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Theorem issubassa2 21666
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 = (algScβ€˜π‘Š)
issubassa2.l 𝐿 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
issubassa2 ((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) β†’ (𝑆 ∈ 𝐿 ↔ ran 𝐴 βŠ† 𝑆))

Proof of Theorem issubassa2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issubassa2.a . . . . 5 𝐴 = (algScβ€˜π‘Š)
2 eqid 2731 . . . . 5 (1rβ€˜π‘Š) = (1rβ€˜π‘Š)
3 eqid 2731 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
41, 2, 3rnascl 21665 . . . 4 (π‘Š ∈ AssAlg β†’ ran 𝐴 = ((LSpanβ€˜π‘Š)β€˜{(1rβ€˜π‘Š)}))
54ad2antrr 723 . . 3 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ ran 𝐴 = ((LSpanβ€˜π‘Š)β€˜{(1rβ€˜π‘Š)}))
6 issubassa2.l . . . 4 𝐿 = (LSubSpβ€˜π‘Š)
7 assalmod 21635 . . . . 5 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
87ad2antrr 723 . . . 4 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ π‘Š ∈ LMod)
9 simpr 484 . . . 4 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ 𝑆 ∈ 𝐿)
102subrg1cl 20471 . . . . 5 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (1rβ€˜π‘Š) ∈ 𝑆)
1110ad2antlr 724 . . . 4 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ (1rβ€˜π‘Š) ∈ 𝑆)
126, 3, 8, 9, 11lspsnel5a 20752 . . 3 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ ((LSpanβ€˜π‘Š)β€˜{(1rβ€˜π‘Š)}) βŠ† 𝑆)
135, 12eqsstrd 4021 . 2 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ 𝑆 ∈ 𝐿) β†’ ran 𝐴 βŠ† 𝑆)
14 subrgsubg 20468 . . . 4 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Š))
1514ad2antlr 724 . . 3 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ 𝑆 ∈ (SubGrpβ€˜π‘Š))
16 simplll 772 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ π‘Š ∈ AssAlg)
17 simprl 768 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
18 eqid 2731 . . . . . . . . . 10 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
1918subrgss 20463 . . . . . . . . 9 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
2019ad2antlr 724 . . . . . . . 8 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
2120sselda 3983 . . . . . . 7 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
2221adantrl 713 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
23 eqid 2731 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
24 eqid 2731 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
25 eqid 2731 . . . . . . 7 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
26 eqid 2731 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
271, 23, 24, 18, 25, 26asclmul1 21660 . . . . . 6 ((π‘Š ∈ AssAlg ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ ((π΄β€˜π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘₯( ·𝑠 β€˜π‘Š)𝑦))
2816, 17, 22, 27syl3anc 1370 . . . . 5 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ ((π΄β€˜π‘₯)(.rβ€˜π‘Š)𝑦) = (π‘₯( ·𝑠 β€˜π‘Š)𝑦))
29 simpllr 773 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ 𝑆 ∈ (SubRingβ€˜π‘Š))
30 simplr 766 . . . . . . . 8 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ran 𝐴 βŠ† 𝑆)
311, 23, 24asclfn 21655 . . . . . . . . . 10 𝐴 Fn (Baseβ€˜(Scalarβ€˜π‘Š))
3231a1i 11 . . . . . . . . 9 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ 𝐴 Fn (Baseβ€˜(Scalarβ€˜π‘Š)))
33 fnfvelrn 7083 . . . . . . . . 9 ((𝐴 Fn (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π΄β€˜π‘₯) ∈ ran 𝐴)
3432, 33sylan 579 . . . . . . . 8 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π΄β€˜π‘₯) ∈ ran 𝐴)
3530, 34sseldd 3984 . . . . . . 7 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π΄β€˜π‘₯) ∈ 𝑆)
3635adantrr 714 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ (π΄β€˜π‘₯) ∈ 𝑆)
37 simprr 770 . . . . . 6 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ 𝑦 ∈ 𝑆)
3825subrgmcl 20475 . . . . . 6 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π΄β€˜π‘₯) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) β†’ ((π΄β€˜π‘₯)(.rβ€˜π‘Š)𝑦) ∈ 𝑆)
3929, 36, 37, 38syl3anc 1370 . . . . 5 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ ((π΄β€˜π‘₯)(.rβ€˜π‘Š)𝑦) ∈ 𝑆)
4028, 39eqeltrrd 2833 . . . 4 ((((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑆)) β†’ (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑆)
4140ralrimivva 3199 . . 3 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ 𝑆 (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑆)
4223, 24, 18, 26, 6islss4 20718 . . . . 5 (π‘Š ∈ LMod β†’ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrpβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ 𝑆 (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑆)))
437, 42syl 17 . . . 4 (π‘Š ∈ AssAlg β†’ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrpβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ 𝑆 (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑆)))
4443ad2antrr 723 . . 3 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrpβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ 𝑆 (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑆)))
4515, 41, 44mpbir2and 710 . 2 (((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) ∧ ran 𝐴 βŠ† 𝑆) β†’ 𝑆 ∈ 𝐿)
4613, 45impbida 798 1 ((π‘Š ∈ AssAlg ∧ 𝑆 ∈ (SubRingβ€˜π‘Š)) β†’ (𝑆 ∈ 𝐿 ↔ ran 𝐴 βŠ† 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060   βŠ† wss 3949  {csn 4629  ran crn 5678   Fn wfn 6539  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  .rcmulr 17203  Scalarcsca 17205   ·𝑠 cvsca 17206  SubGrpcsubg 19037  1rcur 20076  SubRingcsubrg 20458  LModclmod 20615  LSubSpclss 20687  LSpanclspn 20727  AssAlgcasa 21625  algSccascl 21627
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 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-sbg 18861  df-subg 19040  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-subrng 20435  df-subrg 20460  df-lmod 20617  df-lss 20688  df-lsp 20728  df-assa 21628  df-ascl 21630
This theorem is referenced by:  rnasclassa  21669  aspval2  21672
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