MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sraassa Structured version   Visualization version   GIF version

Theorem sraassa 20829
Description: The subring algebra over a commutative ring is an associative algebra. (Contributed by Mario Carneiro, 6-Oct-2015.)
Hypothesis
Ref Expression
sraassa.a 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
Assertion
Ref Expression
sraassa ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Proof of Theorem sraassa
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sraassa.a . . . 4 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
21a1i 11 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3 eqid 2737 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
43subrgss 19801 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
54adantl 485 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
62, 5srabase 20215 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
72, 5srasca 20218 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
8 eqid 2737 . . . 4 (𝑊s 𝑆) = (𝑊s 𝑆)
98subrgbas 19809 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
109adantl 485 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
112, 5sravsca 20219 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
122, 5sramulr 20217 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = (.r𝐴))
131sralmod 20224 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1413adantl 485 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15 crngring 19574 . . . 4 (𝑊 ∈ CRing → 𝑊 ∈ Ring)
1615adantr 484 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17 eqidd 2738 . . . 4 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
182, 5sraaddg 20216 . . . . 5 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1918oveqdr 7241 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2012oveqdr 7241 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)𝑦) = (𝑥(.r𝐴)𝑦))
2117, 6, 19, 20ringpropd 19600 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ Ring ↔ 𝐴 ∈ Ring))
2216, 21mpbid 235 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ Ring)
238subrgcrng 19804 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) ∈ CRing)
2416adantr 484 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
255adantr 484 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑆 ⊆ (Base‘𝑊))
26 simpr1 1196 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥𝑆)
2725, 26sseldd 3902 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
28 simpr2 1197 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
29 simpr3 1198 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
30 eqid 2737 . . . 4 (.r𝑊) = (.r𝑊)
313, 30ringass 19582 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
3224, 27, 28, 29, 31syl13anc 1374 . 2 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
33 eqid 2737 . . . . 5 (mulGrp‘𝑊) = (mulGrp‘𝑊)
3433crngmgp 19570 . . . 4 (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
3534ad2antrr 726 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
3633, 3mgpbas 19510 . . . 4 (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
3733, 30mgpplusg 19508 . . . 4 (.r𝑊) = (+g‘(mulGrp‘𝑊))
3836, 37cmn12 19191 . . 3 (((mulGrp‘𝑊) ∈ CMnd ∧ (𝑦 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r𝑊)(𝑥(.r𝑊)𝑧)) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
3935, 28, 27, 29, 38syl13anc 1374 . 2 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r𝑊)(𝑥(.r𝑊)𝑧)) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
406, 7, 10, 11, 12, 14, 22, 23, 32, 39isassad 20826 1 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wss 3866  cfv 6380  (class class class)co 7213  Basecbs 16760  s cress 16784  +gcplusg 16802  .rcmulr 16803  CMndccmn 19170  mulGrpcmgp 19504  Ringcrg 19562  CRingccrg 19563  SubRingcsubrg 19796  LModclmod 19899  subringAlg csra 20205  AssAlgcasa 20812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-ip 16820  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-grp 18368  df-subg 18540  df-cmn 19172  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-subrg 19798  df-lmod 19901  df-sra 20209  df-assa 20815
This theorem is referenced by:  rlmassa  20830
  Copyright terms: Public domain W3C validator