Theorem ssps 27585

Description: Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ssps.y	|- Y = ( BaseSet ` W )
ssps.s	|- S = ( .sOLD ` U )
ssps.r	|- R = ( .sOLD ` W )
ssps.h	|- H = ( SubSp ` U )

Assertion

Ref	Expression
ssps	|- ( ( U e. NrmCVec /\ W e. H ) -> R = ( S |` ( CC X. Y ) ) )

Proof of Theorem ssps

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . . . 11 |- ( BaseSet ` U ) = ( BaseSet ` U )
2		ssps.s	. . . . . . . . . . 11 |- S = ( .sOLD ` U )
3	1, 2	nvsf 27474	. . . . . . . . . 10 |- ( U e. NrmCVec -> S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) )
4		ffun 6048	. . . . . . . . . 10 |- ( S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) -> Fun S )
5	3, 4	syl 17	. . . . . . . . 9 |- ( U e. NrmCVec -> Fun S )
6		funres 5929	. . . . . . . . 9 |- ( Fun S -> Fun ( S |` ( CC X. Y ) ) )
7	5, 6	syl 17	. . . . . . . 8 |- ( U e. NrmCVec -> Fun ( S |` ( CC X. Y ) ) )
8	7	adantr 481	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. H ) -> Fun ( S |` ( CC X. Y ) ) )
9		ssps.h	. . . . . . . . . 10 |- H = ( SubSp ` U )
10	9	sspnv 27581	. . . . . . . . 9 |- ( ( U e. NrmCVec /\ W e. H ) -> W e. NrmCVec )
11		ssps.y	. . . . . . . . . 10 |- Y = ( BaseSet ` W )
12		ssps.r	. . . . . . . . . 10 |- R = ( .sOLD ` W )
13	11, 12	nvsf 27474	. . . . . . . . 9 |- ( W e. NrmCVec -> R : ( CC X. Y ) --> Y )
14	10, 13	syl 17	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. H ) -> R : ( CC X. Y ) --> Y )
15		ffn 6045	. . . . . . . 8 |- ( R : ( CC X. Y ) --> Y -> R Fn ( CC X. Y ) )
16	14, 15	syl 17	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. H ) -> R Fn ( CC X. Y ) )
17		fnresdm 6000	. . . . . . . . 9 |- ( R Fn ( CC X. Y ) -> ( R |` ( CC X. Y ) ) = R )
18	16, 17	syl 17	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. H ) -> ( R |` ( CC X. Y ) ) = R )
19		eqid 2622	. . . . . . . . . . . 12 |- ( +v ` U ) = ( +v ` U )
20		eqid 2622	. . . . . . . . . . . 12 |- ( +v ` W ) = ( +v ` W )
21		eqid 2622	. . . . . . . . . . . 12 |- ( normCV ` U ) = ( normCV ` U )
22		eqid 2622	. . . . . . . . . . . 12 |- ( normCV ` W ) = ( normCV ` W )
23	19, 20, 2, 12, 21, 22, 9	isssp 27579	. . . . . . . . . . 11 |- ( U e. NrmCVec -> ( W e. H <-> ( W e. NrmCVec /\ ( ( +v ` W ) C_ ( +v ` U ) /\ R C_ S /\ ( normCV ` W ) C_ ( normCV ` U ) ) ) ) )
24	23	simplbda 654	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ W e. H ) -> ( ( +v ` W ) C_ ( +v ` U ) /\ R C_ S /\ ( normCV ` W ) C_ ( normCV ` U ) ) )
25	24	simp2d 1074	. . . . . . . . 9 |- ( ( U e. NrmCVec /\ W e. H ) -> R C_ S )
26		ssres 5424	. . . . . . . . 9 |- ( R C_ S -> ( R |` ( CC X. Y ) ) C_ ( S |` ( CC X. Y ) ) )
27	25, 26	syl 17	. . . . . . . 8 |- ( ( U e. NrmCVec /\ W e. H ) -> ( R |` ( CC X. Y ) ) C_ ( S |` ( CC X. Y ) ) )
