 19663 . . . . 5 (𝑊 ∈ LMod → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
437, 42syl 17 . . . 4 (𝑊 ∈ AssAlg → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
4443ad2antrr 722 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
4515, 41, 44mpbir2and 709 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆𝐿)
4613, 45impbida 797 1 ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  {csn 4557  ran crn 5549   Fn wfn 6343  cfv 6348  (class class class)co 7145  Basecbs 16471  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  SubGrpcsubg 18211  1rcur 19180  SubRingcsubrg 19460  LModclmod 19563  LSubSpclss 19632  LSpanclspn 19672  AssAlgcasa 20010  algSccascl 20012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-subg 18214  df-mgp 19169  df-ur 19181  df-ring 19228  df-subrg 19462  df-lmod 19565  df-lss 19633  df-lsp 19673  df-assa 20013  df-ascl 20015
This theorem is referenced by:  rnasclassa  20052  aspval2  20055
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