Theorem issubassa3 15096

Description: A subring that is also a subspace is a subalgebra. The key theorem is islss3 14800. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
issubassa.s	⊢ 𝑆 = (𝑊 ↾s 𝐴)
issubassa.l	⊢ 𝐿 = (LSubSp‘𝑊)

Assertion

Ref	Expression
issubassa3	⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)

Proof of Theorem issubassa3

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issubassa.s	. . . 4 ⊢ 𝑆 = (𝑊 ↾s 𝐴)
2	1	subrgbas 14622	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑊) → 𝐴 = (Base‘𝑆))
3	2	ad2antrl 494	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝐴 = (Base‘𝑆))
4		eqid 2238	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5	1, 4	ressscag 13590	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐴 ∈ (SubRing‘𝑊)) → (Scalar‘𝑊) = (Scalar‘𝑆))
6	5	adantrr 483	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (Scalar‘𝑊) = (Scalar‘𝑆))
7		eqidd 2239	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)))
8		eqid 2238	. . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
9	1, 8	ressvscag 13591	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝐴 ∈ (SubRing‘𝑊)) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑆))
10	9	adantrr 483	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑆))
11		simprl 535	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝐴 ∈ (SubRing‘𝑊))
12		simpl 109	. . 3 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑊 ∈ AssAlg)
13		eqid 2238	. . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊)
14	1, 13	ressmulrg 13552	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝑊 ∈ AssAlg) → (.r‘𝑊) = (.r‘𝑆))
15	11, 12, 14	syl2anc 415	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (.r‘𝑊) = (.r‘𝑆))
16		assalmod 15090	. . 3 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
17		simpr 110	. . 3 ⊢ ((𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿) → 𝐴 ∈ 𝐿)
18		issubassa.l	. . . 4 ⊢ 𝐿 = (LSubSp‘𝑊)
19	1, 18	lsslmod 14801	. . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐿) → 𝑆 ∈ LMod)
20	16, 17, 19	syl2an 289	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ LMod)
21	1	subrgring 14616	. . 3 ⊢ (𝐴 ∈ (SubRing‘𝑊) → 𝑆 ∈ Ring)
22	21	ad2antrl 494	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ Ring)
23		idd 21	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) → 𝑥 ∈ (Base‘(Scalar‘𝑊))))
24		eqid 2238	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
25	24	subrgss 14614	. . . . . . 7 ⊢ (𝐴 ∈ (SubRing‘𝑊) → 𝐴 ⊆ (Base‘𝑊))
26	25	ad2antrl 494	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝐴 ⊆ (Base‘𝑊))
27	26	sseld 3247	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (𝑦 ∈ 𝐴 → 𝑦 ∈ (Base‘𝑊)))
28	26	sseld 3247	. . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → (𝑧 ∈ 𝐴 → 𝑧 ∈ (Base‘𝑊)))
29	23, 27, 28	3anim123d 1360	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → ((𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))))
30	29	imp 124	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊)))
31		eqid 2238	. . . . 5 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
32	24, 4, 31, 8, 13	assaass 15088	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
33	32	adantlr 481	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
34	30, 33	syldan 282	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥( ·𝑠 ‘𝑊)𝑦)(.r‘𝑊)𝑧) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
35	24, 4, 31, 8, 13	assaassr 15089	. . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r‘𝑊)(𝑥( ·𝑠 ‘𝑊)𝑧)) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
36	35	adantlr 481	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r‘𝑊)(𝑥( ·𝑠 ‘𝑊)𝑧)) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
37	30, 36	syldan 282	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦(.r‘𝑊)(𝑥( ·𝑠 ‘𝑊)𝑧)) = (𝑥( ·𝑠 ‘𝑊)(𝑦(.r‘𝑊)𝑧)))
38	3, 6, 7, 10, 15, 20, 22, 34, 37	isassad 15095	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ (SubRing‘𝑊) ∧ 𝐴 ∈ 𝐿)) → 𝑆 ∈ AssAlg)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ‘cfv 5377  (class class class)co 6085  Basecbs 13404   ↾_s cress 13405  ._rcmulr 13485  Scalarcsca 13487   ·_𝑠 cvsca 13488  Ringcrg 14384  SubRingcsubrg 14609  LModclmod 14707  LSubSpclss 14773  AssAlgcasa 15080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-pre-ltirr 8292  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8363  df-mnf 8364  df-ltxr 8366  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-iress 13412  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13665  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-grp 13861  df-minusg 13862  df-sbg 13863  df-subg 14026  df-mgp 14302  df-ur 14347  df-ring 14386  df-subrg 14611  df-lmod 14709  df-lssm 14774  df-assa 15083

This theorem is used by:  issubassa  15097  rnasclassa  15122