28	18, 27	eqsstr3d 3640	. . . . . . 7 |- ( ( U e. NrmCVec /\ W e. H ) -> R C_ ( S |` ( CC X. Y ) ) )
29	8, 16, 28	3jca 1242	. . . . . 6 |- ( ( U e. NrmCVec /\ W e. H ) -> ( Fun ( S |` ( CC X. Y ) ) /\ R Fn ( CC X. Y ) /\ R C_ ( S |` ( CC X. Y ) ) ) )
30		oprssov 6803	. . . . . 6 |- ( ( ( Fun ( S |` ( CC X. Y ) ) /\ R Fn ( CC X. Y ) /\ R C_ ( S |` ( CC X. Y ) ) ) /\ ( x e. CC /\ y e. Y ) ) -> ( x ( S |` ( CC X. Y ) ) y ) = ( x R y ) )
31	29, 30	sylan 488	. . . . 5 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( x e. CC /\ y e. Y ) ) -> ( x ( S |` ( CC X. Y ) ) y ) = ( x R y ) )
32	31	eqcomd 2628	. . . 4 |- ( ( ( U e. NrmCVec /\ W e. H ) /\ ( x e. CC /\ y e. Y ) ) -> ( x R y ) = ( x ( S |` ( CC X. Y ) ) y ) )
33	32	ralrimivva 2971	. . 3 |- ( ( U e. NrmCVec /\ W e. H ) -> A. x e. CC A. y e. Y ( x R y ) = ( x ( S |` ( CC X. Y ) ) y ) )
34		eqid 2622	. . 3 |- ( CC X. Y ) = ( CC X. Y )
35	33, 34	jctil 560	. 2 |- ( ( U e. NrmCVec /\ W e. H ) -> ( ( CC X. Y ) = ( CC X. Y ) /\ A. x e. CC A. y e. Y ( x R y ) = ( x ( S |` ( CC X. Y ) ) y ) ) )
36		ffn 6045	. . . . . 6 |- ( S : ( CC X. ( BaseSet ` U ) ) --> ( BaseSet ` U ) -> S Fn ( CC X. ( BaseSet ` U ) ) )
37	3, 36	syl 17	. . . . 5 |- ( U e. NrmCVec -> S Fn ( CC X. ( BaseSet ` U ) ) )
38	37	adantr 481	. . . 4 |- ( ( U e. NrmCVec /\ W e. H ) -> S Fn ( CC X. ( BaseSet ` U ) ) )
39		ssid 3624	. . . . 5 |- CC C_ CC
40	1, 11, 9	sspba 27582	. . . . 5 |- ( ( U e. NrmCVec /\ W e. H ) -> Y C_ ( BaseSet ` U ) )
41		xpss12 5225	. . . . 5 |- ( ( CC C_ CC /\ Y C_ ( BaseSet ` U ) ) -> ( CC X. Y ) C_ ( CC X. ( BaseSet ` U ) ) )
42	39, 40, 41	sylancr 695	. . . 4 |- ( ( U e. NrmCVec /\ W e. H ) -> ( CC X. Y ) C_ ( CC X. ( BaseSet ` U ) ) )
43		fnssres 6004	. . . 4 |- ( ( S Fn ( CC X. ( BaseSet ` U ) ) /\ ( CC X. Y ) C_ ( CC X. ( BaseSet ` U ) ) ) -> ( S |` ( CC X. Y ) ) Fn ( CC X. Y ) )
44	38, 42, 43	syl2anc 693	. . 3 |- ( ( U e. NrmCVec /\ W e. H ) -> ( S |` ( CC X. Y ) ) Fn ( CC X. Y ) )
45		eqfnov 6766	. . 3 |- ( ( R Fn ( CC X. Y ) /\ ( S |` ( CC X. Y ) ) Fn ( CC X. Y ) ) -> ( R = ( S |` ( CC X. Y ) ) <-> ( ( CC X. Y ) = ( CC X. Y ) /\ A. x e. CC A. y e. Y ( x R y ) = ( x ( S |` ( CC X. Y ) ) y ) ) ) )
46	16, 44, 45	syl2anc 693	. 2 |- ( ( U e. NrmCVec /\ W e. H ) -> ( R = ( S |` ( CC X. Y ) ) <-> ( ( CC X. Y ) = ( CC X. Y ) /\ A. x e. CC A. y e. Y ( x R y ) = ( x ( S |` ( CC X. Y ) ) y ) ) ) )
47	35, 46	mpbird 247	1 |- ( ( U e. NrmCVec /\ W e. H ) -> R = ( S |` ( CC X. Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   X. cxp 5112   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   norm_CVcnmcv 27445   SubSpcss 27576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-ssp 27577

This theorem is referenced by:  sspsval  27586