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Metamath Proof Explorer

Theorem sspmlem

Description: Lemma for sspm and others. (Contributed by NM, 1-Feb-2008) (New usage is discouraged.)

Ref Expression
Hypotheses sspmlem.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
sspmlem.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
sspmlem.1 ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
sspmlem.2 ⊢ ( 𝑊 ∈ NrmCVec → 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑅 )
sspmlem.3 ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 : ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ⟶ 𝑆 )
Assertion sspmlem ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 sspmlem.y ⊢ 𝑌 = ( BaseSet ‘ 𝑊 )
2 sspmlem.h ⊢ 𝐻 = ( SubSp ‘ 𝑈 )
3 sspmlem.1 ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
4 sspmlem.2 ⊢ ( 𝑊 ∈ NrmCVec → 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑅 )
5 sspmlem.3 ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 : ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ⟶ 𝑆 )
6 ovres ⊢ ( ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
7 6 adantl ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) = ( 𝑥 𝐺 𝑦 ) )
8 3 7 eqtr4d ⊢ ( ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) )
9 8 ralrimivva ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) )
10 eqid ⊢ ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 )
11 9 10 jctil ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) )
12 2 sspnv ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ NrmCVec )
13 ffn ⊢ ( 𝐹 : ( 𝑌 × 𝑌 ) ⟶ 𝑅 → 𝐹 Fn ( 𝑌 × 𝑌 ) )
14 12 4 13 3syl ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 Fn ( 𝑌 × 𝑌 ) )
15 5 ffnd ⊢ ( 𝑈 ∈ NrmCVec → 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
16 15 adantr ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
17 eqid ⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 )
18 17 1 2 sspba ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) )
19 xpss12 ⊢ ( ( 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ∧ 𝑌 ⊆ ( BaseSet ‘ 𝑈 ) ) → ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
20 18 18 19 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) )
21 fnssres ⊢ ( ( 𝐺 Fn ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ∧ ( 𝑌 × 𝑌 ) ⊆ ( ( BaseSet ‘ 𝑈 ) × ( BaseSet ‘ 𝑈 ) ) ) → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) )
22 16 20 21 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) )
23 eqfnov ⊢ ( ( 𝐹 Fn ( 𝑌 × 𝑌 ) ∧ ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) Fn ( 𝑌 × 𝑌 ) ) → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) )
24 14 22 23 syl2anc ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → ( 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) ↔ ( ( 𝑌 × 𝑌 ) = ( 𝑌 × 𝑌 ) ∧ ∀ 𝑥 ∈ 𝑌 ∀ 𝑦 ∈ 𝑌 ( 𝑥 𝐹 𝑦 ) = ( 𝑥 ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) 𝑦 ) ) ) )
25 11 24 mpbird ⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻 ) → 𝐹 = ( 𝐺 ↾ ( 𝑌 × 𝑌 ) ) )