Theorem sspg 8383

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

Hypotheses

Ref	Expression
sspg.y	|- Y = (Base` 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		oprssoprval 4040	. . . . . . 7 |- (((Fun (G |` (Y X. Y)) /\ F Fn (Y X. Y) /\ F (_ (G |` (Y X. Y))) /\ (x e. Y /\ y e. Y)) -> (x(G |` (Y X. Y))y) = (xFy))
2		eqid 1478	. . . . . . . . . . 11 |- (Base` U) = (Base` U)
3		sspg.g	. . . . . . . . . . 11 |- G = (+v` U)
4	2, 3	nvgf 8233	. . . . . . . . . 10 |- (U e. NrmCVec -> G:((Base` U) X. (Base` U))-->(Base` U))
5		ffun 3635	. . . . . . . . . 10 |- (G:((Base` U) X. (Base` U))-->(Base` U) -> Fun G)
6		funres 3557	. . . . . . . . . 10 |- (Fun G -> Fun (G |` (Y X. Y)))
7	4, 5, 6	3syl 20	. . . . . . . . 9 |- (U e. NrmCVec -> Fun (G |` (Y X. Y)))
8	7	adantr 391	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> Fun (G |` (Y X. Y)))
9		sspg.h	. . . . . . . . . 10 |- H = (SubSp` U)
10	9	sspnv 8381	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
11		sspg.y	. . . . . . . . . 10 |- Y = (Base` W)
12		sspg.f	. . . . . . . . . 10 |- F = (+v` W)
13	11, 12	nvgf 8233	. . . . . . . . 9 |- (W e. NrmCVec -> F:(Y X. Y)-->Y)
14		ffn 3633	. . . . . . . . 9 |- (F:(Y X. Y)-->Y -> F Fn (Y X. Y))
15	10, 13, 14	3syl 20	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> F Fn (Y X. Y))
16	10, 13	syl 10	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> F:(Y X. Y)-->Y)
17		fnresdm 3602	. . . . . . . . . 10 |- (F Fn (Y X. Y) -> (F |` (Y X. Y)) = F)
18	16, 14, 17	3syl 20	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (F |` (Y X. Y)) = F)
19		eqid 1478	. . . . . . . . . . . 12 |- (.s` U) = (.s` U)
20		eqid 1478	. . . . . . . . . . . 12 |- (.s` W) = (.s` W)
21		eqid 1478	. . . . . . . . . . . 12 |- (norm` U) = (norm` U)
22		eqid 1478	. . . . . . . . . . . 12 |- (norm` W) = (norm` W)
23	3, 12, 19, 20, 21, 22, 9	isssp 8379	. . . . . . . . . . 11 |- (U e. NrmCVec -> (W e. H <-> (W e. NrmCVec /\ (F (_ G /\ (.s` W) (_ (.s` U) /\ (norm` W) (_ (norm` U)))))
24	23	pm3.27bda 423	. . . . . . . . . 10 |- ((U e. NrmCVec /\ W e. H) -> (F (_ G /\ (.s` W) (_ (.s` U) /\ (norm` W) (_ (norm` U)))
25		3simp1 790	. . . . . . . . . 10 |- ((F (_ G /\ (.s` W) (_ (.s` U) /\ (norm` W) (_ (norm` U)) -> F (_ G)
26		ssres 3391	. . . . . . . . . 10 |- (F (_ G -> (F |` (Y X. Y)) (_ (G |` (Y X. Y)))
27	24, 25, 26	3syl 20	. . . . . . . . 9 |- ((U e. NrmCVec /\ W e. H) -> (F |` (Y X. Y)) (_ (G |` (Y X. Y)))
28	18, 27	eqsstr3d 2099	. . . . . . . 8 |- ((U e. NrmCVec /\ W e. H) -> F (_ (G |` (Y X. Y)))
29	8, 15, 28	3jca 821	. . . . . . 7 |- ((U e. NrmCVec /\ W e. H) -> (Fun (G |` (Y X. Y)) /\ F Fn (Y X. Y) /\ F (_ (G |` (Y X. Y))))
30	1, 29	sylan 450	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (x(G |` (Y X. Y))y) = (xFy))
31	30	eqcomd 1483	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (xFy) = (x(G |` (Y X. Y))y))
32	31	ex 373	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> ((x e. Y /\ y e. Y) -> (xFy) = (x(G |` (Y X. Y))y)))
33	32	r19.21aivv 1723	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y))
34		eqid 1478	. . 3 |- (Y X. Y) = (Y X. Y)
35	33, 34	jctil 292	. 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		eqfnoprval 4022	. . 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))))
37		fnssres 3606	. . . 4 |- ((G Fn ((Base` U) X. (Base` U)) /\ (Y X. Y) (_ ((Base` U) X. (Base` U))) -> (G |` (Y X. Y)) Fn (Y X. Y))
38		ffn 3633	. . . . . 6 |- (G:((Base` U) X. (Base` U))-->(Base` U) -> G Fn ((Base` U) X. (Base` U)))
39	4, 38	syl 10	. . . . 5 |- (U e. NrmCVec -> G Fn ((Base` U) X. (Base` U)))
40	39	adantr 391	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> G Fn ((Base` U) X. (Base` U)))
41		ssxp 3262	. . . . 5 |- ((Y (_ (Base` U) /\ Y (_ (Base` U)) -> (Y X. Y) (_ ((Base` U) X. (Base` U)))
42	2, 11, 9	sspba 8382	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> Y (_ (Base` U))
43	41, 42, 42	sylanc 473	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> (Y X. Y) (_ ((Base` U) X. (Base` U)))
44	37, 40, 43	sylanc 473	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (G |` (Y X. Y)) Fn (Y X. Y))
45	36, 15, 44	sylanc 473	. 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 196	1 |- ((U e. NrmCVec /\ W e. H) -> F = (G |` (Y X. Y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648   (_ wss 2050   X. cxp 3174   |` cres 3178  Fun wfun 3182   Fn wfn 3183  -->wf 3184  ` cfv 3188  (class class class)co 3969  NrmCVeccnv 8199  +vcpv 8200  Basecba 8201  .scns 8202  normcnm 8205  SubSpcss 8376

This theorem is referenced by:  sspgval 8384

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-abl 8096  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215  df-ssp 8377

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