Theorem sspmlem 8391

Description: Lemma for sspm 8393 and others.

Hypotheses

Ref	Expression
sspmlem.y	|- Y = (Base` W)
sspmlem.h	|- H = (SubSp` U)
sspmlem.1	|- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (xFy) = (xGy))
sspmlem.2	|- (W e. NrmCVec -> F:(Y X. Y)-->R)
sspmlem.3	|- (U e. NrmCVec -> G:((Base` U) X. (Base` U))-->S)

Assertion

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

Distinct variable groups:   x,y,F   x,G,y   x,H,y   x,U,y   x,W,y   x,Y,y

Proof of Theorem sspmlem

Step	Hyp	Ref	Expression
1		sspmlem.1	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (xFy) = (xGy))
2		oprvalres 4033	. . . . . . 7 |- ((x e. Y /\ y e. Y) -> (x(G |` (Y X. Y))y) = (xGy))
3	2	adantl 388	. . . . . 6 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (x(G |` (Y X. Y))y) = (xGy))
4	1, 3	eqtr4d 1510	. . . . 5 |- (((U e. NrmCVec /\ W e. H) /\ (x e. Y /\ y e. Y)) -> (xFy) = (x(G |` (Y X. Y))y))
5	4	ex 373	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> ((x e. Y /\ y e. Y) -> (xFy) = (x(G |` (Y X. Y))y)))
6	5	r19.21aivv 1720	. . 3 |- ((U e. NrmCVec /\ W e. H) -> A.x e. Y A.y e. Y (xFy) = (x(G |` (Y X. Y))y))
7		eqid 1475	. . 3 |- (Y X. Y) = (Y X. Y)
8	6, 7	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)))
9		eqfnoprval 4016	. . 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))))
10		sspmlem.h	. . . . 5 |- H = (SubSp` U)
11	10	sspnv 8385	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> W e. NrmCVec)
12		sspmlem.2	. . . 4 |- (W e. NrmCVec -> F:(Y X. Y)-->R)
13		ffn 3627	. . . 4 |- (F:(Y X. Y)-->R -> F Fn (Y X. Y))
14	11, 12, 13	3syl 20	. . 3 |- ((U e. NrmCVec /\ W e. H) -> F Fn (Y X. Y))
15		fnssres 3600	. . . 4 |- ((G Fn ((Base` U) X. (Base` U)) /\ (Y X. Y) (_ ((Base` U) X. (Base` U))) -> (G |` (Y X. Y)) Fn (Y X. Y))
16		sspmlem.3	. . . . . 6 |- (U e. NrmCVec -> G:((Base` U) X. (Base` U))-->S)
17		ffn 3627	. . . . . 6 |- (G:((Base` U) X. (Base` U))-->S -> G Fn ((Base` U) X. (Base` U)))
18	16, 17	syl 10	. . . . 5 |- (U e. NrmCVec -> G Fn ((Base` U) X. (Base` U)))
19	18	adantr 389	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> G Fn ((Base` U) X. (Base` U)))
20		eqid 1475	. . . . . 6 |- (Base` U) = (Base` U)
21		sspmlem.y	. . . . . 6 |- Y = (Base` W)
22	20, 21, 10	sspba 8386	. . . . 5 |- ((U e. NrmCVec /\ W e. H) -> Y (_ (Base` U))
23		ssxp 3256	. . . . . 6 |- ((Y (_ (Base` U) /\ Y (_ (Base` U)) -> (Y X. Y) (_ ((Base` U) X. (Base` U)))
24	23	anidms 434	. . . . 5 |- (Y (_ (Base` U) -> (Y X. Y) (_ ((Base` U) X. (Base` U)))
25	22, 24	syl 10	. . . 4 |- ((U e. NrmCVec /\ W e. H) -> (Y X. Y) (_ ((Base` U) X. (Base` U)))
26	15, 19, 25	sylanc 471	. . 3 |- ((U e. NrmCVec /\ W e. H) -> (G |` (Y X. Y)) Fn (Y X. Y))
27	9, 14, 26	sylanc 471	. 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))))
28	8, 27	mpbird 196	1 |- ((U e. NrmCVec /\ W e. H) -> F = (G |` (Y X. Y)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   (_ wss 2047   X. cxp 3168   |` cres 3172   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  NrmCVeccnv 8203  Basecba 8205  SubSpcss 8380

This theorem is referenced by:  sspm 8393  sspi 8398  sspims 8400

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-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-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-nm 8219  df-ssp 8381

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