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Theorem sralmod 13946
Description: The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
sralmod.a 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
Assertion
Ref Expression
sralmod (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Proof of Theorem sralmod
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sralmod.a . . . 4 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
21a1i 9 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3 eqid 2193 . . . 4 (Base‘𝑊) = (Base‘𝑊)
43subrgss 13718 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5 subrgrcl 13722 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
62, 4, 5srabaseg 13935 . 2 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
72, 4, 5sraaddgg 13936 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝐴))
82, 4, 5srascag 13938 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (Scalar‘𝐴))
92, 4, 5sravscag 13939 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = ( ·𝑠𝐴))
10 eqidd 2194 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (𝑊s 𝑆))
11 eqidd 2194 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
12 id 19 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubRing‘𝑊))
1310, 11, 5, 12ressbasd 12685 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝑆)))
14 eqidd 2194 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝑊))
1510, 14, 12, 5ressplusgd 12746 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g‘(𝑊s 𝑆)))
16 eqid 2193 . . . 4 (𝑊s 𝑆) = (𝑊s 𝑆)
17 eqid 2193 . . . 4 (.r𝑊) = (.r𝑊)
1816, 17ressmulrg 12762 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑊 ∈ Ring) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
195, 18mpdan 421 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
20 eqid 2193 . . 3 (1r𝑊) = (1r𝑊)
2116, 20subrg1 13727 . 2 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) = (1r‘(𝑊s 𝑆)))
2216subrgring 13720 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) ∈ Ring)
235ringgrpd 13501 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
247oveqdr 5946 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2511, 6, 24grppropd 13089 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
2623, 25mpbid 147 . 2 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
2753ad2ant1 1020 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
28 elinel2 3346 . . . 4 (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
29283ad2ant2 1021 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
30 simp3 1001 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
313, 17ringcl 13509 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3227, 29, 30, 31syl3anc 1249 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
335adantr 276 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
34 simpr1 1005 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
3534elin2d 3349 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
36 simpr2 1006 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
37 simpr3 1007 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
38 eqid 2193 . . . 4 (+g𝑊) = (+g𝑊)
393, 38, 17ringdi 13514 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)(𝑦(+g𝑊)𝑧)) = ((𝑥(.r𝑊)𝑦)(+g𝑊)(𝑥(.r𝑊)𝑧)))
4033, 35, 36, 37, 39syl13anc 1251 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)(𝑦(+g𝑊)𝑧)) = ((𝑥(.r𝑊)𝑦)(+g𝑊)(𝑥(.r𝑊)𝑧)))
415adantr 276 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
42 simpr1 1005 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
4342elin2d 3349 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
44 simpr2 1006 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
4544elin2d 3349 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
46 simpr3 1007 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
473, 38, 17ringdir 13515 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g𝑊)𝑦)(.r𝑊)𝑧) = ((𝑥(.r𝑊)𝑧)(+g𝑊)(𝑦(.r𝑊)𝑧)))
4841, 43, 45, 46, 47syl13anc 1251 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g𝑊)𝑦)(.r𝑊)𝑧) = ((𝑥(.r𝑊)𝑧)(+g𝑊)(𝑦(.r𝑊)𝑧)))
493, 17ringass 13512 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
5041, 43, 45, 46, 49syl13anc 1251 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
513, 17, 20ringlidm 13519 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
525, 51sylan 283 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
536, 7, 8, 9, 13, 15, 19, 21, 22, 26, 32, 40, 48, 50, 52islmodd 13789 1 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2164  cin 3152  cfv 5254  (class class class)co 5918  Basecbs 12618  s cress 12619  +gcplusg 12695  .rcmulr 12696  Grpcgrp 13072  1rcur 13455  Ringcrg 13492  SubRingcsubrg 13713  LModclmod 13783  subringAlg csra 13929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-subg 13240  df-mgp 13417  df-ur 13456  df-ring 13494  df-subrg 13715  df-lmod 13785  df-sra 13931
This theorem is referenced by:  rlmlmod  13960
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