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Theorem sralmod 21073
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 11 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3 eqid 2727 . . . 4 (Base‘𝑊) = (Base‘𝑊)
43subrgss 20504 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
52, 4srabase 21056 . 2 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
62, 4sraaddg 21058 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝐴))
72, 4srasca 21062 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (Scalar‘𝐴))
82, 4sravsca 21064 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = ( ·𝑠𝐴))
9 eqid 2727 . . 3 (𝑊s 𝑆) = (𝑊s 𝑆)
109, 3ressbas 17208 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝑆)))
11 eqid 2727 . . 3 (+g𝑊) = (+g𝑊)
129, 11ressplusg 17264 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g‘(𝑊s 𝑆)))
13 eqid 2727 . . 3 (.r𝑊) = (.r𝑊)
149, 13ressmulr 17281 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
15 eqid 2727 . . 3 (1r𝑊) = (1r𝑊)
169, 15subrg1 20514 . 2 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) = (1r‘(𝑊s 𝑆)))
179subrgring 20506 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) ∈ Ring)
18 subrgrcl 20508 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19 ringgrp 20171 . . . 4 (𝑊 ∈ Ring → 𝑊 ∈ Grp)
2018, 19syl 17 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21 eqidd 2728 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
226oveqdr 7442 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2321, 5, 22grppropd 18901 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
2420, 23mpbid 231 . 2 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25183ad2ant1 1131 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26 elinel2 4192 . . . 4 (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
27263ad2ant2 1132 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
28 simp3 1136 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
293, 13ringcl 20183 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3025, 27, 28, 29syl3anc 1369 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3118adantr 480 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
32 simpr1 1192 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
3332elin2d 4195 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
34 simpr2 1193 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
35 simpr3 1194 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
363, 11, 13ringdi 20193 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)(𝑦(+g𝑊)𝑧)) = ((𝑥(.r𝑊)𝑦)(+g𝑊)(𝑥(.r𝑊)𝑧)))
3731, 33, 34, 35, 36syl13anc 1370 . 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)))
3818adantr 480 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
39 simpr1 1192 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
4039elin2d 4195 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41 simpr2 1193 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
4241elin2d 4195 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
43 simpr3 1194 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
443, 11, 13ringdir 20194 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g𝑊)𝑦)(.r𝑊)𝑧) = ((𝑥(.r𝑊)𝑧)(+g𝑊)(𝑦(.r𝑊)𝑧)))
4538, 40, 42, 43, 44syl13anc 1370 . 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)))
463, 13ringass 20186 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
4738, 40, 42, 43, 46syl13anc 1370 . 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)))
483, 13, 15ringlidm 20198 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
4918, 48sylan 579 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
505, 6, 7, 8, 10, 12, 14, 16, 17, 24, 30, 37, 45, 47, 49islmodd 20742 1 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  cin 3943  cfv 6542  (class class class)co 7414  Basecbs 17173  s cress 17202  +gcplusg 17226  .rcmulr 17227  Grpcgrp 18883  1rcur 20114  Ringcrg 20166  SubRingcsubrg 20499  LModclmod 20736  subringAlg csra 21049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11188  ax-resscn 11189  ax-1cn 11190  ax-icn 11191  ax-addcl 11192  ax-addrcl 11193  ax-mulcl 11194  ax-mulrcl 11195  ax-mulcom 11196  ax-addass 11197  ax-mulass 11198  ax-distr 11199  ax-i2m1 11200  ax-1ne0 11201  ax-1rid 11202  ax-rnegex 11203  ax-rrecex 11204  ax-cnre 11205  ax-pre-lttri 11206  ax-pre-lttrn 11207  ax-pre-ltadd 11208  ax-pre-mulgt0 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11274  df-mnf 11275  df-xr 11276  df-ltxr 11277  df-le 11278  df-sub 11470  df-neg 11471  df-nn 12237  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-sets 17126  df-slot 17144  df-ndx 17156  df-base 17174  df-ress 17203  df-plusg 17239  df-mulr 17240  df-sca 17242  df-vsca 17243  df-ip 17244  df-0g 17416  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-grp 18886  df-subg 19071  df-mgp 20068  df-ur 20115  df-ring 20168  df-subrg 20501  df-lmod 20738  df-sra 21051
This theorem is referenced by:  rlmlmod  21089  sraassab  21794  sraassaOLD  21796  sranlm  24594  sralvec  33271  lsssra  33274  evls1maplmhm  33360  algextdeglem2  33376
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