Metamath Proof Explorer

Theorem hdmaprnlem3N

Description: Part of proof of part 12 in Baer p. 49 line 15, T =/= P. Our (`' M `( L{ ( ( Su ) .+b s ) } ) ) is Baer's P, where P* = G(u'+s). (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 )
Assertion hdmaprnlem3N ( 𝜑 → ( 𝑁 ‘ { 𝑣 } ) ≠ ( ◡ 𝑀 ‘ ( 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 1 5 9 lcdlmod ⊢ ( 𝜑 → 𝐶 ∈ LMod )
20 1 2 3 5 15 8 9 13 hdmapcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 )
21 10 eldifad ⊢ ( 𝜑 → 𝑠 ∈ 𝐷 )
22 15 18 lmodvacl ⊢ ( ( 𝐶 ∈ LMod ∧ ( 𝑆 ‘ 𝑢 ) ∈ 𝐷 ∧ 𝑠 ∈ 𝐷 ) → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 )
23 19 20 21 22 syl3anc ⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 )
24 eqid ⊢ ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
25 15 24 6 lspsncl ⊢ ( ( 𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ) → ( 𝐿 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝐶 ) )
26 19 21 25 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ { 𝑠 } ) ∈ ( LSubSp ‘ 𝐶 ) )
27 15 6 lspsnid ⊢ ( ( 𝐶 ∈ LMod ∧ 𝑠 ∈ 𝐷 ) → 𝑠 ∈ ( 𝐿 ‘ { 𝑠 } ) )
28 19 21 27 syl2anc ⊢ ( 𝜑 → 𝑠 ∈ ( 𝐿 ‘ { 𝑠 } ) )
29 1 5 9 lcdlvec ⊢ ( 𝜑 → 𝐶 ∈ LVec )
30 eqid ⊢ ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
31 1 2 9 dvhlmod ⊢ ( 𝜑 → 𝑈 ∈ LMod )
32 3 30 4 lspsncl ⊢ ( ( 𝑈 ∈ LMod ∧ 𝑣 ∈ 𝑉 ) → ( 𝑁 ‘ { 𝑣 } ) ∈ ( LSubSp ‘ 𝑈 ) )
33 31 11 32 syl2anc ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑣 } ) ∈ ( LSubSp ‘ 𝑈 ) )
34 17 30 31 33 13 14 lssneln0 ⊢ ( 𝜑 → 𝑢 ∈ ( 𝑉 ∖ { 0 } ) )
35 1 2 3 17 5 16 15 8 9 34 hdmapnzcl ⊢ ( 𝜑 → ( 𝑆 ‘ 𝑢 ) ∈ ( 𝐷 ∖ { 𝑄 } ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hdmaprnlem1N ⊢ ( 𝜑 → ( 𝐿 ‘ { ( 𝑆 ‘ 𝑢 ) } ) ≠ ( 𝐿 ‘ { 𝑠 } ) )
37 15 16 6 29 35 21 36 lspsnne1 ⊢ ( 𝜑 → ¬ ( 𝑆 ‘ 𝑢 ) ∈ ( 𝐿 ‘ { 𝑠 } ) )
38 15 18 24 19 26 28 20 37 lssvancl2 ⊢ ( 𝜑 → ¬ ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ ( 𝐿 ‘ { 𝑠 } ) )
39 15 6 19 23 21 38 lspsnne2 ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ≠ ( 𝐿 ‘ { 𝑠 } ) )
40 39 necomd ⊢ ( 𝜑 → ( 𝐿 ‘ { 𝑠 } ) ≠ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
41 15 24 6 lspsncl ⊢ ( ( 𝐶 ∈ LMod ∧ ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) ∈ 𝐷 ) → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) )
42 19 23 41 syl2anc ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ( LSubSp ‘ 𝐶 ) )
43 1 7 5 24 9 mapdrn2 ⊢ ( 𝜑 → ran 𝑀 = ( LSubSp ‘ 𝐶 ) )
44 42 43 eleqtrrd ⊢ ( 𝜑 → ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ∈ ran 𝑀 )
45 1 7 9 44 mapdcnvid2 ⊢ ( 𝜑 → ( 𝑀 ‘ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) = ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) )
46 40 12 45 3netr4d ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) ≠ ( 𝑀 ‘ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) )
47 1 7 2 30 9 44 mapdcnvcl ⊢ ( 𝜑 → ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
48 1 2 30 7 9 33 47 mapd11 ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) = ( 𝑀 ‘ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) ↔ ( 𝑁 ‘ { 𝑣 } ) = ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) )
49 48 necon3bid ⊢ ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑣 } ) ) ≠ ( 𝑀 ‘ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) ↔ ( 𝑁 ‘ { 𝑣 } ) ≠ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) ) )
50 46 49 mpbid ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑣 } ) ≠ ( ◡ 𝑀 ‘ ( 𝐿 ‘ { ( ( 𝑆 ‘ 𝑢 ) ✚ 𝑠 ) } ) ) )