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Theorem hdmap1val2 36108
 Description: Value of preliminary map from vectors to functionals in the closed kernel dual space, for nonzero 𝑌. (Contributed by NM, 16-May-2015.)
Hypotheses
Ref Expression
hdmap1val2.h 𝐻 = (LHyp‘𝐾)
hdmap1val2.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1val2.v 𝑉 = (Base‘𝑈)
hdmap1val2.s = (-g𝑈)
hdmap1val2.o 0 = (0g𝑈)
hdmap1val2.n 𝑁 = (LSpan‘𝑈)
hdmap1val2.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1val2.d 𝐷 = (Base‘𝐶)
hdmap1val2.r 𝑅 = (-g𝐶)
hdmap1val2.l 𝐿 = (LSpan‘𝐶)
hdmap1val2.m 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1val2.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1val2.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
hdmap1val2.x (𝜑𝑋𝑉)
hdmap1val2.f (𝜑𝐹𝐷)
hdmap1val2.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
Assertion
Ref Expression
hdmap1val2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))
Distinct variable groups:   𝐶,   𝐷,   ,𝐹   ,𝐿   ,𝑀   ,𝑁   𝑈,   ,𝑉   ,𝑋   ,𝑌   𝜑,
Allowed substitution hints:   𝑅()   𝐻()   𝐼()   𝐾()   ()   𝑊()   0 ()

Proof of Theorem hdmap1val2
StepHypRef Expression
1 hdmap1val2.h . . 3 𝐻 = (LHyp‘𝐾)
2 hdmap1val2.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hdmap1val2.v . . 3 𝑉 = (Base‘𝑈)
4 hdmap1val2.s . . 3 = (-g𝑈)
5 hdmap1val2.o . . 3 0 = (0g𝑈)
6 hdmap1val2.n . . 3 𝑁 = (LSpan‘𝑈)
7 hdmap1val2.c . . 3 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hdmap1val2.d . . 3 𝐷 = (Base‘𝐶)
9 hdmap1val2.r . . 3 𝑅 = (-g𝐶)
10 eqid 2610 . . 3 (0g𝐶) = (0g𝐶)
11 hdmap1val2.l . . 3 𝐿 = (LSpan‘𝐶)
12 hdmap1val2.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
13 hdmap1val2.i . . 3 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14 hdmap1val2.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 hdmap1val2.x . . 3 (𝜑𝑋𝑉)
16 hdmap1val2.f . . 3 (𝜑𝐹𝐷)
17 hdmap1val2.y . . . 4 (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
1817eldifad 3552 . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 36106 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))))
20 eldifsni 4261 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
2120neneqd 2787 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22 iffalse 4045 . . 3 𝑌 = 0 → if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))
2317, 21, 223syl 18 . 2 (𝜑 → if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))
2419, 23eqtrd 2644 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ifcif 4036  {csn 4125  ⟨cotp 4133  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  Basecbs 15695  0gc0g 15923  -gcsg 17247  LSpanclspn 18792  HLchlt 33655  LHypclh 34288  DVecHcdvh 35385  LCDualclcd 35893  mapdcmpd 35931  HDMap1chdma1 36099 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-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-hdmap1 36101 This theorem is referenced by:  hdmap1eq  36109
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