Theorem issubassa2 21665

Description: A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
issubassa2.a	⊢ 𝐴 = (algSc‘𝑊)
issubassa2.l	⊢ 𝐿 = (LSubSp‘𝑊)

Assertion

Ref	Expression
issubassa2	⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))

Proof of Theorem issubassa2

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

Step	Hyp	Ref	Expression
1		issubassa2.a	. . . . 5 ⊢ 𝐴 = (algSc‘𝑊)
2		eqid 2730	. . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊)
3		eqid 2730	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
4	1, 2, 3	rnascl 21664	. . . 4 ⊢ (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
5	4	ad2antrr 722	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
6		issubassa2.l	. . . 4 ⊢ 𝐿 = (LSubSp‘𝑊)
7		assalmod 21634	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
8	7	ad2antrr 722	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ LMod)
9		simpr 483	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ 𝐿)
10	2	subrg1cl 20470	. . . . 5 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) ∈ 𝑆)
11	10	ad2antlr 723	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → (1r‘𝑊) ∈ 𝑆)
12	6, 3, 8, 9, 11	lspsnel5a 20751	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ((LSpan‘𝑊)‘{(1r‘𝑊)}) ⊆ 𝑆)
13	5, 12	eqsstrd 4019	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 ⊆ 𝑆)
14		subrgsubg 20467	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
15	14	ad2antlr 723	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16		simplll 771	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑊 ∈ AssAlg)
17		simprl 767	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	18	subrgss 20462	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
20	19	ad2antlr 723	. . . . . . . 8 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
21	20	sselda 3981	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝑊))
22	21	adantrl 712	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝑊))
23		eqid 2730	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2730	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25		eqid 2730	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
26		eqid 2730	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
27	1, 23, 24, 18, 25, 26	asclmul1 21659	. . . . . 6 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝑊)𝑦))
28	16, 17, 22, 27	syl3anc 1369	. . . . 5 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝑊)𝑦))
29		simpllr 772	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30		simplr 765	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴 ⊆ 𝑆)
31	1, 23, 24	asclfn 21654	. . . . . . . . . 10 ⊢ 𝐴 Fn (Base‘(Scalar‘𝑊))
32	31	a1i 11	. . . . . . . . 9 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝐴 Fn (Base‘(Scalar‘𝑊)))
33		fnfvelrn 7081	. . . . . . . . 9 ⊢ ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
34	32, 33	sylan 578	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
35	30, 34	sseldd 3982	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ 𝑆)
36	35	adantrr 713	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝐴‘𝑥) ∈ 𝑆)
37		simprr 769	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
38	25	subrgmcl 20474	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴‘𝑥) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
39	29, 36, 37, 38	syl3anc 1369	. . . . 5 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
40	28, 39	eqeltrrd 2832	. . . 4 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)
41	40	ralrimivva 3198	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ 𝑆 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)
42	23, 24, 18, 26, 6	islss4 20717	. . . . 5 ⊢ (𝑊 ∈ LMod → (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ 𝑆 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)))
43	7, 42	syl 17	. . . 4 ⊢ (𝑊 ∈ AssAlg → (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ 𝑆 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)))
44	43	ad2antrr 722	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ (SubGrp‘𝑊) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ 𝑆 (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)))
45	15, 41, 44	mpbir2and 709	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ 𝐿)
46	13, 45	impbida 797	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059   ⊆ wss 3947  {csn 4627  ran crn 5676   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411  Basecbs 17148  ._rcmulr 17202  Scalarcsca 17204   ·_𝑠 cvsca 17205  SubGrpcsubg 19036  1_rcur 20075  SubRingcsubrg 20457  LModclmod 20614  LSubSpclss 20686  LSpanclspn 20726  AssAlgcasa 21624  algSccascl 21626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-subrng 20434  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-assa 21627  df-ascl 21629

This theorem is referenced by:  rnasclassa  21668  aspval2  21671