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Theorem hdmap1val2 41802
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 2737 . . 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 3963 . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 41800 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))))
20 eldifsni 4790 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
2120neneqd 2945 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22 iffalse 4534 . . 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 2777 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 395   = wceq 1540  wcel 2108  cdif 3948  ifcif 4525  {csn 4626  cotp 4634  cfv 6561  crio 7387  (class class class)co 7431  Basecbs 17247  0gc0g 17484  -gcsg 18953  LSpanclspn 20969  HLchlt 39351  LHypclh 39986  DVecHcdvh 41080  LCDualclcd 41588  mapdcmpd 41626  HDMap1chdma1 41793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-1st 8014  df-2nd 8015  df-hdmap1 41795
This theorem is referenced by:  hdmap1eq  41803
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