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Theorem hdmap1valc 37410
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 𝑋𝑉 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 37409 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 3619 . . 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 37405 . 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 37409 . . 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 37330 . 2 (𝜑 → (𝐿‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝑔𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑔}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅𝑔)})))))
2319, 22eqtr4d 2688 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐿‘⟨𝑋, 𝐹, 𝑌⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  ifcif 4119  {csn 4210  cotp 4218  cmpt 4762  cfv 5926  crio 6650  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  0gc0g 16147  -gcsg 17471  LSpanclspn 19019  HLchlt 34955  LHypclh 35588  DVecHcdvh 36684  LCDualclcd 37192  mapdcmpd 37230  HDMap1chdma1 37398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-1st 7210  df-2nd 7211  df-hdmap1 37400
This theorem is referenced by:  hdmap1cl  37411  hdmap1eq2  37412  hdmap1eq4N  37413  hdmap1eulem  37430  hdmap1eulemOLDN  37431
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