Metamath Proof Explorer

Theorem hdmaprnN

Description: Part of proof of part 12 in Baer p. 49 line 21, As=B. (Contributed by NM, 30-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmaprn.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmaprn.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmaprn.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmaprn.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
Assertion hdmaprnN ( 𝜑 → ran 𝑆 = 𝐷 )

Proof

Step Hyp Ref Expression
1 hdmaprn.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmaprn.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
3 hdmaprn.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
4 hdmaprn.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
5 hdmaprn.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6 eqid ⊢ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
7 eqid ⊢ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
8 1 6 7 4 5 hdmapfnN ⊢ ( 𝜑 → 𝑆 Fn ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
9 5 adantr ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simpr ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) → 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) )
11 1 6 7 2 3 4 9 10 hdmapcl ⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ) → ( 𝑆 ‘ 𝑠 ) ∈ 𝐷 )
12 11 ralrimiva ⊢ ( 𝜑 → ∀ 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( 𝑆 ‘ 𝑠 ) ∈ 𝐷 )
13 fnfvrnss ⊢ ( ( 𝑆 Fn ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ∧ ∀ 𝑠 ∈ ( Base ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) ( 𝑆 ‘ 𝑠 ) ∈ 𝐷 ) → ran 𝑆 ⊆ 𝐷 )
14 8 12 13 syl2anc ⊢ ( 𝜑 → ran 𝑆 ⊆ 𝐷 )
15 eqid ⊢ ( LSpan ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) ) = ( LSpan ‘ ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 ) )
16 eqid ⊢ ( 0g ‘ 𝐶 ) = ( 0g ‘ 𝐶 )
17 eqid ⊢ ( LSpan ‘ 𝐶 ) = ( LSpan ‘ 𝐶 )
18 eqid ⊢ ( ( mapd ‘ 𝐾 ) ‘ 𝑊 ) = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
19 5 adantr ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝐷 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
20 simpr ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝐷 ) → 𝑠 ∈ 𝐷 )
21 1 6 7 15 2 3 16 17 18 4 19 20 hdmaprnlem17N ⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝐷 ) → 𝑠 ∈ ran 𝑆 )
22 14 21 eqelssd ⊢ ( 𝜑 → ran 𝑆 = 𝐷 )