Metamath Proof Explorer

Theorem hdmaprnlem11N

Description: Lemma for hdmaprnN . Show s is in the range of S . (Contributed by NM, 29-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
hdmaprnlem1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
hdmaprnlem1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
hdmaprnlem1.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
hdmaprnlem1.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
hdmaprnlem1.l ⊢ 𝐿 = ( LSpan ‘ 𝐶 )
hdmaprnlem1.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
hdmaprnlem1.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
hdmaprnlem1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
hdmaprnlem1.se ⊢ ( 𝜑 → 𝑠 ∈ ( 𝐷 ∖ { 𝑄 } ) )
hdmaprnlem1.ve ⊢ ( 𝜑 → 𝑣 ∈ 𝑉 )
hdmaprnlem1.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) = ( 𝐿 ‘ { 𝑠 } ) )
hdmaprnlem1.ue ⊢ ( 𝜑 → 𝑢 ∈ 𝑉 )
hdmaprnlem1.un ⊢ ( 𝜑 → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑣 } ) )
hdmaprnlem1.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
hdmaprnlem1.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
hdmaprnlem1.o ⊢ 0 = ( 0g ‘ 𝑈 )
hdmaprnlem1.a ⊢ ✚ = ( +g ‘ 𝐶 )
hdmaprnlem3e.p ⊢ + = ( +g ‘ 𝑈 )
Assertion hdmaprnlem11N ( 𝜑 → 𝑠 ∈ ran 𝑆 )

Proof

Step Hyp Ref Expression
1 hdmaprnlem1.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 hdmaprnlem1.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3 hdmaprnlem1.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
4 hdmaprnlem1.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
5 hdmaprnlem1.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
6 hdmaprnlem1.l ⊢ 𝐿 = ( LSpan ‘ 𝐶 )
7 hdmaprnlem1.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
8 hdmaprnlem1.s ⊢ 𝑆 = ( ( HDMap ‘ 𝐾 ) ‘ 𝑊 )
9 hdmaprnlem1.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 hdmaprnlem1.se ⊢ ( 𝜑 → 𝑠 ∈ ( 𝐷 ∖ { 𝑄 } ) )
11 hdmaprnlem1.ve ⊢ ( 𝜑 → 𝑣 ∈ 𝑉 )
12 hdmaprnlem1.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) = ( 𝐿 ‘ { 𝑠 } ) )
13 hdmaprnlem1.ue ⊢ ( 𝜑 → 𝑢 ∈ 𝑉 )
14 hdmaprnlem1.un ⊢ ( 𝜑 → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑣 } ) )
15 hdmaprnlem1.d ⊢ 𝐷 = ( Base ‘ 𝐶 )
16 hdmaprnlem1.q ⊢ 𝑄 = ( 0g ‘ 𝐶 )
17 hdmaprnlem1.o ⊢ 0 = ( 0g ‘ 𝑈 )
18 hdmaprnlem1.a ⊢ ✚ = ( +g ‘ 𝐶 )
19 hdmaprnlem3e.p ⊢ + = ( +g ‘ 𝑈 )
20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 hdmaprnlem10N ⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝑉 ( 𝑆 ‘ 𝑡 ) = 𝑠 )
21 1 2 3 8 9 hdmapfnN ⊢ ( 𝜑 → 𝑆 Fn 𝑉 )
22 fvelrnb ⊢ ( 𝑆 Fn 𝑉 → ( 𝑠 ∈ ran 𝑆 ↔ ∃ 𝑡 ∈ 𝑉 ( 𝑆 ‘ 𝑡 ) = 𝑠 ) )
23 21 22 syl ⊢ ( 𝜑 → ( 𝑠 ∈ ran 𝑆 ↔ ∃ 𝑡 ∈ 𝑉 ( 𝑆 ‘ 𝑡 ) = 𝑠 ) )
24 20 23 mpbird ⊢ ( 𝜑 → 𝑠 ∈ ran 𝑆 )