Theorem sralmod 21195

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 2736	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
4	3	subrgss 20573	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
5	2, 4	srabase 21178	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝐴))
6	2, 4	sraaddg 21180	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘𝐴))
7	2, 4	srasca 21184	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) = (Scalar‘𝐴))
8	2, 4	sravsca 21186	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = ( ·𝑠 ‘𝐴))
9		eqid 2736	. . 3 ⊢ (𝑊 ↾s 𝑆) = (𝑊 ↾s 𝑆)
10	9, 3	ressbas 17281	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑆 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝑆)))
11		eqid 2736	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
12	9, 11	ressplusg 17335	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (+g‘𝑊) = (+g‘(𝑊 ↾s 𝑆)))
13		eqid 2736	. . 3 ⊢ (.r‘𝑊) = (.r‘𝑊)
14	9, 13	ressmulr 17352	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (.r‘𝑊) = (.r‘(𝑊 ↾s 𝑆)))
15		eqid 2736	. . 3 ⊢ (1r‘𝑊) = (1r‘𝑊)
16	9, 15	subrg1 20583	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) = (1r‘(𝑊 ↾s 𝑆)))
17	9	subrgring 20575	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ↾s 𝑆) ∈ Ring)
18		subrgrcl 20577	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Ring)
19		ringgrp 20236	. . . 4 ⊢ (𝑊 ∈ Ring → 𝑊 ∈ Grp)
20	18, 19	syl 17	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑊 ∈ Grp)
21		eqidd 2737	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (Base‘𝑊) = (Base‘𝑊))
22	6	oveqdr 7460	. . . 4 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐴)𝑦))
23	21, 5, 22	grppropd 18970	. . 3 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (𝑊 ∈ Grp ↔ 𝐴 ∈ Grp))
24	20, 23	mpbid 232	. 2 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ Grp)
25	18	3ad2ant1 1133	. . 3 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → 𝑊 ∈ Ring)
26		elinel2 4201	. . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) → 𝑥 ∈ (Base‘𝑊))
27	26	3ad2ant2 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))
29	3, 13	ringcl 20248	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
30	25, 27, 28, 29	syl3anc 1372	. 2 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ 𝑥 ∈ (𝑆 ∩ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (𝑥(.r‘𝑊)𝑦) ∈ (Base‘𝑊))
31	18	adantr 480	. . 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)))
33	32	elin2d 4204	. . 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))
36	3, 11, 13	ringdi 20259	. . 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 1373	. 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 480	. . 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)))
40	39	elin2d 4204	. . 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)))
42	41	elin2d 4204	. . 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))
44	3, 11, 13	ringdir 20260	. . 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 1373	. 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 20251	. . 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 1373	. 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 20267	. . 3 ⊢ ((𝑊 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑊)) → ((1r‘𝑊)(.r‘𝑊)𝑥) = 𝑥)
49	18, 48	sylan 580	. 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 20865	1 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ∩ cin 3949  ‘cfv 6560  (class class class)co 7432  Basecbs 17248   ↾_s cress 17275  +_gcplusg 17298  ._rcmulr 17299  Grpcgrp 18952  1_rcur 20179  Ringcrg 20231  SubRingcsubrg 20570  LModclmod 20859  subringAlg csra 21171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-subg 19142  df-mgp 20139  df-ur 20180  df-ring 20233  df-subrg 20571  df-lmod 20861  df-sra 21173

This theorem is referenced by:  rlmlmod  21211  sraassab  21889  sraassaOLD  21891  evls1maplmhm  22382  sranlm  24706  sralvec  33637  lsssra  33640  fldextrspunlsplem  33724  fldextrspunlsp  33725  fldextrspunlem1  33726  fldextrspunfld  33727  algextdeglem2  33760