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Theorem sralmod 13910
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 2189 . . . 4 (Base‘𝑊) = (Base‘𝑊)
43subrgss 13682 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5 subrgrcl 13686 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
62, 4, 5srabaseg 13899 . 2 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
72, 4, 5sraaddgg 13900 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝐴))
82, 4, 5srascag 13902 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (Scalar‘𝐴))
92, 4, 5sravscag 13903 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = ( ·𝑠𝐴))
10 eqidd 2190 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (𝑊s 𝑆))
11 eqidd 2190 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
12 id 19 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubRing‘𝑊))
1310, 11, 5, 12ressbasd 12659 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝑆)))
14 eqidd 2190 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝑊))
1510, 14, 12, 5ressplusgd 12720 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g‘(𝑊s 𝑆)))
16 eqid 2189 . . . 4 (𝑊s 𝑆) = (𝑊s 𝑆)
17 eqid 2189 . . . 4 (.r𝑊) = (.r𝑊)
1816, 17ressmulrg 12736 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑊 ∈ Ring) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
195, 18mpdan 421 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
20 eqid 2189 . . 3 (1r𝑊) = (1r𝑊)
2116, 20subrg1 13691 . 2 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) = (1r‘(𝑊s 𝑆)))
2216subrgring 13684 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) ∈ Ring)
235ringgrpd 13465 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
247oveqdr 5934 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2511, 6, 24grppropd 13063 . . 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 3342 . . . 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 13473 . . 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 3345 . . 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 2189 . . . 4 (+g𝑊) = (+g𝑊)
393, 38, 17ringdi 13478 . . 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 3345 . . 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 3345 . . 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 13479 . . 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 13476 . . 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 13483 . . 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 13753 1 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  cin 3148  cfv 5242  (class class class)co 5906  Basecbs 12592  s cress 12593  +gcplusg 12669  .rcmulr 12670  Grpcgrp 13046  1rcur 13419  Ringcrg 13456  SubRingcsubrg 13677  LModclmod 13747  subringAlg csra 13893
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7949  ax-resscn 7950  ax-1cn 7951  ax-1re 7952  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-addcom 7958  ax-addass 7960  ax-i2m1 7963  ax-0lt1 7964  ax-0id 7966  ax-rnegex 7967  ax-pre-ltirr 7970  ax-pre-lttrn 7972  ax-pre-ltadd 7974
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-riota 5861  df-ov 5909  df-oprab 5910  df-mpo 5911  df-pnf 8042  df-mnf 8043  df-ltxr 8045  df-inn 8969  df-2 9027  df-3 9028  df-4 9029  df-5 9030  df-6 9031  df-7 9032  df-8 9033  df-ndx 12595  df-slot 12596  df-base 12598  df-sets 12599  df-iress 12600  df-plusg 12682  df-mulr 12683  df-sca 12685  df-vsca 12686  df-ip 12687  df-0g 12843  df-mgm 12913  df-sgrp 12959  df-mnd 12972  df-grp 13049  df-subg 13213  df-mgp 13381  df-ur 13420  df-ring 13458  df-subrg 13679  df-lmod 13749  df-sra 13895
This theorem is referenced by:  rlmlmod  13924
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