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Theorem issubassa2 19267
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 2621 . . . . 5 (1r𝑊) = (1r𝑊)
3 eqid 2621 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
41, 2, 3rnascl 19265 . . . 4 (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
54ad2antrr 761 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r𝑊)}))
6 issubassa2.l . . . 4 𝐿 = (LSubSp‘𝑊)
7 assalmod 19241 . . . . 5 (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
87ad2antrr 761 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑊 ∈ LMod)
9 simpr 477 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → 𝑆𝐿)
102subrg1cl 18712 . . . . 5 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) ∈ 𝑆)
1110ad2antlr 762 . . . 4 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → (1r𝑊) ∈ 𝑆)
126, 3, 8, 9, 11lspsnel5a 18918 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ((LSpan‘𝑊)‘{(1r𝑊)}) ⊆ 𝑆)
135, 12eqsstrd 3620 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆𝐿) → ran 𝐴𝑆)
14 subrgsubg 18710 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
1514ad2antlr 762 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16 simplll 797 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑊 ∈ AssAlg)
17 simprl 793 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18 eqid 2621 . . . . . . . . . 10 (Base‘𝑊) = (Base‘𝑊)
1918subrgss 18705 . . . . . . . . 9 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
2019ad2antlr 762 . . . . . . . 8 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆 ⊆ (Base‘𝑊))
2120sselda 3584 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑦𝑆) → 𝑦 ∈ (Base‘𝑊))
2221adantrl 751 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦 ∈ (Base‘𝑊))
23 eqid 2621 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25 eqid 2621 . . . . . . 7 (.r𝑊) = (.r𝑊)
26 eqid 2621 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
271, 23, 24, 18, 25, 26asclmul1 19261 . . . . . 6 ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
2816, 17, 22, 27syl3anc 1323 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) = (𝑥( ·𝑠𝑊)𝑦))
29 simpllr 798 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30 simplr 791 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴𝑆)
311, 23, 24asclfn 19258 . . . . . . . . . 10 𝐴 Fn (Base‘(Scalar‘𝑊))
3231a1i 11 . . . . . . . . 9 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝐴 Fn (Base‘(Scalar‘𝑊)))
33 fnfvelrn 6314 . . . . . . . . 9 ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3432, 33sylan 488 . . . . . . . 8 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ ran 𝐴)
3530, 34sseldd 3585 . . . . . . 7 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴𝑥) ∈ 𝑆)
3635adantrr 752 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝐴𝑥) ∈ 𝑆)
37 simprr 795 . . . . . 6 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → 𝑦𝑆)
3825subrgmcl 18716 . . . . . 6 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴𝑥) ∈ 𝑆𝑦𝑆) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
3929, 36, 37, 38syl3anc 1323 . . . . 5 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → ((𝐴𝑥)(.r𝑊)𝑦) ∈ 𝑆)
4028, 39eqeltrrd 2699 . . . 4 ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦𝑆)) → (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4140ralrimivva 2965 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)
4223, 24, 18, 26, 6islss4 18884 . . . . 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 761 . . 3 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → (𝑆𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦𝑆 (𝑥( ·𝑠𝑊)𝑦) ∈ 𝑆)))
4515, 41, 44mpbir2and 956 . 2 (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴𝑆) → 𝑆𝐿)
4613, 45impbida 876 1 ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆𝐿 ↔ ran 𝐴𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3556  {csn 4150  ran crn 5077   Fn wfn 5844  cfv 5849  (class class class)co 6607  Basecbs 15784  .rcmulr 15866  Scalarcsca 15868   ·𝑠 cvsca 15869  SubGrpcsubg 17512  1rcur 18425  SubRingcsubrg 18700  LModclmod 18787  LSubSpclss 18854  LSpanclspn 18893  AssAlgcasa 19231  algSccascl 19233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-ndx 15787  df-slot 15788  df-base 15789  df-sets 15790  df-ress 15791  df-plusg 15878  df-mulr 15879  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-subg 17515  df-mgp 18414  df-ur 18426  df-ring 18473  df-subrg 18702  df-lmod 18789  df-lss 18855  df-lsp 18894  df-assa 19234  df-ascl 19236
This theorem is referenced by:  aspval2  19269
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