Metamath Proof Explorer


Theorem mapdh6lem2N

Description: Lemma for mapdh6N . Part (6) in Baer p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q 𝑄 = ( 0g𝐶 )
mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
mapdh.h 𝐻 = ( LHyp ‘ 𝐾 )
mapdh.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdh.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdh.v 𝑉 = ( Base ‘ 𝑈 )
mapdh.s = ( -g𝑈 )
mapdhc.o 0 = ( 0g𝑈 )
mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdh.d 𝐷 = ( Base ‘ 𝐶 )
mapdh.r 𝑅 = ( -g𝐶 )
mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
mapdhc.f ( 𝜑𝐹𝐷 )
mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdh.p + = ( +g𝑈 )
mapdh.a = ( +g𝐶 )
mapdhe6.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe6.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
mapdhe6.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
mapdh6.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
mapdh6.fg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
mapdh6.fe ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
Assertion mapdh6lem2N ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝐸 ) } ) )

Proof

Step Hyp Ref Expression
1 mapdh.q 𝑄 = ( 0g𝐶 )
2 mapdh.i 𝐼 = ( 𝑥 ∈ V ↦ if ( ( 2nd𝑥 ) = 0 , 𝑄 , ( 𝐷 ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 2nd𝑥 ) } ) ) = ( 𝐽 ‘ { } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( ( 1st ‘ ( 1st𝑥 ) ) ( 2nd𝑥 ) ) } ) ) = ( 𝐽 ‘ { ( ( 2nd ‘ ( 1st𝑥 ) ) 𝑅 ) } ) ) ) ) )
3 mapdh.h 𝐻 = ( LHyp ‘ 𝐾 )
4 mapdh.m 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5 mapdh.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
6 mapdh.v 𝑉 = ( Base ‘ 𝑈 )
7 mapdh.s = ( -g𝑈 )
8 mapdhc.o 0 = ( 0g𝑈 )
9 mapdh.n 𝑁 = ( LSpan ‘ 𝑈 )
10 mapdh.c 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
11 mapdh.d 𝐷 = ( Base ‘ 𝐶 )
12 mapdh.r 𝑅 = ( -g𝐶 )
13 mapdh.j 𝐽 = ( LSpan ‘ 𝐶 )
14 mapdh.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
15 mapdhc.f ( 𝜑𝐹𝐷 )
16 mapdh.mn ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐹 } ) )
17 mapdhcl.x ( 𝜑𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
18 mapdh.p + = ( +g𝑈 )
19 mapdh.a = ( +g𝐶 )
20 mapdhe6.y ( 𝜑𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 mapdhe6.z ( 𝜑𝑍 ∈ ( 𝑉 ∖ { 0 } ) )
22 mapdhe6.xn ( 𝜑 → ¬ 𝑋 ∈ ( 𝑁 ‘ { 𝑌 , 𝑍 } ) )
23 mapdh6.yz ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
24 mapdh6.fg ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 )
25 mapdh6.fe ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 )
26 eqid ( LSubSp ‘ 𝑈 ) = ( LSubSp ‘ 𝑈 )
27 3 5 14 dvhlmod ( 𝜑𝑈 ∈ LMod )
28 20 eldifad ( 𝜑𝑌𝑉 )
29 6 26 9 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉 ) → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
30 27 28 29 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) )
31 21 eldifad ( 𝜑𝑍𝑉 )
32 6 26 9 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑍𝑉 ) → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
33 27 31 32 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) )
34 eqid ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
35 26 34 lsmcl ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { 𝑌 } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝑍 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
36 27 30 33 35 syl3anc ( 𝜑 → ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
37 17 eldifad ( 𝜑𝑋𝑉 )
38 6 18 lmodvacl ( ( 𝑈 ∈ LMod ∧ 𝑌𝑉𝑍𝑉 ) → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
39 27 28 31 38 syl3anc ( 𝜑 → ( 𝑌 + 𝑍 ) ∈ 𝑉 )
40 6 7 lmodvsubcl ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ∧ ( 𝑌 + 𝑍 ) ∈ 𝑉 ) → ( 𝑋 ( 𝑌 + 𝑍 ) ) ∈ 𝑉 )
41 27 37 39 40 syl3anc ( 𝜑 → ( 𝑋 ( 𝑌 + 𝑍 ) ) ∈ 𝑉 )
42 6 26 9 lspsncl ( ( 𝑈 ∈ LMod ∧ ( 𝑋 ( 𝑌 + 𝑍 ) ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
43 27 41 42 syl2anc ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) )
44 6 26 9 lspsncl ( ( 𝑈 ∈ LMod ∧ 𝑋𝑉 ) → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
45 27 37 44 syl2anc ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) )
46 26 34 lsmcl ( ( 𝑈 ∈ LMod ∧ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ∈ ( LSubSp ‘ 𝑈 ) ∧ ( 𝑁 ‘ { 𝑋 } ) ∈ ( LSubSp ‘ 𝑈 ) ) → ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
47 27 43 45 46 syl3anc ( 𝜑 → ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ∈ ( LSubSp ‘ 𝑈 ) )
48 3 4 5 26 14 36 47 mapdin ( 𝜑 → ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) ∩ ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) )
49 eqid ( LSSum ‘ 𝐶 ) = ( LSSum ‘ 𝐶 )
50 3 4 5 26 34 10 49 14 30 33 mapdlsm ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) ) )
51 3 5 14 dvhlvec ( 𝜑𝑈 ∈ LVec )
52 6 8 9 51 28 21 37 23 22 lspindp2 ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ∧ ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) )
53 52 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 53 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) ∈ 𝐷 )
55 24 54 eqeltrrd ( 𝜑𝐺𝐷 )
56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 55 53 mapdheq ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑌 ⟩ ) = 𝐺 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) ) )
57 24 56 mpbid ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐺 ) } ) ) )
58 57 simpld ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
59 6 8 9 51 20 31 37 23 22 lspindp1 ( 𝜑 → ( ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑋 , 𝑍 } ) ) )
60 59 simpld ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑍 } ) )
61 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 60 mapdhcl ( 𝜑 → ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) ∈ 𝐷 )
62 25 61 eqeltrrd ( 𝜑𝐸𝐷 )
63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 62 60 mapdheq ( 𝜑 → ( ( 𝐼 ‘ ⟨ 𝑋 , 𝐹 , 𝑍 ⟩ ) = 𝐸 ↔ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) ) )
64 25 63 mpbid ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 𝐸 ) } ) ) )
65 64 simpld ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) = ( 𝐽 ‘ { 𝐸 } ) )
66 58 65 oveq12d ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) )
67 50 66 eqtrd ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) = ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) )
68 3 4 5 26 34 10 49 14 43 45 mapdlsm ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) )
69 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 mapdh6lem1N ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) = ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) )
70 69 16 oveq12d ( 𝜑 → ( ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ) ( LSSum ‘ 𝐶 ) ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) )
71 68 70 eqtrd ( 𝜑 → ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) = ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) )
72 67 71 ineq12d ( 𝜑 → ( ( 𝑀 ‘ ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ) ∩ ( 𝑀 ‘ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) )
73 48 72 eqtrd ( 𝜑 → ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) )
74 6 7 8 34 9 51 37 22 23 20 21 18 baerlem5b ( 𝜑 → ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) = ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) )
75 74 fveq2d ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝑀 ‘ ( ( ( 𝑁 ‘ { 𝑌 } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑍 } ) ) ∩ ( ( 𝑁 ‘ { ( 𝑋 ( 𝑌 + 𝑍 ) ) } ) ( LSSum ‘ 𝑈 ) ( 𝑁 ‘ { 𝑋 } ) ) ) ) )
76 3 10 14 lcdlvec ( 𝜑𝐶 ∈ LVec )
77 3 4 5 6 9 10 11 13 14 15 16 37 28 55 58 31 62 65 22 mapdindp ( 𝜑 → ¬ 𝐹 ∈ ( 𝐽 ‘ { 𝐺 , 𝐸 } ) )
78 3 4 5 6 9 10 11 13 14 55 58 28 31 62 65 23 mapdncol ( 𝜑 → ( 𝐽 ‘ { 𝐺 } ) ≠ ( 𝐽 ‘ { 𝐸 } ) )
79 3 4 5 6 9 10 11 13 14 55 58 8 1 20 mapdn0 ( 𝜑𝐺 ∈ ( 𝐷 ∖ { 𝑄 } ) )
80 3 4 5 6 9 10 11 13 14 62 65 8 1 21 mapdn0 ( 𝜑𝐸 ∈ ( 𝐷 ∖ { 𝑄 } ) )
81 11 12 1 49 13 76 15 77 78 79 80 19 baerlem5b ( 𝜑 → ( 𝐽 ‘ { ( 𝐺 𝐸 ) } ) = ( ( ( 𝐽 ‘ { 𝐺 } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐸 } ) ) ∩ ( ( 𝐽 ‘ { ( 𝐹 𝑅 ( 𝐺 𝐸 ) ) } ) ( LSSum ‘ 𝐶 ) ( 𝐽 ‘ { 𝐹 } ) ) ) )
82 73 75 81 3eqtr4d ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑌 + 𝑍 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝐸 ) } ) )