Theorem sralmod 13548

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 9	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3		eqid 2177	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 13354	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5		subrgrcl 13358	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
6	2, 4, 5	srabaseg 13537	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
7	2, 4, 5	sraaddgg 13538	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝐴))
8	2, 4, 5	srascag 13540	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
9	2, 4, 5	sravscag 13541	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
10		eqidd 2178	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆))
11		eqidd 2178	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
12		id 19	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubRing‘𝑊))
13	10, 11, 5, 12	ressbasd 12530	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝑆)))
14		eqidd 2178	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝑊))
15	10, 14, 12, 5	ressplusgd 12590	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘(𝑊 ↾s 𝑆)))
16		eqid 2177	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
17		eqid 2177	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
18	16, 17	ressmulrg 12606	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑊 ∈ Ring) → (.r‘𝑊) = (.r‘(𝑊 ↾s 𝑆)))
19	5, 18	mpdan 421	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = (.r‘(𝑊 ↾s 𝑆)))
20		eqid 2177	. . 3 ⊢ (1r‘𝑊) = (1r‘𝑊)
21	16, 20	subrg1 13363	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘(𝑊 ↾s 𝑆)))
22	16	subrgring 13356	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) ∈ Ring)
23	5	ringgrpd 13199	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
24	7	oveqdr 5906	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
25	11, 6, 24	grppropd 12900	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
26	23, 25	mpbid 147	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
27	5	3ad2ant1 1018	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
28		elinel2 3324	. . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
29	28	3ad2ant2 1019	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
30		simp3 999	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
31	3, 17	ringcl 13207	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
32	27, 29, 30, 31	syl3anc 1238	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
33	5	adantr 276	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
34		simpr1 1003	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
35	34	elin2d 3327	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
36		simpr2 1004	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
37		simpr3 1005	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
38		eqid 2177	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
39	3, 38, 17	ringdi 13212	. . 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)))
40	33, 35, 36, 37, 39	syl13anc 1240	. 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)))
41	5	adantr 276	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
42		simpr1 1003	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
43	42	elin2d 3327	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
44		simpr2 1004	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
45	44	elin2d 3327	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
46		simpr3 1005	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
47	3, 38, 17	ringdir 13213	. . 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)))
48	41, 43, 45, 46, 47	syl13anc 1240	. 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)))
49	3, 17	ringass 13210	. . 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)))
50	41, 43, 45, 46, 49	syl13anc 1240	. 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)))
51	3, 17, 20	ringlidm 13217	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
52	5, 51	sylan 283	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
53	6, 7, 8, 9, 13, 15, 19, 21, 22, 26, 32, 40, 48, 50, 52	islmodd 13394	1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148   ∩ cin 3130  ‘cfv 5218  (class class class)co 5878  Basecbs 12465   ↾_s cress 12466  +_gcplusg 12539  ._rcmulr 12540  Grpcgrp 12884  1_rcur 13153  Ringcrg 13190  SubRingcsubrg 13349  LModclmod 13388  subringAlg csra 13531

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-pre-ltirr 7926  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-ltxr 8000  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473  df-plusg 12552  df-mulr 12553  df-sca 12555  df-vsca 12556  df-ip 12557  df-0g 12713  df-mgm 12782  df-sgrp 12815  df-mnd 12825  df-grp 12887  df-subg 13040  df-mgp 13142  df-ur 13154  df-ring 13192  df-subrg 13351  df-lmod 13390  df-sra 13533

This theorem is referenced by:  rlmlmod  13561