Theorem ssps 9728

Description: Scalar multiplication on a subspace is a restriction of scalar multiplication on the parent space.

Hypotheses

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

Step	Hyp	Ref	Expression
1		oprssoprv 4964	. . . . . . 7 |- (((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) = (xRy))
2		eqid 1884	. . . . . . . . . . 11 |- (BaseSet` U) = (BaseSet` U)
3		ssps.s	. . . . . . . . . . 11 |- S = (.s` U)
4	2, 3	nvsf 9570	. . . . . . . . . 10 |- (U e. NrmCVec -> S:(CC X. (BaseSet` U))-->(BaseSet` U))
5		ffun 4565	. . . . . . . . . 10 |- (S:(CC X. (BaseSet` U))-->(BaseSet` U) -> Fun S)
6		funres 4459	. . . . . . . . . 10 |- (Fun S -> Fun (S |` (CC X. Y)))
7	4, 5, 6	3syl 24	. . . . . . . . 9 |- (U e. NrmCVec -> Fun (S |` (CC X. Y)))
8	7	adantr 425	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> Fun (S |` (CC X. Y)))
9		ssps.h	. . . . . . . . . 10 |- H = (SubSp` U)
10	9	sspnv 9724	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
11		ssps.y	. . . . . . . . . 10 |- Y = (BaseSet` W)
12		ssps.r	. . . . . . . . . 10 |- R = (.s` W)
13	11, 12	nvsf 9570	. . . . . . . . 9 |- (W e. NrmCVec -> R:(CC X. Y)-->Y)
14		ffn 4562	. . . . . . . . 9 |- (R:(CC X. Y)-->Y -> R Fn (CC X. Y))
15	10, 13, 14	3syl 24	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> R Fn (CC X. Y))
16	10, 13	syl 12	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> R:(CC X. Y)-->Y)
17		fnresdm 4522	. . . . . . . . . 10 |- (R Fn (CC X. Y) -> (R |` (CC X. Y)) = R)
18	16, 14, 17	3syl 24	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (R |` (CC X. Y)) = R)
19		eqid 1884	. . . . . . . . . . . 12 |- (+v` U) = (+v` U)
20		eqid 1884	. . . . . . . . . . . 12 |- (+v` W) = (+v` W)
21		eqid 1884	. . . . . . . . . . . 12 |- (norm` U) = (norm` U)
22		eqid 1884	. . . . . . . . . . . 12 |- (norm` W) = (norm` W)
23	19, 20, 3, 12, 21, 22, 9	isssp 9722	. . . . . . . . . . 11 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ ((+v` W) C_ (+v` U) /\ R C_ S /\ (norm` W) C_ (norm` U)))))
24	23	simplbda 465	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> ((+v` W) C_ (+v` U) /\ R C_ S /\ (norm` W) C_ (norm` U)))
25		simp2 877	. . . . . . . . . 10 |- (((+v` W) C_ (+v` U) /\ R C_ S /\ (norm` W) C_ (norm` U)) -> R C_ S)
26		ssres 4239	. . . . . . . . . 10 |- (R C_ S -> (R |` (CC X. Y)) C_ (S |` (CC X. Y)))
27	24, 25, 26	3syl 24	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (R |` (CC X. Y)) C_ (S |` (CC X. Y)))
28	18, 27	eqsstr3d 2652	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> R C_ (S |` (CC X. Y)))
29	8, 15, 28	3jca 1050	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (Fun (S |` (CC X. Y)) /\ R Fn (CC X. Y) /\ R C_ (S |` (CC X. Y))))
30	1, 29	sylan 497	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. CC /\ y e. Y)) -> (x(S |` (CC X. Y))y) = (xRy))
31	30	eqcomd 1889	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ (x e. CC /\ y e. Y)) -> (xRy) = (x(S |` (CC X. Y))y))
32	31	ex 402	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> ((x e. CC /\ y e. Y) -> (xRy) = (x(S |` (CC X. Y))y)))
33	32	r19.21aivv 2183	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. CC A.y e. Y (xRy) = (x(S |` (CC X. Y))y))
34		eqid 1884	. . 3 |- (CC X. Y) = (CC X. Y)
35	33, 34	jctil 316	. 2 |- ((U e. NrmCVec /\ W e. H) -> ((CC X. Y) = (CC X. Y) /\ A.x e. CC A.y e. Y (xRy) = (x(S |` (CC X. Y))y)))
36		ffn 4562	. . . . . 6 |- (S:(CC X. (BaseSet` U))-->(BaseSet` U) -> S Fn (CC X. (BaseSet` U)))
37	4, 36	syl 12	. . . . 5 |- (U e. NrmCVec -> S Fn (CC X. (BaseSet` U)))
38	37	adantr 425	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> S Fn (CC X. (BaseSet` U)))
39		ssid 2634	. . . . . 6 |- CC C_ CC
40	39	a1i 8	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> CC C_ CC)
41	2, 11, 9	sspba 9725	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> Y C_ (BaseSet` U))
42		xpss12 4089	. . . . 5 |- ((CC C_ CC /\ Y C_ (BaseSet` U)) -> (CC X. Y) C_ (CC X. (BaseSet` U)))
43	40, 41, 42	syl11anc 524	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> (CC X. Y) C_ (CC X. (BaseSet` U)))
44		fnssres 4526	. . . 4 |- ((S Fn (CC X. (BaseSet` U)) /\ (CC X. Y) C_ (CC X. (BaseSet` U))) -> (S |` (CC X. Y)) Fn (CC X. Y))
45	38, 43, 44	syl11anc 524	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (S |` (CC X. Y)) Fn (CC X. Y))
46		eqfnoprv 4945	. . 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 (xRy) = (x(S |` (CC X. Y))y))))
47	15, 45, 46	syl11anc 524	. 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 (xRy) = (x(S |` (CC X. Y))y))))
48	35, 47	mpbird 213	1 |- ((U e. NrmCVec /\ W e. H) -> R = (S |` (CC X. Y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593   X. cxp 3984   |` cres 3988  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  SubSpcss 9719

This theorem is referenced by:  sspsval 9729

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-gid 9317  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ssp 9720

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