Theorem issubassa2 21930

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 2735	. . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊)
3		eqid 2735	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
4	1, 2, 3	rnascl 21929	. . . 4 ⊢ (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
5	4	ad2antrr 726	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
6		issubassa2.l	. . . 4 ⊢ 𝐿 = (LSubSp‘𝑊)
7		assalmod 21898	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
8	7	ad2antrr 726	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ LMod)
9		simpr 484	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ 𝐿)
10	2	subrg1cl 20597	. . . . 5 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) ∈ 𝑆)
11	10	ad2antlr 727	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → (1r‘𝑊) ∈ 𝑆)
12	6, 3, 8, 9, 11	ellspsn5 21012	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ((LSpan‘𝑊)‘{(1r‘𝑊)}) ⊆ 𝑆)
13	5, 12	eqsstrd 4034	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 ⊆ 𝑆)
14		subrgsubg 20594	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
15	14	ad2antlr 727	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ (SubGrp‘𝑊))
16		simplll 775	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑊 ∈ AssAlg)
17		simprl 771	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	18	subrgss 20589	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
20	19	ad2antlr 727	. . . . . . . 8 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
21	20	sselda 3995	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝑊))
22	21	adantrl 716	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝑊))
23		eqid 2735	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2735	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25		eqid 2735	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
26		eqid 2735	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
27	1, 23, 24, 18, 25, 26	asclmul1 21924	. . . . . 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 1370	. . . . 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 776	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30		simplr 769	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴 ⊆ 𝑆)
31	1, 23, 24	asclfn 21919	. . . . . . . . . 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 7100	. . . . . . . . 9 ⊢ ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
34	32, 33	sylan 580	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
35	30, 34	sseldd 3996	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ 𝑆)
36	35	adantrr 717	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝐴‘𝑥) ∈ 𝑆)
37		simprr 773	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
38	25	subrgmcl 20601	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴‘𝑥) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
39	29, 36, 37, 38	syl3anc 1370	. . . . 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 3200	. . 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 20978	. . . . 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 726	. . 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 713	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ∈ 𝐿)
46	13, 45	impbida 801	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963  {csn 4631  ran crn 5690   Fn wfn 6558  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  ._rcmulr 17299  Scalarcsca 17301   ·_𝑠 cvsca 17302  SubGrpcsubg 19151  1_rcur 20199  SubRingcsubrg 20586  LModclmod 20875  LSubSpclss 20947  LSpanclspn 20987  AssAlgcasa 21888  algSccascl 21890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-assa 21891  df-ascl 21893

This theorem is referenced by:  rnasclassa  21933  aspval2  21936