| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mapdpg.h | . . . 4
⊢ 𝐻 = (LHyp‘𝐾) | 
| 2 |  | mapdpg.m | . . . 4
⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | 
| 3 |  | mapdpg.u | . . . 4
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | 
| 4 |  | mapdpg.v | . . . 4
⊢ 𝑉 = (Base‘𝑈) | 
| 5 |  | mapdpg.s | . . . 4
⊢  − =
(-g‘𝑈) | 
| 6 |  | mapdpg.z | . . . 4
⊢  0 =
(0g‘𝑈) | 
| 7 |  | mapdpg.n | . . . 4
⊢ 𝑁 = (LSpan‘𝑈) | 
| 8 |  | mapdpg.c | . . . 4
⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | 
| 9 |  | mapdpg.f | . . . 4
⊢ 𝐹 = (Base‘𝐶) | 
| 10 |  | mapdpg.r | . . . 4
⊢ 𝑅 = (-g‘𝐶) | 
| 11 |  | mapdpg.j | . . . 4
⊢ 𝐽 = (LSpan‘𝐶) | 
| 12 |  | mapdpg.k | . . . . 5
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 13 | 12 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 14 |  | mapdpg.x | . . . . 5
⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 15 | 14 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 16 |  | mapdpg.y | . . . . 5
⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 17 | 16 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 18 |  | mapdpg.g | . . . . 5
⊢ (𝜑 → 𝐺 ∈ 𝐹) | 
| 19 | 18 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝐺 ∈ 𝐹) | 
| 20 |  | mapdpg.ne | . . . . 5
⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | 
| 21 | 20 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | 
| 22 |  | mapdpg.e | . . . . 5
⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | 
| 23 | 22 | 3ad2ant1 1134 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | 
| 24 |  | simp2l 1200 | . . . . 5
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ ∈ 𝐹) | 
| 25 |  | simp3l 1202 | . . . . 5
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | 
| 26 | 24, 25 | jca 511 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) | 
| 27 |  | simp2r 1201 | . . . . 5
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → 𝑖 ∈ 𝐹) | 
| 28 |  | simp3r 1203 | . . . . 5
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) | 
| 29 | 27, 28 | jca 511 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | 
| 30 |  | eqid 2737 | . . . 4
⊢
(Scalar‘𝑈) =
(Scalar‘𝑈) | 
| 31 |  | eqid 2737 | . . . 4
⊢
(Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈)) | 
| 32 |  | eqid 2737 | . . . 4
⊢ (
·𝑠 ‘𝐶) = ( ·𝑠
‘𝐶) | 
| 33 |  | eqid 2737 | . . . 4
⊢
(0g‘(Scalar‘𝑈)) =
(0g‘(Scalar‘𝑈)) | 
| 34 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33 | mapdpglem26 41700 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})ℎ = (𝑢( ·𝑠
‘𝐶)𝑖)) | 
| 35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33 | mapdpglem27 41701 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})(𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖))) | 
| 36 |  | reeanv 3229 | . . 3
⊢
(∃𝑢 ∈
((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖))) ↔ (∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ ∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})(𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) | 
| 37 | 34, 35, 36 | sylanbrc 583 | . 2
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) | 
| 38 | 13 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | 
| 39 | 15 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 𝑋 ∈ (𝑉 ∖ { 0 })) | 
| 40 | 17 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 𝑌 ∈ (𝑉 ∖ { 0 })) | 
| 41 | 19 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 𝐺 ∈ 𝐹) | 
| 42 | 21 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | 
| 43 | 23 | 3ad2ant1 1134 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | 
| 44 |  | simp12l 1287 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → ℎ ∈ 𝐹) | 
| 45 |  | simp13l 1289 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | 
| 46 | 44, 45 | jca 511 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) | 
| 47 |  | simp12r 1288 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 𝑖 ∈ 𝐹) | 
| 48 |  | simp13r 1290 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) | 
| 49 | 47, 48 | jca 511 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | 
| 50 |  | eldifi 4131 | . . . . . . 7
⊢ (𝑣 ∈
((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) → 𝑣 ∈ (Base‘(Scalar‘𝑈))) | 
| 51 | 50 | adantl 481 | . . . . . 6
⊢ ((𝑢 ∈
((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) → 𝑣 ∈ (Base‘(Scalar‘𝑈))) | 
| 52 | 51 | 3ad2ant2 1135 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 𝑣 ∈ (Base‘(Scalar‘𝑈))) | 
| 53 |  | simp3l 1202 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → ℎ = (𝑢( ·𝑠
‘𝐶)𝑖)) | 
| 54 |  | simp3r 1203 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖))) | 
| 55 |  | eldifi 4131 | . . . . . . 7
⊢ (𝑢 ∈
((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) → 𝑢 ∈ (Base‘(Scalar‘𝑈))) | 
| 56 | 55 | adantr 480 | . . . . . 6
⊢ ((𝑢 ∈
((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) → 𝑢 ∈ (Base‘(Scalar‘𝑈))) | 
| 57 | 56 | 3ad2ant2 1135 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → 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 41705 | . . . 4
⊢ (((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ∧ (𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) ∧ (ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖)))) → ℎ = 𝑖) | 
| 59 | 58 | 3exp 1120 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ((𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))}) ∧ 𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})) → ((ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖))) → ℎ = 𝑖))) | 
| 60 | 59 | rexlimdvv 3212 | . 2
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → (∃𝑢 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})∃𝑣 ∈ ((Base‘(Scalar‘𝑈)) ∖
{(0g‘(Scalar‘𝑈))})(ℎ = (𝑢( ·𝑠
‘𝐶)𝑖) ∧ (𝐺𝑅ℎ) = (𝑣( ·𝑠
‘𝐶)(𝐺𝑅𝑖))) → ℎ = 𝑖)) | 
| 61 | 37, 60 | mpd 15 | 1
⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖) |