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Theorem sralmod 19008
 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 2610 . . . 4 (Base‘𝑊) = (Base‘𝑊)
43subrgss 18604 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
52, 4srabase 18999 . 2 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
62, 4sraaddg 19000 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g𝐴))
72, 4srasca 19002 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) = (Scalar‘𝐴))
82, 4sravsca 19003 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = ( ·𝑠𝐴))
9 eqid 2610 . . 3 (𝑊s 𝑆) = (𝑊s 𝑆)
109, 3ressbas 15757 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊s 𝑆)))
11 eqid 2610 . . 3 (+g𝑊) = (+g𝑊)
129, 11ressplusg 15818 . 2 (𝑆 ∈ (SubRing‘𝑊) → (+g𝑊) = (+g‘(𝑊s 𝑆)))
13 eqid 2610 . . 3 (.r𝑊) = (.r𝑊)
149, 13ressmulr 15829 . 2 (𝑆 ∈ (SubRing‘𝑊) → (.r𝑊) = (.r‘(𝑊s 𝑆)))
15 eqid 2610 . . 3 (1r𝑊) = (1r𝑊)
169, 15subrg1 18613 . 2 (𝑆 ∈ (SubRing‘𝑊) → (1r𝑊) = (1r‘(𝑊s 𝑆)))
179subrgring 18606 . 2 (𝑆 ∈ (SubRing‘𝑊) → (𝑊s 𝑆) ∈ Ring)
18 subrgrcl 18608 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19 ringgrp 18375 . . . 4 (𝑊 ∈ Ring → 𝑊 ∈ Grp)
2018, 19syl 17 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21 eqidd 2611 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
226oveqdr 6573 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2321, 5, 22grppropd 17260 . . 3 (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
2420, 23mpbid 221 . 2 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25183ad2ant1 1075 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26 inss2 3796 . . . . 5 (𝑆 ∩ (Base‘𝑊)) ⊆ (Base‘𝑊)
2726sseli 3564 . . . 4 (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
28273ad2ant2 1076 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
29 simp3 1056 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
303, 13ringcl 18384 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3125, 28, 29, 30syl3anc 1318 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r𝑊)𝑦) ∈ (Base‘𝑊))
3218adantr 480 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
33 simpr1 1060 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
3426, 33sseldi 3566 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
35 simpr2 1061 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
36 simpr3 1062 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
373, 11, 13ringdi 18389 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)(𝑦(+g𝑊)𝑧)) = ((𝑥(.r𝑊)𝑦)(+g𝑊)(𝑥(.r𝑊)𝑧)))
3832, 34, 35, 36, 37syl13anc 1320 . 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)))
3918adantr 480 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
40 simpr1 1060 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
4126, 40sseldi 3566 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
42 simpr2 1061 . . . 4 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
4326, 42sseldi 3566 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
44 simpr3 1062 . . 3 ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
453, 11, 13ringdir 18390 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g𝑊)𝑦)(.r𝑊)𝑧) = ((𝑥(.r𝑊)𝑧)(+g𝑊)(𝑦(.r𝑊)𝑧)))
4639, 41, 43, 44, 45syl13anc 1320 . 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)))
473, 13ringass 18387 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
4839, 41, 43, 44, 47syl13anc 1320 . 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)))
493, 13, 15ringlidm 18394 . . 3 ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
5018, 49sylan 487 . 2 ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r𝑊)(.r𝑊)𝑥) = 𝑥)
515, 6, 7, 8, 10, 12, 14, 16, 17, 24, 31, 38, 46, 48, 50islmodd 18692 1 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  .rcmulr 15769  Grpcgrp 17245  1rcur 18324  Ringcrg 18370  SubRingcsubrg 18599  LModclmod 18686  subringAlg csra 18989 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-sra 18993 This theorem is referenced by:  rlmlmod  19026  sraassa  19146  sranlm  22298
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