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Theorem mapdpglem32 41102
Description: Lemma for mapdpg 41103. Uniqueness part - consolidate hypotheses in mapdpglem31 41100. (Contributed by NM, 23-Mar-2015.)
Hypotheses
Ref Expression
mapdpg.h 𝐻 = (LHypβ€˜πΎ)
mapdpg.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
mapdpg.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
mapdpg.v 𝑉 = (Baseβ€˜π‘ˆ)
mapdpg.s βˆ’ = (-gβ€˜π‘ˆ)
mapdpg.z 0 = (0gβ€˜π‘ˆ)
mapdpg.n 𝑁 = (LSpanβ€˜π‘ˆ)
mapdpg.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
mapdpg.f 𝐹 = (Baseβ€˜πΆ)
mapdpg.r 𝑅 = (-gβ€˜πΆ)
mapdpg.j 𝐽 = (LSpanβ€˜πΆ)
mapdpg.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
mapdpg.x (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
mapdpg.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
mapdpg.g (πœ‘ β†’ 𝐺 ∈ 𝐹)
mapdpg.ne (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
mapdpg.e (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))
Assertion
Ref Expression
mapdpglem32 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ β„Ž = 𝑖)
Distinct variable groups:   𝐢,β„Ž   β„Ž,𝐹   β„Ž,𝐺   β„Ž,𝐽   β„Ž,𝑀   β„Ž,𝑁   𝑅,β„Ž   βˆ’ ,β„Ž   π‘ˆ,β„Ž   β„Ž,𝑋   β„Ž,π‘Œ   β„Ž,𝑖
Allowed substitution hints:   πœ‘(β„Ž,𝑖)   𝐢(𝑖)   𝑅(𝑖)   π‘ˆ(𝑖)   𝐹(𝑖)   𝐺(𝑖)   𝐻(β„Ž,𝑖)   𝐽(𝑖)   𝐾(β„Ž,𝑖)   𝑀(𝑖)   βˆ’ (𝑖)   𝑁(𝑖)   𝑉(β„Ž,𝑖)   π‘Š(β„Ž,𝑖)   𝑋(𝑖)   π‘Œ(𝑖)   0 (β„Ž,𝑖)

Proof of Theorem mapdpglem32
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapdpg.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 mapdpg.m . . . 4 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
3 mapdpg.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
4 mapdpg.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
5 mapdpg.s . . . 4 βˆ’ = (-gβ€˜π‘ˆ)
6 mapdpg.z . . . 4 0 = (0gβ€˜π‘ˆ)
7 mapdpg.n . . . 4 𝑁 = (LSpanβ€˜π‘ˆ)
8 mapdpg.c . . . 4 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
9 mapdpg.f . . . 4 𝐹 = (Baseβ€˜πΆ)
10 mapdpg.r . . . 4 𝑅 = (-gβ€˜πΆ)
11 mapdpg.j . . . 4 𝐽 = (LSpanβ€˜πΆ)
12 mapdpg.k . . . . 5 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
13123ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
14 mapdpg.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
15143ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
16 mapdpg.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
17163ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
18 mapdpg.g . . . . 5 (πœ‘ β†’ 𝐺 ∈ 𝐹)
19183ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ 𝐺 ∈ 𝐹)
20 mapdpg.ne . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
21203ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
22 mapdpg.e . . . . 5 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))
23223ad2ant1 1131 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))
24 simp2l 1197 . . . . 5 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ β„Ž ∈ 𝐹)
25 simp3l 1199 . . . . 5 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})))
2624, 25jca 511 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))
27 simp2r 1198 . . . . 5 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ 𝑖 ∈ 𝐹)
28 simp3r 1200 . . . . 5 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))
2927, 28jca 511 . . . 4 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))
30 eqid 2727 . . . 4 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
31 eqid 2727 . . . 4 (Baseβ€˜(Scalarβ€˜π‘ˆ)) = (Baseβ€˜(Scalarβ€˜π‘ˆ))
32 eqid 2727 . . . 4 ( ·𝑠 β€˜πΆ) = ( ·𝑠 β€˜πΆ)
33 eqid 2727 . . . 4 (0gβ€˜(Scalarβ€˜π‘ˆ)) = (0gβ€˜(Scalarβ€˜π‘ˆ))
341, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33mapdpglem26 41095 . . 3 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ βˆƒπ‘’ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖))
351, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 26, 29, 30, 31, 32, 33mapdpglem27 41096 . . 3 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})(πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))
36 reeanv 3221 . . 3 (βˆƒπ‘’ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})(β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖))) ↔ (βˆƒπ‘’ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})(πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖))))
3734, 35, 36sylanbrc 582 . 2 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ βˆƒπ‘’ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})(β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖))))
38133ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
39153ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
40173ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
41193ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ 𝐺 ∈ 𝐹)
42213ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
