Metamath Proof Explorer


Theorem mapdpglem12

Description: Lemma for mapdpg . TODO: Can some commonality with mapdpglem6 through mapdpglem11 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-Mar-2015)

Ref Expression
Hypotheses mapdpglem.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdpglem.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.v 𝑉 = ( Base ‘ 𝑈 )
mapdpglem.s = ( -g𝑈 )
mapdpglem.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdpglem.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdpglem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdpglem.x ( 𝜑𝑋𝑉 )
mapdpglem.y ( 𝜑𝑌𝑉 )
mapdpglem1.p = ( LSSum ‘ 𝐶 )
mapdpglem2.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdpglem3.f 𝐹 = ( Base ‘ 𝐶 )
mapdpglem3.te ( 𝜑𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
mapdpglem3.a 𝐴 = ( Scalar ‘ 𝑈 )
mapdpglem3.b 𝐵 = ( Base ‘ 𝐴 )
mapdpglem3.t · = ( ·𝑠𝐶 )
mapdpglem3.r 𝑅 = ( -g𝐶 )
mapdpglem3.g ( 𝜑𝐺𝐹 )
mapdpglem3.e ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
mapdpglem4.q 𝑄 = ( 0g𝑈 )
mapdpglem.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
mapdpglem4.jt ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
mapdpglem4.z 0 = ( 0g𝐴 )
mapdpglem4.g4 ( 𝜑𝑔𝐵 )
mapdpglem4.z4 ( 𝜑𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
mapdpglem4.t4 ( 𝜑𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
mapdpglem4.xn ( 𝜑𝑋𝑄 )
mapdpglem12.yn ( 𝜑𝑌𝑄 )
mapdpglem12.g0 ( 𝜑𝑧 = ( 0g𝐶 ) )
Assertion mapdpglem12 ( 𝜑𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )

Proof

Step Hyp Ref Expression
1 mapdpglem.h 𝐻 = ( LHyp ‘ 𝐾 )
2 mapdpglem.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3 mapdpglem.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 mapdpglem.v 𝑉 = ( Base ‘ 𝑈 )
5 mapdpglem.s = ( -g𝑈 )
6 mapdpglem.n 𝑁 = ( LSpan ‘ 𝑈 )
7 mapdpglem.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
8 mapdpglem.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 mapdpglem.x ( 𝜑𝑋𝑉 )
10 mapdpglem.y ( 𝜑𝑌𝑉 )
11 mapdpglem1.p = ( LSSum ‘ 𝐶 )
12 mapdpglem2.j 𝐽 = ( LSpan ‘ 𝐶 )
13 mapdpglem3.f 𝐹 = ( Base ‘ 𝐶 )
14 mapdpglem3.te ( 𝜑𝑡 ∈ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ) )
15 mapdpglem3.a 𝐴 = ( Scalar ‘ 𝑈 )
16 mapdpglem3.b 𝐵 = ( Base ‘ 𝐴 )
17 mapdpglem3.t · = ( ·𝑠𝐶 )
18 mapdpglem3.r 𝑅 = ( -g𝐶 )
19 mapdpglem3.g ( 𝜑𝐺𝐹 )
20 mapdpglem3.e ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
21 mapdpglem4.q 𝑄 = ( 0g𝑈 )
22 mapdpglem.ne ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
23 mapdpglem4.jt ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { 𝑡 } ) )
24 mapdpglem4.z 0 = ( 0g𝐴 )
25 mapdpglem4.g4 ( 𝜑𝑔𝐵 )
26 mapdpglem4.z4 ( 𝜑𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) )
27 mapdpglem4.t4 ( 𝜑𝑡 = ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) )
28 mapdpglem4.xn ( 𝜑𝑋𝑄 )
29 mapdpglem12.yn ( 𝜑𝑌𝑄 )
30 mapdpglem12.g0 ( 𝜑𝑧 = ( 0g𝐶 ) )
31 1 7 8 lcdlmod ( 𝜑𝐶 ∈ LMod )
32 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
33 eqid ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
34 1 3 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
35 4 32 6 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
36 34 9 35 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
37 1 2 3 32 7 33 8 36 mapdcl2 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
38 13 12 lspsnid ( ( 𝐶 ∈ LMod ∧ 𝐺𝐹 ) → 𝐺 ∈ ( 𝐽 ‘ { 𝐺 } ) )
39 31 19 38 syl2anc ( 𝜑𝐺 ∈ ( 𝐽 ‘ { 𝐺 } ) )
40 39 20 eleqtrrd ( 𝜑𝐺 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
41 1 3 15 16 7 13 17 33 8 37 25 40 lcdlssvscl ( 𝜑 → ( 𝑔 · 𝐺 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
42 eqid ( 0g𝐶 ) = ( 0g𝐶 )
43 42 33 lss0cl ( ( 𝐶 ∈ LMod ∧ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ) → ( 0g𝐶 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
44 31 37 43 syl2anc ( 𝜑 → ( 0g𝐶 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
45 30 44 eqeltrd ( 𝜑𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
46 18 33 lssvsubcl ( ( ( 𝐶 ∈ LMod ∧ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝐶 ) ) ∧ ( ( 𝑔 · 𝐺 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ∧ 𝑧 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) ) → ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
47 31 37 41 45 46 syl22anc ( 𝜑 → ( ( 𝑔 · 𝐺 ) 𝑅 𝑧 ) ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )
48 27 47 eqeltrd ( 𝜑𝑡 ∈ ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) )