# Metamath Proof Explorer

## Theorem mapdpglem5N

Description: Lemma for mapdpg . (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 } ) )
Assertion mapdpglem5N ( 𝜑𝑡 ≠ ( 0g𝐶 ) )

### 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 eqid ( LSAtoms ‘ 𝑈 ) = ( LSAtoms ‘ 𝑈 )
25 eqid ( LSAtoms ‘ 𝐶 ) = ( LSAtoms ‘ 𝐶 )
26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 mapdpglem4N ( 𝜑 → ( 𝑋 𝑌 ) ≠ 𝑄 )
27 1 3 8 dvhlmod ( 𝜑𝑈 ∈ LMod )
28 4 5 lmodvsubcl ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉𝑌𝑉 ) → ( 𝑋 𝑌 ) ∈ 𝑉 )
29 27 9 10 28 syl3anc ( 𝜑 → ( 𝑋 𝑌 ) ∈ 𝑉 )
30 4 6 21 24 27 29 lsatspn0 ( 𝜑 → ( ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ∈ ( LSAtoms ‘ 𝑈 ) ↔ ( 𝑋 𝑌 ) ≠ 𝑄 ) )
31 26 30 mpbird ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ∈ ( LSAtoms ‘ 𝑈 ) )
32 1 2 3 24 7 25 8 31 mapdat ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) ∈ ( LSAtoms ‘ 𝐶 ) )
33 23 32 eqeltrrd ( 𝜑 → ( 𝐽 ‘ { 𝑡 } ) ∈ ( LSAtoms ‘ 𝐶 ) )
34 eqid ( 0g𝐶 ) = ( 0g𝐶 )
35 1 7 8 lcdlmod ( 𝜑𝐶 ∈ LMod )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 mapdpglem2a ( 𝜑𝑡𝐹 )
37 13 12 34 25 35 36 lsatspn0 ( 𝜑 → ( ( 𝐽 ‘ { 𝑡 } ) ∈ ( LSAtoms ‘ 𝐶 ) ↔ 𝑡 ≠ ( 0g𝐶 ) ) )
38 33 37 mpbid ( 𝜑𝑡 ≠ ( 0g𝐶 ) )