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Theorem hdmap1valc 39045
 Description: Connect the value of the preliminary map from vectors to functionals 𝐼 to the hypothesis 𝐿 used by earlier theorems. Note: the 𝑋 ∈ (𝑉 ∖ { 0 }) hypothesis could be the more general 𝑋 ∈ 𝑉 but the former will be easier to use. TODO: use the 𝐼 function directly in those theorems, so this theorem becomes unnecessary? TODO: The hdmap1cbv 39044 is probably unnecessary, but it would mean different \$d's later on. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1valc.h 𝐻 = (LHyp‘𝐾)
hdmap1valc.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1valc.v 𝑉 = (Base‘𝑈)
hdmap1valc.s = (-g𝑈)
hdmap1valc.o 0 = (0g𝑈)
hdmap1valc.n 𝑁 = (LSpan‘𝑈)
hdmap1valc.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1valc.d 𝐷 = (Base‘𝐶)
hdmap1valc.r 𝑅 = (-g𝐶)
hdmap1valc.q 𝑄 = (0g𝐶)
hdmap1valc.j 𝐽 = (LSpan‘𝐶)
hdmap1valc.m 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1valc.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1valc.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
hdmap1valc.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap1valc.f (𝜑𝐹𝐷)
hdmap1valc.y (𝜑𝑌𝑉)
hdmap1valc.l 𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
Assertion
Ref Expression
hdmap1valc (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))
Distinct variable groups:   𝑥, 0   𝑥,,𝐷   ,𝐽,𝑥   ,𝑀,𝑥   ,,𝑥   ,𝑁,𝑥   𝑅,,𝑥   𝑥,𝑄
Allowed substitution hints:   𝜑(𝑥,)   𝐶(𝑥,)   𝑄()   𝑈(𝑥,)   𝐹(𝑥,)   𝐻(𝑥,)   𝐼(𝑥,)   𝐾(𝑥,)   𝐿(𝑥,)   𝑉(𝑥,)   𝑊(𝑥,)   𝑋(𝑥,)   𝑌(𝑥,)   0 ()

Proof of Theorem hdmap1valc
Dummy variables 𝑤 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hdmap1valc.h . . 3 𝐻 = (LHyp‘𝐾)
2 hdmap1valc.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hdmap1valc.v . . 3 𝑉 = (Base‘𝑈)
4 hdmap1valc.s . . 3 = (-g𝑈)
5 hdmap1valc.o . . 3 0 = (0g𝑈)
6 hdmap1valc.n . . 3 𝑁 = (LSpan‘𝑈)
7 hdmap1valc.c . . 3 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hdmap1valc.d . . 3 𝐷 = (Base‘𝐶)
9 hdmap1valc.r . . 3 𝑅 = (-g𝐶)
10 hdmap1valc.q . . 3 𝑄 = (0g𝐶)
11 hdmap1valc.j . . 3 𝐽 = (LSpan‘𝐶)
12 hdmap1valc.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
13 hdmap1valc.i . . 3 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14 hdmap1valc.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 hdmap1valc.x . . . 4 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
1615eldifad 3932 . . 3 (𝜑𝑋𝑉)
17 hdmap1valc.f . . 3 (𝜑𝐹𝐷)
18 hdmap1valc.y . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18hdmap1val 39040 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝑔𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)})))))
20 hdmap1valc.l . . . 4 𝐿 = (𝑥 ∈ V ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
2120hdmap1cbv 39044 . . 3 𝐿 = (𝑤 ∈ V ↦ if((2nd𝑤) = 0 , 𝑄, (𝑔𝐷 ((𝑀‘(𝑁‘{(2nd𝑤)})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑤)) (2nd𝑤))})) = (𝐽‘{((2nd ‘(1st𝑤))𝑅𝑔)})))))
2210, 21, 16, 17, 18mapdhval 38966 . 2 (𝜑 → (𝐿‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝑔𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)})))))
2319, 22eqtr4d 2862 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3481   ∖ cdif 3917  ifcif 4451  {csn 4551  ⟨cotp 4559   ↦ cmpt 5133  ‘cfv 6344  ℩crio 7107  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  Basecbs 16486  0gc0g 16716  -gcsg 18108  LSpanclspn 19746  HLchlt 36592  LHypclh 37226  DVecHcdvh 38320  LCDualclcd 38828  mapdcmpd 38866  HDMap1chdma1 39033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-ot 4560  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-1st 7685  df-2nd 7686  df-hdmap1 39035 This theorem is referenced by:  hdmap1cl  39046  hdmap1eq2  39047  hdmap1eq4N  39048  hdmap1eulem  39064  hdmap1eulemOLDN  39065
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