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Theorem hdmap1val2 38816
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 2818 . . 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 3945 . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 38814 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))))
20 eldifsni 4714 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
2120neneqd 3018 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22 iffalse 4472 . . 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 2853 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 396   = wceq 1528  wcel 2105  cdif 3930  ifcif 4463  {csn 4557  cotp 4565  cfv 6348  crio 7102  (class class class)co 7145  Basecbs 16471  0gc0g 16701  -gcsg 18043  LSpanclspn 19672  HLchlt 36366  LHypclh 37000  DVecHcdvh 38094  LCDualclcd 38602  mapdcmpd 38640  HDMap1chdma1 38807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-ot 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-hdmap1 38809
This theorem is referenced by:  hdmap1eq  38817
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