Theorem sralmod 21223

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.

Step	Hyp	Ref	Expression
1		sralmod.a	. . . 4 ⊢ 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
2	1	a1i 11	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3		eqid 2752	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 20590	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	2, 4	srabase 21213	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
6	2, 4	sraaddg 21214	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝐴))
7	2, 4	srasca 21216	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8	2, 4	sravsca 21217	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
9		eqid 2752	. . 3 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
10	9, 3	ressbas 17244	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝑆)))
11		eqid 2752	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
12	9, 11	ressplusg 17292	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘(𝑊 ↾s 𝑆)))
13		eqid 2752	. . 3 ⊢ (.r‘𝑊) = (.r‘𝑊)
14	9, 13	ressmulr 17308	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = (.r‘(𝑊 ↾s 𝑆)))
15		eqid 2752	. . 3 ⊢ (1r‘𝑊) = (1r‘𝑊)
16	9, 15	subrg1 20600	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘(𝑊 ↾s 𝑆)))
17	9	subrgring 20592	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) ∈ Ring)
18		subrgrcl 20594	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19		ringgrp 20256	. . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ Grp)
20	18, 19	syl 17	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21		eqidd 2753	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
22	6	oveqdr 7409	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
23	21, 5, 22	grppropd 18965	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
24	20, 23	mpbid 234	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25	18	3ad2ant1 1142	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26		elinel2 4145	. . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
27	26	3ad2ant2 1143	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
28		simp3 1147	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
29	3, 13	ringcl 20268	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
30	25, 27, 28, 29	syl3anc 1382	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
31	18	adantr 483	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
32		simpr1 1204	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
33	32	elin2d 4148	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
34		simpr2 1205	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
35		simpr3 1206	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
36	3, 11, 13	ringdi 20279	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)(𝑦(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑊)𝑦)(+g‘𝑊)(𝑥(.r‘𝑊)𝑧)))
37	31, 33, 34, 35, 36	syl13anc 1383	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑥(.r‘𝑊)(𝑦(+g‘𝑊)𝑧)) = ((𝑥(.r‘𝑊)𝑦)(+g‘𝑊)(𝑥(.r‘𝑊)𝑧)))
38	18	adantr 483	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
39		simpr1 1204	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
40	39	elin2d 4148	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
41		simpr2 1205	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
42	41	elin2d 4148	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
43		simpr3 1206	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
44	3, 11, 13	ringdir 20280	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g‘𝑊)𝑦)(.r‘𝑊)𝑧) = ((𝑥(.r‘𝑊)𝑧)(+g‘𝑊)(𝑦(.r‘𝑊)𝑧)))
45	38, 40, 42, 43, 44	syl13anc 1383	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(+g‘𝑊)𝑦)(.r‘𝑊)𝑧) = ((𝑥(.r‘𝑊)𝑧)(+g‘𝑊)(𝑦(.r‘𝑊)𝑧)))
46	3, 13	ringass 20271	. . 3 ⊢ ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
47	38, 40, 42, 43, 46	syl13anc 1383	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥(.r‘𝑊)(𝑦(.r‘𝑊)𝑧)))
48	3, 13, 15	ringlidm 20287	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
49	18, 48	sylan 588	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
50	5, 6, 7, 8, 10, 12, 14, 16, 17, 24, 30, 37, 45, 47, 49	islmodd 20902	1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ∩ cin 3894  ‘cfv 6506  (class class class)co 7381  Basecbs 17217   ↾_s cress 17238  +_gcplusg 17258  ._rcmulr 17259  Grpcgrp 18947  1_rcur 20199  Ringcrg 20251  SubRingcsubrg 20587  LModclmod 20896  subringAlg csra 21207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-nel 3052  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-er 8662  df-en 8913  df-dom 8914  df-sdom 8915  df-pnf 11204  df-mnf 11205  df-xr 11206  df-ltxr 11207  df-le 11208  df-sub 11402  df-neg 11403  df-nn 12197  df-2 12266  df-3 12267  df-4 12268  df-5 12269  df-6 12270  df-7 12271  df-8 12272  df-sets 17172  df-slot 17190  df-ndx 17202  df-base 17218  df-ress 17239  df-plusg 17271  df-mulr 17272  df-sca 17274  df-vsca 17275  df-ip 17276  df-0g 17442  df-mgm 18646  df-sgrp 18725  df-mnd 18741  df-grp 18950  df-subg 19137  df-mgp 20159  df-ur 20200  df-ring 20253  df-subrg 20588  df-lmod 20898  df-sra 21209

This theorem is referenced by:  rlmlmod  21239  sraassab  21889  evls1maplmhm  22409  sranlm  24713  sralvec  33826  lsssra  33829  fldextrspunlsplem  33914  fldextrspunlsp  33915  fldextrspunlem1  33916  fldextrspunfld  33917  algextdeglem2  33959