Theorem ssps 8389

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

Hypotheses

Ref	Expression
ssps.y	|- Y = (Base` 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		oprssoprval 4034	. . . . . . 7 |- (((Fun (S |` (CC X. Y)) /\ R Fn (CC X. Y) /\ R (_ (S |` (CC X. Y))) /\ (x e. CC /\ y e. Y)) -> (x(S |` (CC X. Y))y) = (xRy))
2		eqid 1475	. . . . . . . . . . 11 |- (Base` U) = (Base` U)
3		ssps.s	. . . . . . . . . . 11 |- S = (.s` U)
4	2, 3	nvsf 8238	. . . . . . . . . 10 |- (U e. NrmCVec -> S:(CC X. (Base` U))-->(Base` U))
5		ffun 3629	. . . . . . . . . 10 |- (S:(CC X. (Base` U))-->(Base` U) -> Fun S)
6		funres 3551	. . . . . . . . . 10 |- (Fun S -> Fun (S |` (CC X. Y)))
7	4, 5, 6	3syl 20	. . . . . . . . 9 |- (U e. NrmCVec -> Fun (S |` (CC X. Y)))
8	7	adantr 389	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> Fun (S |` (CC X. Y)))
9		ssps.h	. . . . . . . . . 10 |- H = (SubSp` U)
10	9	sspnv 8385	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
11		ssps.y	. . . . . . . . . 10 |- Y = (Base` W)
12		ssps.r	. . . . . . . . . 10 |- R = (.s` W)
13	11, 12	nvsf 8238	. . . . . . . . 9 |- (W e. NrmCVec -> R:(CC X. Y)-->Y)
14		ffn 3627	. . . . . . . . 9 |- (R:(CC X. Y)-->Y -> R Fn (CC X. Y))
15	10, 13, 14	3syl 20	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> R Fn (CC X. Y))
16	10, 13	syl 10	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> R:(CC X. Y)-->Y)
17		fnresdm 3596	. . . . . . . . . 10 |- (R Fn (CC X. Y) -> (R |` (CC X. Y)) = R)
18	16, 14, 17	3syl 20	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (R |` (CC X. Y)) = R)
19		eqid 1475	. . . . . . . . . . . 12 |- (+v` U) = (+v` U)
20		eqid 1475	. . . . . . . . . . . 12 |- (+v` W) = (+v` W)
21		eqid 1475	. . . . . . . . . . . 12 |- (norm` U) = (norm` U)
22		eqid 1475	. . . . . . . . . . . 12 |- (norm` W) = (norm` W)
23	19, 20, 3, 12, 21, 22, 9	isssp 8383	. . . . . . . . . . 11 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ ((+v` W) (_ (+v` U) /\ R (_ S /\ (norm` W) (_ (norm` U)))))
24	23	pm3.27bda 421	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> ((+v` W) (_ (+v` U) /\ R (_ S /\ (norm` W) (_ (norm` U)))
25		3simp2 789	. . . . . . . . . 10 |- (((+v` W) (_ (+v` U) /\ R (_ S /\ (norm` W) (_ (norm` U)) -> R (_ S)
26		ssres 3385	. . . . . . . . . 10 |- (R (_ S -> (R |` (CC X. Y)) (_ (S |` (CC X. Y)))
27	24, 25, 26	3syl 20	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (R |` (CC X. Y)) (_ (S |` (CC X. Y)))
28	18, 27	eqsstr3d 2096	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> R (_ (S |` (CC X. Y)))
29	8, 15, 28	3jca 819	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (Fun (S |` (CC X. Y)) /\ R Fn (CC X. Y) /\ R (_ (S |` (CC X. Y))))
30	1, 29	sylan 448	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. CC /\ y e. Y)) -> (x(S |` (CC X. Y))y) = (xRy))
31	30	eqcomd 1480	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ (x e. CC /\ y e. Y)) -> (xRy) = (x(S |` (CC X. Y))y))
32	31	ex 373	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> ((x e. CC /\ y e. Y) -> (xRy) = (x(S |` (CC X. Y))y)))
33	32	r19.21aivv 1720	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. CC A.y e. Y (xRy) = (x(S |` (CC X. Y))y))
34		eqid 1475	. . 3 |- (CC X. Y) = (CC X. Y)
35	33, 34	jctil 292	. 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		eqfnoprval 4016	. . 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))))
37		fnssres 3600	. . . 4 |- ((S Fn (CC X. (Base` U)) /\ (CC X. Y) (_ (CC X. (Base` U))) -> (S |` (CC X. Y)) Fn (CC X. Y))
38		ffn 3627	. . . . . 6 |- (S:(CC X. (Base` U))-->(Base` U) -> S Fn (CC X. (Base` U)))
39	4, 38	syl 10	. . . . 5 |- (U e. NrmCVec -> S Fn (CC X. (Base` U)))
40	39	adantr 389	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> S Fn (CC X. (Base` U)))
41		ssxp 3256	. . . . 5 |- ((CC (_ CC /\ Y (_ (Base` U)) -> (CC X. Y) (_ (CC X. (Base` U)))
42		ssid 2080	. . . . . 6 |- CC (_ CC
43	42	a1i 8	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> CC (_ CC)
44	2, 11, 9	sspba 8386	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> Y (_ (Base` U))
45	41, 43, 44	sylanc 471	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> (CC X. Y) (_ (CC X. (Base` U)))
46	37, 40, 45	sylanc 471	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (S |` (CC X. Y)) Fn (CC X. Y))
47	36, 15, 46	sylanc 471	. 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 196	1 |- ((U e. NrmCVec /\ W e. H) -> R = (S |` (CC X. Y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   X. cxp 3168   |` cres 3172  Fun wfun 3176   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  normcnm 8209  SubSpcss 8380

This theorem is referenced by:  sspsval 8390

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219  df-ssp 8381

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