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Theorem issubassa3 21760
Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 20806. (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 20483 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 = (Baseβ€˜π‘†))
32ad2antrl 725 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 = (Baseβ€˜π‘†))
4 eqid 2726 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
51, 4resssca 17297 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
65ad2antrl 725 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
7 eqidd 2727 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
8 eqid 2726 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
91, 8ressvsca 17298 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
109ad2antrl 725 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
11 eqid 2726 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
121, 11ressmulr 17261 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
1312ad2antrl 725 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
14 assalmod 21755 . . 3 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
15 simpr 484 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿) β†’ 𝐴 ∈ 𝐿)
16 issubassa.l . . . 4 𝐿 = (LSubSpβ€˜π‘Š)
171, 16lsslmod 20807 . . 3 ((π‘Š ∈ LMod ∧ 𝐴 ∈ 𝐿) β†’ 𝑆 ∈ LMod)
1814, 15, 17syl2an 595 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ LMod)
191subrgring 20476 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 ∈ Ring)
2019ad2antrl 725 . 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 2726 . . . . . . . 8 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2322subrgss 20474 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2423ad2antrl 725 . . . . . 6 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2524sseld 3976 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ (Baseβ€˜π‘Š)))
2624sseld 3976 . . . . 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 406 . . 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 2726 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3022, 4, 29, 8, 11assaass 21753 . . . 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 712 . . 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 590 . 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 21754 . . . 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 712 . . 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 590 . 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 21759 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153   β†Ύs cress 17182  .rcmulr 17207  Scalarcsca 17209   ·𝑠 cvsca 17210  Ringcrg 20138  SubRingcsubrg 20469  LModclmod 20706  LSubSpclss 20778  AssAlgcasa 21745
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-mgp 20040  df-ur 20087  df-ring 20140  df-subrg 20471  df-lmod 20708  df-lss 20779  df-assa 21748
This theorem is referenced by:  issubassa  21761  rnasclassa  21789
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