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Theorem sralmod 20656
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 2736 . . . 4 (Base‘𝑊) = (Base‘𝑊)
43subrgss 20223 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
52, 4srabase 20640 . 2 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
62, 4sraaddg 20642 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝐴))
72, 4srasca 20646 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (Scalar‘𝐴))
82, 4sravsca 20648 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = ( ·𝑠𝐴))
9 eqid 2736 . . 3 (𝑊s 𝑆) = (𝑊s 𝑆)
109, 3ressbas 17118 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝑆)))
11 eqid 2736 . . 3 (+g𝑊) = (+g𝑊)
129, 11ressplusg 17171 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g‘(𝑊s 𝑆)))
13 eqid 2736 . . 3 (.r𝑊) = (.r𝑊)
149, 13ressmulr 17188 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
15 eqid 2736 . . 3 (1r𝑊) = (1r𝑊)
169, 15subrg1 20232 . 2 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) = (1r‘(𝑊s 𝑆)))
179subrgring 20225 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) ∈ Ring)
18 subrgrcl 20227 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19 ringgrp 19969 . . . 4 (𝑊 ∈ Ring → 𝑊 ∈ Grp)
2018, 19syl 17 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21 eqidd 2737 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
226oveqdr 7385 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2321, 5, 22grppropd 18765 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
2420, 23mpbid 231 . 2 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25183ad2ant1 1133 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26 elinel2 4156 . . . 4 (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
27263ad2ant2 1134 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
28 simp3 1138 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
293, 13ringcl 19981 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3025, 27, 28, 29syl3anc 1371 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3118adantr 481 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
32 simpr1 1194 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
3332elin2d 4159 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
34 simpr2 1195 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
35 simpr3 1196 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
363, 11, 13ringdi 19987 . . 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 1372 . 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 481 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
39 simpr1 1194 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
4039elin2d 4159 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41 simpr2 1195 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
4241elin2d 4159 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
43 simpr3 1196 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
443, 11, 13ringdir 19988 . . 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 1372 . 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 19984 . . 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 1372 . 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 19992 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
4918, 48sylan 580 . 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 20328 1 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  cin 3909  cfv 6496  (class class class)co 7357  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Grpcgrp 18748  1rcur 19913  Ringcrg 19964  SubRingcsubrg 20218  LModclmod 20322  subringAlg csra 20629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-subg 18925  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-sra 20633
This theorem is referenced by:  rlmlmod  20674  sraassa  21273  sranlm  24048  sralvec  32289
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