Theorem sspmlem 8387

Description: Lemma for sspm 8389 and others.

Hypotheses

Ref	Expression
sspmlem.y	⊢ Y = (Base ‘W)
sspmlem.h	⊢ H = (SubSp ‘U)
sspmlem.1	⊢ (((U ∈ NrmCVec ⋀ W ∈ H) ⋀ (x ∈ Y ⋀ y ∈ Y)) → (xFy) = (xGy))
sspmlem.2	⊢ (W ∈ NrmCVec → F:(Y × Y)–→R)
sspmlem.3	⊢ (U ∈ NrmCVec → G:((Base ‘U) × (Base ‘U))–→S)

Assertion

Ref	Expression
sspmlem	⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → F = (G ↾ (Y × 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 ∈ NrmCVec ⋀ W ∈ H) ⋀ (x ∈ Y ⋀ y ∈ Y)) → (xFy) = (xGy))
2		oprvalres 4039	. . . . . . 7 ⊢ ((x ∈ Y ⋀ y ∈ Y) → (x(G ↾ (Y × Y))y) = (xGy))
3	2	adantl 390	. . . . . 6 ⊢ (((U ∈ NrmCVec ⋀ W ∈ H) ⋀ (x ∈ Y ⋀ y ∈ Y)) → (x(G ↾ (Y × Y))y) = (xGy))
4	1, 3	eqtr4d 1513	. . . . 5 ⊢ (((U ∈ NrmCVec ⋀ W ∈ H) ⋀ (x ∈ Y ⋀ y ∈ Y)) → (xFy) = (x(G ↾ (Y × Y))y))
5	4	ex 373	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → ((x ∈ Y ⋀ y ∈ Y) → (xFy) = (x(G ↾ (Y × Y))y)))
6	5	r19.21aivv 1723	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → ∀x ∈ Y ∀y ∈ Y (xFy) = (x(G ↾ (Y × Y))y))
7		eqid 1478	. . 3 ⊢ (Y × Y) = (Y × Y)
8	6, 7	jctil 292	. 2 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → ((Y × Y) = (Y × Y) ⋀ ∀x ∈ Y ∀y ∈ Y (xFy) = (x(G ↾ (Y × Y))y)))
9		eqfnoprval 4022	. . 3 ⊢ ((F Fn (Y × Y) ⋀ (G ↾ (Y × Y)) Fn (Y × Y)) → (F = (G ↾ (Y × Y)) ↔ ((Y × Y) = (Y × Y) ⋀ ∀x ∈ Y ∀y ∈ Y (xFy) = (x(G ↾ (Y × Y))y))))
10		sspmlem.h	. . . . 5 ⊢ H = (SubSp ‘U)
11	10	sspnv 8381	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → W ∈ NrmCVec)
12		sspmlem.2	. . . 4 ⊢ (W ∈ NrmCVec → F:(Y × Y)–→R)
13		ffn 3633	. . . 4 ⊢ (F:(Y × Y)–→R → F Fn (Y × Y))
14	11, 12, 13	3syl 20	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → F Fn (Y × Y))
15		fnssres 3606	. . . 4 ⊢ ((G Fn ((Base ‘U) × (Base ‘U)) ⋀ (Y × Y) ⊆ ((Base ‘U) × (Base ‘U))) → (G ↾ (Y × Y)) Fn (Y × Y))
16		sspmlem.3	. . . . . 6 ⊢ (U ∈ NrmCVec → G:((Base ‘U) × (Base ‘U))–→S)
17		ffn 3633	. . . . . 6 ⊢ (G:((Base ‘U) × (Base ‘U))–→S → G Fn ((Base ‘U) × (Base ‘U)))
18	16, 17	syl 10	. . . . 5 ⊢ (U ∈ NrmCVec → G Fn ((Base ‘U) × (Base ‘U)))
19	18	adantr 391	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → G Fn ((Base ‘U) × (Base ‘U)))
20		eqid 1478	. . . . . 6 ⊢ (Base ‘U) = (Base ‘U)
21		sspmlem.y	. . . . . 6 ⊢ Y = (Base ‘W)
22	20, 21, 10	sspba 8382	. . . . 5 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → Y ⊆ (Base ‘U))
23		ssxp 3262	. . . . . 6 ⊢ ((Y ⊆ (Base ‘U) ⋀ Y ⊆ (Base ‘U)) → (Y × Y) ⊆ ((Base ‘U) × (Base ‘U)))
24	23	anidms 436	. . . . 5 ⊢ (Y ⊆ (Base ‘U) → (Y × Y) ⊆ ((Base ‘U) × (Base ‘U)))
25	22, 24	syl 10	. . . 4 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → (Y × Y) ⊆ ((Base ‘U) × (Base ‘U)))
26	15, 19, 25	sylanc 473	. . 3 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → (G ↾ (Y × Y)) Fn (Y × Y))
27	9, 14, 26	sylanc 473	. 2 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → (F = (G ↾ (Y × Y)) ↔ ((Y × Y) = (Y × Y) ⋀ ∀x ∈ Y ∀y ∈ Y (xFy) = (x(G ↾ (Y × Y))y))))
28	8, 27	mpbird 196	1 ⊢ ((U ∈ NrmCVec ⋀ W ∈ H) → F = (G ↾ (Y × Y)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648   ⊆ wss 2050   × cxp 3174   ↾ cres 3178   Fn wfn 3183  –→wf 3184   ‘cfv 3188  (class class class)co 3969  NrmCVeccnv 8199  Basecba 8201  SubSpcss 8376

This theorem is referenced by:  sspm 8389  sspi 8394  sspims 8396

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-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-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-nm 8215  df-ssp 8377

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