Metamath Proof Explorer

Theorem hdmaprnlem10N

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 hdmaprnlem10N ( 𝜑 → ∃ 𝑡 ∈ 𝑉 ( 𝑆 ‘ 𝑡 ) = 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 hdmaprnlem3eN ⊢ ( 𝜑 → ∃ 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) )
21 9 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
22 10 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑠 ∈ ( 𝐷 ∖ { 𝑄 } ) )
23 11 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑣 ∈ 𝑉 )
24 12 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) = ( 𝐿 ‘ { 𝑠 } ) )
25 13 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑢 ∈ 𝑉 )
26 14 adantr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → ¬ 𝑢 ∈ ( 𝑁 ‘ { 𝑣 } ) )
27 simprl ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) )
28 1 2 3 4 5 6 7 8 21 22 23 24 25 26 15 16 17 18 27 hdmaprnlem4tN ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑡 ∈ 𝑉 )
29 simprr ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) )
30 1 2 3 4 5 6 7 8 21 22 23 24 25 26 15 16 17 18 27 19 29 hdmaprnlem9N ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → 𝑠 = ( 𝑆 ‘ 𝑡 ) )
31 30 eqcomd ⊢ ( ( 𝜑 ∧ ( 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) ) ) → ( 𝑆 ‘ 𝑡 ) = 𝑠 )
32 20 28 31 reximssdv ⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝑉 ( 𝑆 ‘ 𝑡 ) = 𝑠 )