Theorem sspg 9726

Description: Vector addition on a subspace is a restriction of vector addition on the parent space.

Hypotheses

Ref	Expression
sspg.y	|- Y = (BaseSet` W)
sspg.g	|- G = (+v` U)
sspg.f	|- F = (+v` W)
sspg.h	|- H = (SubSp` U)

Assertion

Ref	Expression
sspg	|- ((U e. NrmCVec /\ W e. H) -> F = (G |` (Y X. Y)))

Proof of Theorem sspg

Step	Hyp	Ref	Expression
1		oprssoprv 4964	. . . . . . 7 |- (((Fun (G |` (Y X. Y)) /\ F Fn (Y X. Y) /\ F C_ (G |` (Y X. Y))) /\ (x e. Y /\ y e. Y)) -> (x(G |` (Y X. Y))y) = (xFy))
2		eqid 1884	. . . . . . . . . . 11 |- (BaseSet` U) = (BaseSet` U)
3		sspg.g	. . . . . . . . . . 11 |- G = (+v` U)
4	2, 3	nvgf 9569	. . . . . . . . . 10 |- (U e. NrmCVec -> G:((BaseSet` U) X. (BaseSet` U))-->(BaseSet` U))
5		ffun 4565	. . . . . . . . . 10 |- (G:((BaseSet` U) X. (BaseSet` U))-->(BaseSet` U) -> Fun G)
6		funres 4459	. . . . . . . . . 10 |- (Fun G -> Fun (G |` (Y X. Y)))
7	4, 5, 6	3syl 24	. . . . . . . . 9 |- (U e. NrmCVec -> Fun (G |` (Y X. Y)))
8	7	adantr 425	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> Fun (G |` (Y X. Y)))
9		sspg.h	. . . . . . . . . 10 |- H = (SubSp` U)
10	9	sspnv 9724	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
11		sspg.y	. . . . . . . . . 10 |- Y = (BaseSet` W)
12		sspg.f	. . . . . . . . . 10 |- F = (+v` W)
13	11, 12	nvgf 9569	. . . . . . . . 9 |- (W e. NrmCVec -> F:(Y X. Y)-->Y)
14		ffn 4562	. . . . . . . . 9 |- (F:(Y X. Y)-->Y -> F Fn (Y X. Y))
15	10, 13, 14	3syl 24	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> F Fn (Y X. Y))
16	10, 13	syl 12	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> F:(Y X. Y)-->Y)
17		fnresdm 4522	. . . . . . . . . 10 |- (F Fn (Y X. Y) -> (F |` (Y X. Y)) = F)
18	16, 14, 17	3syl 24	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (F |` (Y X. Y)) = F)
19		eqid 1884	. . . . . . . . . . . 12 |- (.s` U) = (.s` U)
20		eqid 1884	. . . . . . . . . . . 12 |- (.s` W) = (.s` W)
21		eqid 1884	. . . . . . . . . . . 12 |- (norm` U) = (norm` U)
22		eqid 1884	. . . . . . . . . . . 12 |- (norm` W) = (norm` W)
23	3, 12, 19, 20, 21, 22, 9	isssp 9722	. . . . . . . . . . 11 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ (F C_ G /\ (.s` W) C_ (.s` U) /\ (norm` W) C_ (norm` U)))))
24	23	simplbda 465	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> (F C_ G /\ (.s` W) C_ (.s` U) /\ (norm` W) C_ (norm` U)))
25		simp1 876	. . . . . . . . . 10 |- ((F C_ G /\ (.s` W) C_ (.s` U) /\ (norm` W) C_ (norm` U)) -> F C_ G)
26		ssres 4239	. . . . . . . . . 10 |- (F C_ G -> (F |` (Y X. Y)) C_ (G |` (Y X. Y)))
27	24, 25, 26	3syl 24	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (F |` (Y X. Y)) C_ (G |` (Y X. Y)))
28	18, 27	eqsstr3d 2652	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> F C_ (G |` (Y X. Y)))
29	8, 15, 28	3jca 1050	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (Fun (G |` (Y X. Y)) /\ F Fn (Y X. Y) /\ F C_ (G |` (Y X. Y))))
30	1, 29	sylan 497	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (x(G |` (Y X. Y))y) = (xFy))
31	30	eqcomd 1889	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (xFy) = (x(G |` (Y X. Y))y))
32	31	ex 402	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> ((x e. Y /\ y e. Y) -> (xFy) = (x(G |` (Y X. Y))y)))
33	32	r19.21aivv 2183	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y))
34		eqid 1884	. . 3 |- (Y X. Y) = (Y X. Y)
35	33, 34	jctil 316	. 2 |- ((U e. NrmCVec /\ W e. H) -> ((Y X. Y) = (Y X. Y) /\ A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y)))
36		ffn 4562	. . . . . 6 |- (G:((BaseSet` U) X. (BaseSet` U))-->(BaseSet` U) -> G Fn ((BaseSet` U) X. (BaseSet` U)))
37	4, 36	syl 12	. . . . 5 |- (U e. NrmCVec -> G Fn ((BaseSet` U) X. (BaseSet` U)))
38	37	adantr 425	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> G Fn ((BaseSet` U) X. (BaseSet` U)))
39	2, 11, 9	sspba 9725	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> Y C_ (BaseSet` U))
40		xpss12 4089	. . . . 5 |- ((Y C_ (BaseSet` U) /\ Y C_ (BaseSet` U)) -> (Y X. Y) C_ ((BaseSet` U) X. (BaseSet` U)))
41	39, 39, 40	syl11anc 524	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> (Y X. Y) C_ ((BaseSet` U) X. (BaseSet` U)))
42		fnssres 4526	. . . 4 |- ((G Fn ((BaseSet` U) X. (BaseSet` U)) /\ (Y X. Y) C_ ((BaseSet` U) X. (BaseSet` U))) -> (G |` (Y X. Y)) Fn (Y X. Y))
43	38, 41, 42	syl11anc 524	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (G |` (Y X. Y)) Fn (Y X. Y))
44		eqfnoprv 4945	. . 3 |- ((F Fn (Y X. Y) /\ (G |` (Y X. Y)) Fn (Y X. Y)) -> (F = (G |` (Y X. Y)) <-> ((Y X. Y) = (Y X. Y) /\ A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y))))
45	15, 43, 44	syl11anc 524	. 2 |- ((U e. NrmCVec /\ W e. H) -> (F = (G |` (Y X. Y)) <-> ((Y X. Y) = (Y X. Y) /\ A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y))))
46	35, 45	mpbird 213	1 |- ((U e. NrmCVec /\ W e. H) -> F = (G |` (Y 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  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  SubSpcss 9719

This theorem is referenced by:  sspgval 9727

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-grp 9316  df-gid 9317  df-abl 9408  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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