# Metamath Proof Explorer

## Theorem mapdpglem17N

Description: Lemma for mapdpg . Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015) (New usage is discouraged.)

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 )
mapdpglem17.ep 𝐸 = ( ( ( invr𝐴 ) ‘ 𝑔 ) · 𝑧 )
Assertion mapdpglem17N ( 𝜑𝐸𝐹 )

### 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 mapdpglem17.ep 𝐸 = ( ( ( invr𝐴 ) ‘ 𝑔 ) · 𝑧 )
31 1 3 8 dvhlvec ( 𝜑𝑈 ∈ LVec )
32 15 lvecdrng ( 𝑈 ∈ LVec → 𝐴 ∈ DivRing )
33 31 32 syl ( 𝜑𝐴 ∈ DivRing )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 mapdpglem11 ( 𝜑𝑔0 )
35 eqid ( invr𝐴 ) = ( invr𝐴 )
36 16 24 35 drnginvrcl ( ( 𝐴 ∈ DivRing ∧ 𝑔𝐵𝑔0 ) → ( ( invr𝐴 ) ‘ 𝑔 ) ∈ 𝐵 )
37 33 25 34 36 syl3anc ( 𝜑 → ( ( invr𝐴 ) ‘ 𝑔 ) ∈ 𝐵 )
38 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
39 eqid ( LSubSp ‘ 𝐶 ) = ( LSubSp ‘ 𝐶 )
40 1 3 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
41 4 38 6 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
42 40 10 41 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
43 1 2 3 38 7 39 8 42 mapdcl2 ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) )
44 13 39 lssss ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ∈ ( LSubSp ‘ 𝐶 ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ 𝐹 )
45 43 44 syl ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ⊆ 𝐹 )
46 45 26 sseldd ( 𝜑𝑧𝐹 )
47 1 3 15 16 7 13 17 8 37 46 lcdvscl ( 𝜑 → ( ( ( invr𝐴 ) ‘ 𝑔 ) · 𝑧 ) ∈ 𝐹 )
48 30 47 eqeltrid ( 𝜑𝐸𝐹 )