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Theorem mapdpglem25 36004
Description: Lemma for mapdpg 36013. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-Mar-2015.)
Hypotheses
Ref Expression
mapdpg.h 𝐻 = (LHyp‘𝐾)
mapdpg.m 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpg.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpg.v 𝑉 = (Base‘𝑈)
mapdpg.s = (-g𝑈)
mapdpg.z 0 = (0g𝑈)
mapdpg.n 𝑁 = (LSpan‘𝑈)
mapdpg.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpg.f 𝐹 = (Base‘𝐶)
mapdpg.r 𝑅 = (-g𝐶)
mapdpg.j 𝐽 = (LSpan‘𝐶)
mapdpg.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
mapdpg.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
mapdpg.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
mapdpg.g (𝜑𝐺𝐹)
mapdpg.ne (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpg.e (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpgem25.h1 (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))
mapdpgem25.i1 (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
Assertion
Ref Expression
mapdpglem25 (𝜑 → ((𝐽‘{}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅)}) = (𝐽‘{(𝐺𝑅𝑖)})))

Proof of Theorem mapdpglem25
StepHypRef Expression
1 mapdpgem25.h1 . . . . 5 (𝜑 → (𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))))
21simprd 478 . . . 4 (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)})))
32simpld 474 . . 3 (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}))
4 mapdpgem25.i1 . . . . 5 (𝜑 → (𝑖𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
54simprd 478 . . . 4 (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
65simpld 474 . . 3 (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))
73, 6eqtr3d 2646 . 2 (𝜑 → (𝐽‘{}) = (𝐽‘{𝑖}))
82simprd 478 . . 3 (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅)}))
95simprd 478 . . 3 (𝜑 → (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))
108, 9eqtr3d 2646 . 2 (𝜑 → (𝐽‘{(𝐺𝑅)}) = (𝐽‘{(𝐺𝑅𝑖)}))
117, 10jca 553 1 (𝜑 → ((𝐽‘{}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅)}) = (𝐽‘{(𝐺𝑅𝑖)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  cdif 3537  {csn 4125  cfv 5804  (class class class)co 6549  Basecbs 15695  0gc0g 15923  -gcsg 17247  LSpanclspn 18792  HLchlt 33655  LHypclh 34288  DVecHcdvh 35385  LCDualclcd 35893  mapdcmpd 35931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-cleq 2603
This theorem is referenced by:  mapdpglem26  36005  mapdpglem27  36006
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