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Theorem issubassa3 21420
Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 20570. (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 20328 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 = (Baseβ€˜π‘†))
32ad2antrl 727 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 = (Baseβ€˜π‘†))
4 eqid 2733 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
51, 4resssca 17288 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
65ad2antrl 727 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘†))
7 eqidd 2734 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
8 eqid 2733 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
91, 8ressvsca 17289 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
109ad2antrl 727 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘†))
11 eqid 2733 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
121, 11ressmulr 17252 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
1312ad2antrl 727 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (.rβ€˜π‘Š) = (.rβ€˜π‘†))
14 assalmod 21415 . . 3 (π‘Š ∈ AssAlg β†’ π‘Š ∈ LMod)
15 simpr 486 . . 3 ((𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿) β†’ 𝐴 ∈ 𝐿)
16 issubassa.l . . . 4 𝐿 = (LSubSpβ€˜π‘Š)
171, 16lsslmod 20571 . . 3 ((π‘Š ∈ LMod ∧ 𝐴 ∈ 𝐿) β†’ 𝑆 ∈ LMod)
1814, 15, 17syl2an 597 . 2 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝑆 ∈ LMod)
191subrgring 20322 . . 3 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 ∈ Ring)
2019ad2antrl 727 . 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 2733 . . . . . . . 8 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2322subrgss 20320 . . . . . . 7 (𝐴 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2423ad2antrl 727 . . . . . 6 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ 𝐴 βŠ† (Baseβ€˜π‘Š))
2524sseld 3982 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ (Baseβ€˜π‘Š)))
2624sseld 3982 . . . . 5 ((π‘Š ∈ AssAlg ∧ (𝐴 ∈ (SubRingβ€˜π‘Š) ∧ 𝐴 ∈ 𝐿)) β†’ (𝑧 ∈ 𝐴 β†’ 𝑧 ∈ (Baseβ€˜π‘Š)))
2721, 25, 263anim123d 1444 . . . 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 408 . . 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 2733 . . . . 5 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3022, 4, 29, 8, 11assaass 21413 . . . 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 714 . . 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 592 . 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 21414 . . . 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 714 . . 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 592 . 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 21419 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144   β†Ύs cress 17173  .rcmulr 17198  Scalarcsca 17200   ·𝑠 cvsca 17201  Ringcrg 20056  SubRingcsubrg 20315  LModclmod 20471  LSubSpclss 20542  AssAlgcasa 21405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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-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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-mgp 19988  df-ur 20005  df-ring 20058  df-subrg 20317  df-lmod 20473  df-lss 20543  df-assa 21408
This theorem is referenced by:  issubassa  21421  rnasclassa  21449
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