Theorem issubassa2 21867

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 2739	. . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊)
3		eqid 2739	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
4	1, 2, 3	rnascl 21866	. . . 4 ⊢ (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
5	4	ad2antrr 732	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
6		issubassa2.l	. . . 4 ⊢ 𝐿 = (LSubSp‘𝑊)
7		assalmod 21835	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
8	7	ad2antrr 732	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ LMod)
9		simpr 485	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ 𝐿)
10	2	subrg1cl 20552	. . . . 5 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) ∈ 𝑆)
11	10	ad2antlr 733	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → (1r‘𝑊) ∈ 𝑆)
12	6, 3, 8, 9, 11	ellspsn5 20986	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ((LSpan‘𝑊)‘{(1r‘𝑊)}) ⊆ 𝑆)
13	5, 12	eqsstrd 3949	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 ⊆ 𝑆)
14		subrgsubg 20549	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
15	14	ad2antlr 733	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16		simplll 780	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑊 ∈ AssAlg)
17		simprl 776	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18		eqid 2739	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	18	subrgss 20544	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
20	19	ad2antlr 733	. . . . . . . 8 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
21	20	sselda 3915	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝑊))
22	21	adantrl 722	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝑊))
23		eqid 2739	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2739	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25		eqid 2739	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
26		eqid 2739	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
27	1, 23, 24, 18, 25, 26	asclmul1 21861	. . . . . 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 1379	. . . . 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 781	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30		simplr 774	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴 ⊆ 𝑆)
31	1, 23, 24	asclfn 21855	. . . . . . . . . 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 7021	. . . . . . . . 9 ⊢ ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
34	32, 33	sylan 586	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
35	30, 34	sseldd 3916	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ 𝑆)
36	35	adantrr 723	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝐴‘𝑥) ∈ 𝑆)
37		simprr 778	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
38	25	subrgmcl 20556	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴‘𝑥) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
39	29, 36, 37, 38	syl3anc 1379	. . . . 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 2840	. . . 4 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)
41	40	ralrimivva 3182	. . 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 20952	. . . . 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 732	. . 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 719	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ 𝐿)
46	13, 45	impbida 806	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883  {csn 4555  ran crn 5619   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  SubGrpcsubg 19087  1_rcur 20153  SubRingcsubrg 20541  LModclmod 20850  LSubSpclss 20921  LSpanclspn 20961  AssAlgcasa 21825  algSccascl 21827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904  df-sbg 18905  df-subg 19090  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-subrng 20518  df-subrg 20542  df-lmod 20852  df-lss 20922  df-lsp 20962  df-assa 21828  df-ascl 21830

This theorem is referenced by:  rnasclassa  21870  aspval2  21873