Theorem issubassa2 21446

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 2733	. . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊)
3		eqid 2733	. . . . 5 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊)
4	1, 2, 3	rnascl 21445	. . . 4 ⊢ (𝑊 ∈ AssAlg → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
5	4	ad2antrr 725	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 = ((LSpan‘𝑊)‘{(1r‘𝑊)}))
6		issubassa2.l	. . . 4 ⊢ 𝐿 = (LSubSp‘𝑊)
7		assalmod 21415	. . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod)
8	7	ad2antrr 725	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑊 ∈ LMod)
9		simpr 486	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → 𝑆 ∈ 𝐿)
10	2	subrg1cl 20327	. . . . 5 ⊢ (𝑆 ∈ (SubRing‘𝑊) → (1r‘𝑊) ∈ 𝑆)
11	10	ad2antlr 726	. . . 4 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → (1r‘𝑊) ∈ 𝑆)
12	6, 3, 8, 9, 11	lspsnel5a 20607	. . 3 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ((LSpan‘𝑊)‘{(1r‘𝑊)}) ⊆ 𝑆)
13	5, 12	eqsstrd 4021	. 2 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ 𝑆 ∈ 𝐿) → ran 𝐴 ⊆ 𝑆)
14		subrgsubg 20325	. . . 4 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ∈ (SubGrp‘𝑊))
15	14	ad2antlr 726	. . 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 2733	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
19	18	subrgss 20320	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
20	19	ad2antlr 726	. . . . . . . 8 ⊢ (((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) → 𝑆 ⊆ (Base‘𝑊))
21	20	sselda 3983	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ (Base‘𝑊))
22	21	adantrl 715	. . . . . 6 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ (Base‘𝑊))
23		eqid 2733	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
24		eqid 2733	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
25		eqid 2733	. . . . . . 7 ⊢ (.r‘𝑊) = (.r‘𝑊)
26		eqid 2733	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
27	1, 23, 24, 18, 25, 26	asclmul1 21440	. . . . . 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 1372	. . . . 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 21435	. . . . . . . . . 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 7083	. . . . . . . . 9 ⊢ ((𝐴 Fn (Base‘(Scalar‘𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
34	32, 33	sylan 581	. . . . . . . 8 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ ran 𝐴)
35	30, 34	sseldd 3984	. . . . . . 7 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑊))) → (𝐴‘𝑥) ∈ 𝑆)
36	35	adantrr 716	. . . . . 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 20331	. . . . . 6 ⊢ ((𝑆 ∈ (SubRing‘𝑊) ∧ (𝐴‘𝑥) ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝐴‘𝑥)(.r‘𝑊)𝑦) ∈ 𝑆)
39	29, 36, 37, 38	syl3anc 1372	. . . . 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 2835	. . . 4 ⊢ ((((𝑊 ∈ AssAlg ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ ran 𝐴 ⊆ 𝑆) ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ 𝑆)) → (𝑥( ·𝑠 ‘𝑊)𝑦) ∈ 𝑆)
41	40	ralrimivva 3201	. . 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 20573	. . . . 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 725	. . 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 712	. 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 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949  {csn 4629  ran crn 5678   Fn wfn 6539  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  ._rcmulr 17198  Scalarcsca 17200   ·_𝑠 cvsca 17201  SubGrpcsubg 19000  1_rcur 20004  SubRingcsubrg 20315  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  AssAlgcasa 21405  algSccascl 21407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-subg 19003  df-mgp 19988  df-ur 20005  df-ring 20058  df-subrg 20317  df-lmod 20473  df-lss 20543  df-lsp 20583  df-assa 21408  df-ascl 21410

This theorem is referenced by:  rnasclassa  21449  aspval2  21452