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Theorem issubassa2 21842
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 2725 . . . . 5 (1r𝑊) = (1r𝑊)
3 eqid 2725 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
41, 2, 3rnascl 21841 . . . 4 (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
54ad2antrr 724 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
6 issubassa2.l . . . 4 𝐿 = (LSubSp‘𝑊)
7 assalmod 21811 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
87ad2antrr 724 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑊 ∈ LMod)
9 simpr 483 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑆𝐿)
102subrg1cl 20531 . . . . 5 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) ∈ 𝑆)
1110ad2antlr 725 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → (1r𝑊) ∈ 𝑆)
126, 3, 8, 9, 11lspsnel5a 20892 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ((LSpan‘𝑊)‘{(1r𝑊)}) ⊆ 𝑆)
135, 12eqsstrd 4015 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴𝑆)
14 subrgsubg 20528 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
1514ad2antlr 725 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16 simplll 773 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑊 ∈ AssAlg)
17 simprl 769 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18 eqid 2725 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
1918subrgss 20523 . . . . . . . . 9 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
2019ad2antlr 725 . . . . . . . 8 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ⊆ (Base‘𝑊))
2120sselda 3976 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝑊))
2221adantrl 714 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦 ∈ (Base‘𝑊))
23 eqid 2725 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2725 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25 eqid 2725 . . . . . . 7 (.r𝑊) = (.r𝑊)
26 eqid 2725 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
271, 23, 24, 18, 25, 26asclmul1 21836 . . . . . 6 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
2816, 17, 22, 27syl3anc 1368 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
29 simpllr 774 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30 simplr 767 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴𝑆)
311, 23, 24asclfn 21831 . . . . . . . . . 10 𝐴 Fn (Base‘(Scalar‘𝑊))
3231a1i 11 . . . . . . . . 9 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝐴 Fn (Base‘(Scalar‘𝑊)))
33 fnfvelrn 7089 . . . . . . . . 9 ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3432, 33sylan 578 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3530, 34sseldd 3977 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ 𝑆)
3635adantrr 715 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝐴𝑥) ∈ 𝑆)
37 simprr 771 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦𝑆)
3825subrgmcl 20535 . . . . . 6 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴𝑥) ∈ 𝑆𝑦𝑆) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
3929, 36, 37, 38syl3anc 1368 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
4028, 39eqeltrrd 2826 . . . 4 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4140ralrimivva 3190 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4223, 24, 18, 26, 6islss4 20858 . . . . 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 724 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
4515, 41, 44mpbir2and 711 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆𝐿)
4613, 45impbida 799 1 ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wss 3944  {csn 4630  ran crn 5679   Fn wfn 6544  cfv 6549  (class class class)co 7419  Basecbs 17183  .rcmulr 17237  Scalarcsca 17239   ·𝑠 cvsca 17240  SubGrpcsubg 19083  1rcur 20133  SubRingcsubrg 20518  LModclmod 20755  LSubSpclss 20827  LSpanclspn 20867  AssAlgcasa 21801  algSccascl 21803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-subrng 20495  df-subrg 20520  df-lmod 20757  df-lss 20828  df-lsp 20868  df-assa 21804  df-ascl 21806
This theorem is referenced by:  rnasclassa  21845  aspval2  21848
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