Theorem sralmod 14727

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 2234	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 14471	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5		subrgrcl 14475	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
6	2, 4, 5	srabaseg 14716	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
7	2, 4, 5	sraaddgg 14717	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝐴))
8	2, 4, 5	srascag 14719	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
9	2, 4, 5	sravscag 14720	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
10		eqidd 2235	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆))
11		eqidd 2235	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
12		id 19	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubRing‘𝑊))
13	10, 11, 5, 12	ressbasd 13367	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝑆)))
14		eqidd 2235	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝑊))
15	10, 14, 12, 5	ressplusgd 13429	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘(𝑊 ↾s 𝑆)))
16		eqid 2234	. . . 4 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
17		eqid 2234	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
18	16, 17	ressmulrg 13445	. . 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 2234	. . 3 ⊢ (1r‘𝑊) = (1r‘𝑊)
21	16, 20	subrg1 14480	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘(𝑊 ↾s 𝑆)))
22	16	subrgring 14473	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) ∈ Ring)
23	5	ringgrpd 14251	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
24	7	oveqdr 6086	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
25	11, 6, 24	grppropd 13775	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
26	23, 25	mpbid 147	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
27	5	3ad2ant1 1045	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
28		elinel2 3410	. . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
29	28	3ad2ant2 1046	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
30		simp3 1026	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑦 ∈ (Base‘𝑊))
31	3, 17	ringcl 14259	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
32	27, 29, 30, 31	syl3anc 1274	. 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 1030	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
35	34	elin2d 3413	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
36		simpr2 1031	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
37		simpr3 1032	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
38		eqid 2234	. . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊)
39	3, 38, 17	ringdi 14264	. . 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 1276	. 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 1030	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)))
43	42	elin2d 3413	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
44		simpr2 1031	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)))
45	44	elin2d 3413	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
46		simpr3 1032	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
47	3, 38, 17	ringdir 14265	. . 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 1276	. 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 14262	. . 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 1276	. 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 14269	. . 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 14570	1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205   ∩ cin 3213  ‘cfv 5357  (class class class)co 6058  Basecbs 13299   ↾_s cress 13300  +_gcplusg 13377  ._rcmulr 13378  Grpcgrp 13758  1_rcur 14205  Ringcrg 14242  SubRingcsubrg 14466  LModclmod 14564  subringAlg csra 14710

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-ndx 13302  df-slot 13303  df-base 13305  df-sets 13306  df-iress 13307  df-plusg 13390  df-mulr 13391  df-sca 13393  df-vsca 13394  df-ip 13395  df-0g 13558  df-mgm 13622  df-sgrp 13668  df-mnd 13681  df-grp 13761  df-subg 13926  df-mgp 14163  df-ur 14206  df-ring 14244  df-subrg 14468  df-lmod 14566  df-sra 14712

This theorem is referenced by:  rlmlmod  14741