Metamath Proof Explorer

Theorem hdmaprnlem7N

Description: Part of proof of part 12 in Baer p. 49 line 19, s-St e. G(u'+s) = P*. (Contributed by NM, 27-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 )
hdmaprnlem1.t2 ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) )
hdmaprnlem1.p ⊢ + = ( +g ‘ 𝑈 )
hdmaprnlem1.pt ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) )
Assertion hdmaprnlem7N ( 𝜑 → ( 𝑠 ( -g ‘ 𝐶 ) ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 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 hdmaprnlem1.t2 ⊢ ( 𝜑 → 𝑡 ∈ ( ( 𝑁 ‘ { 𝑣 } ) ∖ { 0 } ) )
20 hdmaprnlem1.p ⊢ + = ( +g ‘ 𝑈 )
21 hdmaprnlem1.pt ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑢 + 𝑡 ) } ) ) )
22 eqid ⊢ ( -g ‘ 𝐶 ) = ( -g ‘ 𝐶 )
23 1 5 9 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
24 lmodabl ⊢ ( 𝐶 ∈ LMod → 𝐶 ∈ Abel )
25 23 24 syl ⊢ ( 𝜑 → 𝐶 ∈ Abel )
26 1 2 3 5 15 8 9 13 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 )
27 10 eldifad ⊢ ( 𝜑 → 𝑠 ∈ 𝐷 )
28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 hdmaprnlem4tN ⊢ ( 𝜑 → 𝑡 ∈ 𝑉 )
29 1 2 3 5 15 8 9 28 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 )
30 15 18 22 25 26 27 29 25 26 27 29 ablpnpcan ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ( -g ‘ 𝐶 ) ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ) = ( 𝑠 ( -g ‘ 𝐶 ) ( 𝑆 ‘ 𝑡 ) ) )
31 15 18 lmodvacl ⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 )
32 23 26 27 31 syl3anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 )
33 eqid ⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
34 15 33 6 lspsncl ⊢ ( ( 𝐶 ∈ LMod ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) )
35 23 32 34 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) )
36 15 6 lspsnid ⊢ ( ( 𝐶 ∈ LMod ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
37 23 32 36 syl2anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
38 15 18 lmodvacl ⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 ∧ ( 𝑆 ‘ 𝑡 ) ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ 𝐷 )
39 23 26 29 38 syl3anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ 𝐷 )
40 15 6 lspsnid ⊢ ( ( 𝐶 ∈ LMod ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) } ) )
41 23 39 40 syl2anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) } ) )
42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 hdmaprnlem6N ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) = ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) } ) )
43 41 42 eleqtrrd ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
44 22 33 lssvsubcl ⊢ ( ( ( 𝐶 ∈ LMod ∧ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) ) ∧ ( ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) → ( ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ( -g ‘ 𝐶 ) ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
45 23 35 37 43 44 syl22anc ⊢ ( 𝜑 → ( ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ( -g ‘ 𝐶 ) ( ( 𝑆 ‘ 𝑢 ) ✚ ( 𝑆 ‘ 𝑡 ) ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
46 30 45 eqeltrrd ⊢ ( 𝜑 → ( 𝑠 ( -g ‘ 𝐶 ) ( 𝑆 ‘ 𝑡 ) ) ∈ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )