Metamath Proof Explorer

Theorem mapdpglem32

Description: Lemma for mapdpg . Uniqueness part - consolidate hypotheses in mapdpglem31 . (Contributed by NM, 23-Mar-2015)

Ref Expression
Hypotheses mapdpg.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
mapdpg.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
mapdpg.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
mapdpg.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
mapdpg.s ⊢ − = ( -g ‘ 𝑈 )
mapdpg.z ⊢ 0 = ( 0g ‘ 𝑈 )
mapdpg.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
mapdpg.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
mapdpg.f ⊢ 𝐹 = ( Base ‘ 𝐶 )
mapdpg.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
mapdpg.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
mapdpg.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
mapdpg.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
mapdpg.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
mapdpg.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
mapdpg.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
mapdpg.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
Assertion mapdpglem32 ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ℎ = 𝑖 )

Proof

Step Hyp Ref Expression
1 mapdpg.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2 mapdpg.m ⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
3 mapdpg.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
4 mapdpg.v ⊢ 𝑉 = ( Base ‘ 𝑈 )
5 mapdpg.s ⊢ − = ( -g ‘ 𝑈 )
6 mapdpg.z ⊢ 0 = ( 0g ‘ 𝑈 )
7 mapdpg.n ⊢ 𝑁 = ( LSpan ‘ 𝑈 )
8 mapdpg.c ⊢ 𝐶 = ( ( LCDual ‘ 𝐾 ) ‘ 𝑊 )
9 mapdpg.f ⊢ 𝐹 = ( Base ‘ 𝐶 )
10 mapdpg.r ⊢ 𝑅 = ( -g ‘ 𝐶 )
11 mapdpg.j ⊢ 𝐽 = ( LSpan ‘ 𝐶 )
12 mapdpg.k ⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
13 mapdpg.x ⊢ ( 𝜑 → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
14 mapdpg.y ⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
15 mapdpg.g ⊢ ( 𝜑 → 𝐺 ∈ 𝐹 )
16 mapdpg.ne ⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
17 mapdpg.e ⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
18 12 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 13 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
20 14 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
21 15 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → 𝐺 ∈ 𝐹 )
22 16 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
23 17 3ad2ant1 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
24 simp2l ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ℎ ∈ 𝐹 )
25 simp3l ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) )
26 24 25 jca ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( ℎ ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) )
27 simp2r ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → 𝑖 ∈ 𝐹 )
28 simp3r ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) )
29 27 28 jca ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( 𝑖 ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) )
30 eqid ⊢ ( Scalar ‘ 𝑈 ) = ( Scalar ‘ 𝑈 )
31 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑈 ) ) = ( Base ‘ ( Scalar ‘ 𝑈 ) )
32 eqid ⊢ ( ·𝑠 ‘ 𝐶 ) = ( ·𝑠 ‘ 𝐶 )
33 eqid ⊢ ( 0g ‘ ( Scalar ‘ 𝑈 ) ) = ( 0g ‘ ( Scalar ‘ 𝑈 ) )
34 1 2 3 4 5 6 7 8 9 10 11 18 19 20 21 22 23 26 29 30 31 32 33 mapdpglem26 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ∃ 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) )
35 1 2 3 4 5 6 7 8 9 10 11 18 19 20 21 22 23 26 29 30 31 32 33 mapdpglem27 ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ∃ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) )
36 reeanv ⊢ ( ∃ 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∃ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ↔ ( ∃ 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ∃ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) )
37 34 35 36 sylanbrc ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ∃ 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∃ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) )
38 18 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
39 19 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝑋 ∈ ( 𝑉 ∖ { 0 } ) )
40 20 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) )
41 21 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝐺 ∈ 𝐹 )
42 22 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) )
43 23 3ad2ant1 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = ( 𝐽 ‘ { 𝐺 } ) )
44 simp12l ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ℎ ∈ 𝐹 )
45 simp13l ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) )
46 44 45 jca ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( ℎ ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ) )
47 simp12r ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝑖 ∈ 𝐹 )
48 simp13r ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) )
49 47 48 jca ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( 𝑖 ∈ 𝐹 ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) )
50 eldifi ⊢ ( 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) → 𝑣 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
51 50 adantl ⊢ ( ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) → 𝑣 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
52 51 3ad2ant2 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝑣 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
53 simp3l ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) )
54 simp3r ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) )
55 eldifi ⊢ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) → 𝑢 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
56 55 adantr ⊢ ( ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) → 𝑢 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
57 56 3ad2ant2 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → 𝑢 ∈ ( Base ‘ ( Scalar ‘ 𝑈 ) ) )
58 1 2 3 4 5 6 7 8 9 10 11 38 39 40 41 42 43 46 49 30 31 32 33 52 53 54 57 mapdpglem31 ⊢ ( ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) ∧ ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) ∧ ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) ) → ℎ = 𝑖 )
59 58 3exp ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( ( 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∧ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ) → ( ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) → ℎ = 𝑖 ) ) )
60 59 rexlimdvv ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ( ∃ 𝑢 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ∃ 𝑣 ∈ ( ( Base ‘ ( Scalar ‘ 𝑈 ) ) ∖ { ( 0g ‘ ( Scalar ‘ 𝑈 ) ) } ) ( ℎ = ( 𝑢 ( ·𝑠 ‘ 𝐶 ) 𝑖 ) ∧ ( 𝐺 𝑅 ℎ ) = ( 𝑣 ( ·𝑠 ‘ 𝐶 ) ( 𝐺 𝑅 𝑖 ) ) ) → ℎ = 𝑖 ) )
61 37 60 mpd ⊢ ( ( 𝜑 ∧ ( ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹 ) ∧ ( ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { ℎ } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 ℎ ) } ) ) ∧ ( ( 𝑀 ‘ ( 𝑁 ‘ { 𝑌 } ) ) = ( 𝐽 ‘ { 𝑖 } ) ∧ ( 𝑀 ‘ ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) ) = ( 𝐽 ‘ { ( 𝐺 𝑅 𝑖 ) } ) ) ) ) → ℎ = 𝑖 )