Metamath Proof Explorer


Theorem lflsub

Description: Property of a linear functional. ( lnfnaddi analog.) (Contributed by NM, 18-Apr-2014)

Ref Expression
Hypotheses lflsub.d 𝐷 = ( Scalar ‘ 𝑊 )
lflsub.m 𝑀 = ( -g𝐷 )
lflsub.v 𝑉 = ( Base ‘ 𝑊 )
lflsub.a = ( -g𝑊 )
lflsub.f 𝐹 = ( LFnl ‘ 𝑊 )
Assertion lflsub ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 𝑌 ) ) = ( ( 𝐺𝑋 ) 𝑀 ( 𝐺𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 lflsub.d 𝐷 = ( Scalar ‘ 𝑊 )
2 lflsub.m 𝑀 = ( -g𝐷 )
3 lflsub.v 𝑉 = ( Base ‘ 𝑊 )
4 lflsub.a = ( -g𝑊 )
5 lflsub.f 𝐹 = ( LFnl ‘ 𝑊 )
6 simp1 ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝑊 ∈ LMod )
7 simp3l ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝑋𝑉 )
8 1 lmodring ( 𝑊 ∈ LMod → 𝐷 ∈ Ring )
9 8 3ad2ant1 ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝐷 ∈ Ring )
10 ringgrp ( 𝐷 ∈ Ring → 𝐷 ∈ Grp )
11 9 10 syl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝐷 ∈ Grp )
12 eqid ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
13 eqid ( 1r𝐷 ) = ( 1r𝐷 )
14 12 13 ringidcl ( 𝐷 ∈ Ring → ( 1r𝐷 ) ∈ ( Base ‘ 𝐷 ) )
15 9 14 syl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 1r𝐷 ) ∈ ( Base ‘ 𝐷 ) )
16 eqid ( invg𝐷 ) = ( invg𝐷 )
17 12 16 grpinvcl ( ( 𝐷 ∈ Grp ∧ ( 1r𝐷 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ∈ ( Base ‘ 𝐷 ) )
18 11 15 17 syl2anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ∈ ( Base ‘ 𝐷 ) )
19 simp3r ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝑌𝑉 )
20 eqid ( ·𝑠𝑊 ) = ( ·𝑠𝑊 )
21 3 1 20 12 lmodvscl ( ( 𝑊 ∈ LMod ∧ ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑌𝑉 ) → ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ∈ 𝑉 )
22 6 18 19 21 syl3anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ∈ 𝑉 )
23 eqid ( +g𝑊 ) = ( +g𝑊 )
24 3 23 lmodcom ( ( 𝑊 ∈ LMod ∧ 𝑋𝑉 ∧ ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ∈ 𝑉 ) → ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) = ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ( +g𝑊 ) 𝑋 ) )
25 6 7 22 24 syl3anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) = ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ( +g𝑊 ) 𝑋 ) )
26 25 fveq2d ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) ) = ( 𝐺 ‘ ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ( +g𝑊 ) 𝑋 ) ) )
27 simp2 ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝐺𝐹 )
28 eqid ( +g𝐷 ) = ( +g𝐷 )
29 eqid ( .r𝐷 ) = ( .r𝐷 )
30 3 23 1 20 12 28 29 5 lfli ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ∈ ( Base ‘ 𝐷 ) ∧ 𝑌𝑉𝑋𝑉 ) ) → ( 𝐺 ‘ ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ( +g𝑊 ) 𝑋 ) ) = ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( .r𝐷 ) ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) )
31 6 27 18 19 7 30 syl113anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ( +g𝑊 ) 𝑋 ) ) = ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( .r𝐷 ) ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) )
32 1 12 3 5 lflcl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹𝑌𝑉 ) → ( 𝐺𝑌 ) ∈ ( Base ‘ 𝐷 ) )
33 32 3adant3l ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺𝑌 ) ∈ ( Base ‘ 𝐷 ) )
34 12 29 13 16 9 33 ringnegl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( .r𝐷 ) ( 𝐺𝑌 ) ) = ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) )
35 34 oveq1d ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( .r𝐷 ) ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) = ( ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) )
36 ringabl ( 𝐷 ∈ Ring → 𝐷 ∈ Abel )
37 9 36 syl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → 𝐷 ∈ Abel )
38 12 16 grpinvcl ( ( 𝐷 ∈ Grp ∧ ( 𝐺𝑌 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ∈ ( Base ‘ 𝐷 ) )
39 11 33 38 syl2anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ∈ ( Base ‘ 𝐷 ) )
40 1 12 3 5 lflcl ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹𝑋𝑉 ) → ( 𝐺𝑋 ) ∈ ( Base ‘ 𝐷 ) )
41 40 3adant3r ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺𝑋 ) ∈ ( Base ‘ 𝐷 ) )
42 12 28 ablcom ( ( 𝐷 ∈ Abel ∧ ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺𝑋 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
43 37 39 41 42 syl3anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
44 35 43 eqtrd ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( .r𝐷 ) ( 𝐺𝑌 ) ) ( +g𝐷 ) ( 𝐺𝑋 ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
45 26 31 44 3eqtrd ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
46 3 23 4 1 20 16 13 lmodvsubval2 ( ( 𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉 ) → ( 𝑋 𝑌 ) = ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) )
47 6 7 19 46 syl3anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝑋 𝑌 ) = ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) )
48 47 fveq2d ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 𝑌 ) ) = ( 𝐺 ‘ ( 𝑋 ( +g𝑊 ) ( ( ( invg𝐷 ) ‘ ( 1r𝐷 ) ) ( ·𝑠𝑊 ) 𝑌 ) ) ) )
49 12 28 16 2 grpsubval ( ( ( 𝐺𝑋 ) ∈ ( Base ‘ 𝐷 ) ∧ ( 𝐺𝑌 ) ∈ ( Base ‘ 𝐷 ) ) → ( ( 𝐺𝑋 ) 𝑀 ( 𝐺𝑌 ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
50 41 33 49 syl2anc ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( ( 𝐺𝑋 ) 𝑀 ( 𝐺𝑌 ) ) = ( ( 𝐺𝑋 ) ( +g𝐷 ) ( ( invg𝐷 ) ‘ ( 𝐺𝑌 ) ) ) )
51 45 48 50 3eqtr4d ( ( 𝑊 ∈ LMod ∧ 𝐺𝐹 ∧ ( 𝑋𝑉𝑌𝑉 ) ) → ( 𝐺 ‘ ( 𝑋 𝑌 ) ) = ( ( 𝐺𝑋 ) 𝑀 ( 𝐺𝑌 ) ) )