Theorem issubassa2 21834

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 2729	. . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊)
3		eqid 2729	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
4	1, 2, 3	rnascl 21833	. . . 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 21802	. . . . 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 20500	. . . . 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 20934	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ((LSpan‘𝑊)‘{(1r‘𝑊)}) ⊆ 𝑆)
13	5, 12	eqsstrd 3978	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 ⊆ 𝑆)
14		subrgsubg 20497	. . . 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 774	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑊 ∈ AssAlg)
17		simprl 770	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ (Base‘(Scalar‘𝑊)))
18		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	18	subrgss 20492	. . . . . . . . 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 3943	. . . . . . 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 2729	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2729	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25		eqid 2729	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
26		eqid 2729	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
27	1, 23, 24, 18, 25, 26	asclmul1 21828	. . . . . 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 1373	. . . . 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 775	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑆 ∈ (SubRing‘𝑊))
30		simplr 768	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → ran 𝐴 ⊆ 𝑆)
31	1, 23, 24	asclfn 21823	. . . . . . . . . 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 7034	. . . . . . . . 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 3944	. . . . . . 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 772	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
38	25	subrgmcl 20504	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴‘𝑥) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
39	29, 36, 37, 38	syl3anc 1373	. . . . 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 2829	. . . 4 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)
41	40	ralrimivva 3178	. . 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 20900	. . . . 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 800	1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑆 ∈ 𝐿 ↔ ran 𝐴 ⊆ 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911  {csn 4585  ran crn 5632   Fn wfn 6494  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  SubGrpcsubg 19034  1_rcur 20101  SubRingcsubrg 20489  LModclmod 20798  LSubSpclss 20869  LSpanclspn 20909  AssAlgcasa 21792  algSccascl 21794

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-0g 17380  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-minusg 18851  df-sbg 18852  df-subg 19037  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-subrng 20466  df-subrg 20490  df-lmod 20800  df-lss 20870  df-lsp 20910  df-assa 21795  df-ascl 21797

This theorem is referenced by:  rnasclassa  21837  aspval2  21840