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Theorem issubassa3 21801
Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 20845. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
issubassa.s 𝑆 = (π‘Š β†Ύs 𝐴)
issubassa.l 𝐿 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
issubassa3 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ AssAlg)
Proof of Theorem issubassa3
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issubassa.s . . . 4 𝑆 = (π‘Š β†Ύs 𝐴)
21subrgbas 20522 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 = (Baseβ€˜π‘†))
32ad2antrl 726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 = (Baseβ€˜π‘†))
4 eqid 2725 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
51, 4resssca 17321 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
65ad2antrl 726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
7 eqidd 2726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
8 eqid 2725 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
91, 8ressvsca 17322 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
109ad2antrl 726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
11 eqid 2725 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
121, 11ressmulr 17285 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
1312ad2antrl 726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
14 assalmod 21796 . . 3 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
15 simpr 483 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿) β†’ 𝐴 ∈ 𝐿)
16 issubassa.l . . . 4 𝐿 = (LSubSpβ€˜π‘Š)
171, 16lsslmod 20846 . . 3 ((π‘Š ∈ LMod ∧ 𝐴 ∈ 𝐿) β†’ 𝑆 ∈ LMod)
1814, 15, 17syl2an 594 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ LMod)
191subrgring 20515 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 ∈ Ring)
2019ad2antrl 726 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ Ring)
21 idd 24 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
22 eqid 2725 . . . . . . . 8 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2322subrgss 20513 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2423ad2antrl 726 . . . . . 6 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2524sseld 3971 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ (Baseβ€˜π‘Š)))
2624sseld 3971 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ (Baseβ€˜π‘Š)))
2721, 25, 263anim123d 1439 . . . 4 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ ((π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) β†’ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))))
2827imp 405 . . 3 (((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) β†’ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š)))
29 eqid 2725 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3022, 4, 29, 8, 11assaass 21794 . . . 4 ((π‘Š ∈ AssAlg ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
3130adantlr 713 . . 3 (((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
3228, 31syldan 589 . 2 (((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
3322, 4, 29, 8, 11assaassr 21795 . . . 4 ((π‘Š ∈ AssAlg ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)𝑧)) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
3433adantlr 713 . . 3 (((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)𝑧)) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
3528, 34syldan 589 . 2 (((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)𝑧)) = (π‘₯( ·𝑠 β€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
363, 6, 7, 10, 13, 18, 20, 32, 35isassad 21800 1 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177   β†Ύs cress 17206  .rcmulr 17231  Scalarcsca 17233   ·𝑠 cvsca 17234  Ringcrg 20175  SubRingcsubrg 20508  LModclmod 20745  LSubSpclss 20817  AssAlgcasa 21786
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-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-ress 17207  df-plusg 17243  df-mulr 17244  df-sca 17246  df-vsca 17247  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896  df-sbg 18897  df-subg 19080  df-mgp 20077  df-ur 20124  df-ring 20177  df-subrg 20510  df-lmod 20747  df-lss 20818  df-assa 21789
This theorem is referenced by:  issubassa  21802  rnasclassa  21830
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