43233ad2ant1 1131 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (π‘€β€˜(π‘β€˜{𝑋})) = (π½β€˜{𝐺}))
44 simp12l 1284 . . . . . 6 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ β„Ž ∈ 𝐹)
45 simp13l 1286 . . . . . 6 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})))
4644, 45jca 511 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (β„Ž ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)}))))
47 simp12r 1285 . . . . . 6 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ 𝑖 ∈ 𝐹)
48 simp13r 1287 . . . . . 6 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))
4947, 48jca 511 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (𝑖 ∈ 𝐹 ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)}))))
50 eldifi 4122 . . . . . . 7 (𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) β†’ 𝑣 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
5150adantl 481 . . . . . 6 ((𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) β†’ 𝑣 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
52513ad2ant2 1132 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ 𝑣 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
53 simp3l 1199 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖))
54 simp3r 1200 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))
55 eldifi 4122 . . . . . . 7 (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) β†’ 𝑒 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
5655adantr 480 . . . . . 6 ((𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) β†’ 𝑒 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
57563ad2ant2 1132 . . . . 5 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ 𝑒 ∈ (Baseβ€˜(Scalarβ€˜π‘ˆ)))
581, 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, 57mapdpglem31 41100 . . . 4 (((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) ∧ (𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) ∧ (β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖)))) β†’ β„Ž = 𝑖)
59583exp 1117 . . 3 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ ((𝑒 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})) β†’ ((β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖))) β†’ β„Ž = 𝑖)))
6059rexlimdvv 3205 . 2 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ (βˆƒπ‘’ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜π‘ˆ)) βˆ– {(0gβ€˜(Scalarβ€˜π‘ˆ))})(β„Ž = (𝑒( ·𝑠 β€˜πΆ)𝑖) ∧ (πΊπ‘…β„Ž) = (𝑣( ·𝑠 β€˜πΆ)(𝐺𝑅𝑖))) β†’ β„Ž = 𝑖))
6137, 60mpd 15 1 ((πœ‘ ∧ (β„Ž ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΊπ‘…β„Ž)})) ∧ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{𝑖}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(𝐺𝑅𝑖)})))) β†’ β„Ž = 𝑖)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065   βˆ– cdif 3941  {csn 4624  β€˜cfv 6542  (class class class)co 7414  Basecbs 17165  Scalarcsca 17221   ·𝑠 cvsca 17222  0gc0g 17406  -gcsg 18877  LSpanclspn 20837  HLchlt 38746  LHypclh 39381  DVecHcdvh 40475  LCDualclcd 40983  mapdcmpd 41021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-riotaBAD 38349
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7677  df-om 7863  df-1st 7985  df-2nd 7986  df-tpos 8223  df-undef 8270  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-n0 12489  df-z 12575  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-ress 17195  df-plusg 17231  df-mulr 17232  df-sca 17234  df-vsca 17235  df-0g 17408  df-mre 17551  df-mrc 17552  df-acs 17554  df-proset 18272  df-poset 18290  df-plt 18307  df-lub 18323  df-glb 18324  df-join 18325  df-meet 18326  df-p0 18402  df-p1 18403  df-lat 18409  df-clat 18476  df-mgm 18585  df-sgrp 18664  df-mnd 18680  df-submnd 18726  df-grp 18878  df-minusg 18879  df-sbg 18880  df-subg 19062  df-cntz 19252  df-oppg 19281  df-lsm 19575  df-cmn 19721  df-abl 19722  df-mgp 20059  df-rng 20077  df-ur 20106  df-ring 20159  df-oppr 20255  df-dvdsr 20278  df-unit 20279  df-invr 20309  df-dvr 20322  df-drng 20608  df-lmod 20727  df-lss 20798  df-lsp 20838  df-lvec 20970  df-lsatoms 38372  df-lshyp 38373  df-lcv 38415  df-lfl 38454  df-lkr 38482  df-ldual 38520  df-oposet 38572  df-ol 38574  df-oml 38575  df-covers 38662  df-ats 38663  df-atl 38694  df-cvlat 38718  df-hlat 38747  df-llines 38895  df-lplanes 38896  df-lvols 38897  df-lines 38898  df-psubsp 38900  df-pmap 38901  df-padd 39193  df-lhyp 39385  df-laut 39386  df-ldil 39501  df-ltrn 39502  df-trl 39556  df-tgrp 40140  df-tendo 40152  df-edring 40154  df-dveca 40400  df-disoa 40426  df-dvech 40476  df-dib 40536  df-dic 40570  df-dih 40626  df-doch 40745  df-djh 40792  df-lcdual 40984  df-mapd 41022
This theorem is referenced by:  mapdpg  41103
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