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Theorem hdmap1val2 37407
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 2651 . . 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 3619 . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 37405 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))))
20 eldifsni 4353 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
2120neneqd 2828 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22 iffalse 4128 . . 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 2685 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 1523  wcel 2030  cdif 3604  ifcif 4119  {csn 4210  cotp 4218  cfv 5926  crio 6650  (class class class)co 6690  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:  hdmap1eq  37408
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