MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubassa2 Structured version   Visualization version   GIF version

Theorem issubassa2 21176
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 2736 . . . . 5 (1r𝑊) = (1r𝑊)
3 eqid 2736 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
41, 2, 3rnascl 21175 . . . 4 (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
54ad2antrr 723 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
6 issubassa2.l . . . 4 𝐿 = (LSubSp‘𝑊)
7 assalmod 21147 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
87ad2antrr 723 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑊 ∈ LMod)
9 simpr 485 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑆𝐿)
102subrg1cl 20111 . . . . 5 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) ∈ 𝑆)
1110ad2antlr 724 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → (1r𝑊) ∈ 𝑆)
126, 3, 8, 9, 11lspsnel5a 20338 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ((LSpan‘𝑊)‘{(1r𝑊)}) ⊆ 𝑆)
135, 12eqsstrd 3968 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴𝑆)
14 subrgsubg 20109 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
1514ad2antlr 724 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16 simplll 772 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑊 ∈ AssAlg)
17 simprl 768 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18 eqid 2736 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
1918subrgss 20104 . . . . . . . . 9 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
2019ad2antlr 724 . . . . . . . 8 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ⊆ (Base‘𝑊))
2120sselda 3930 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝑊))
2221adantrl 713 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦 ∈ (Base‘𝑊))
23 eqid 2736 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2736 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25 eqid 2736 . . . . . . 7 (.r𝑊) = (.r𝑊)
26 eqid 2736 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
271, 23, 24, 18, 25, 26asclmul1 21170 . . . . . 6 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
2816, 17, 22, 27syl3anc 1370 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
29 simpllr 773 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30 simplr 766 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴𝑆)
311, 23, 24asclfn 21165 . . . . . . . . . 10 𝐴 Fn (Base‘(Scalar‘𝑊))
3231a1i 11 . . . . . . . . 9 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝐴 Fn (Base‘(Scalar‘𝑊)))
33 fnfvelrn 6997 . . . . . . . . 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 3931 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ 𝑆)
3635adantrr 714 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝐴𝑥) ∈ 𝑆)
37 simprr 770 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦𝑆)
3825subrgmcl 20115 . . . . . 6 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴𝑥) ∈ 𝑆𝑦𝑆) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
3929, 36, 37, 38syl3anc 1370 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
4028, 39eqeltrrd 2838 . . . 4 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4140ralrimivva 3193 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4223, 24, 18, 26, 6islss4 20304 . . . . 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 723 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
4515, 41, 44mpbir2and 710 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆𝐿)
4613, 45impbida 798 1 ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  wss 3896  {csn 4570  ran crn 5608   Fn wfn 6460  cfv 6465  (class class class)co 7316  Basecbs 16986  .rcmulr 17037  Scalarcsca 17039   ·𝑠 cvsca 17040  SubGrpcsubg 18822  1rcur 19809  SubRingcsubrg 20099  LModclmod 20203  LSubSpclss 20273  LSpanclspn 20313  AssAlgcasa 21137  algSccascl 21139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-er 8547  df-en 8783  df-dom 8784  df-sdom 8785  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-3 12116  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-plusg 17049  df-mulr 17050  df-0g 17226  df-mgm 18400  df-sgrp 18449  df-mnd 18460  df-grp 18653  df-minusg 18654  df-sbg 18655  df-subg 18825  df-mgp 19793  df-ur 19810  df-ring 19857  df-subrg 20101  df-lmod 20205  df-lss 20274  df-lsp 20314  df-assa 21140  df-ascl 21142
This theorem is referenced by:  rnasclassa  21179  aspval2  21182
  Copyright terms: Public domain W3C validator