Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1val2 Structured version   Visualization version   GIF version

Theorem hdmap1val2 39054
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 2822 . . 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 3920 . . 3 (𝜑𝑌𝑉)
191, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18hdmap1val 39052 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , (0g𝐶), (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐿‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐿‘{(𝐹𝑅)})))))
20 eldifsni 4696 . . . 4 (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌0 )
2120neneqd 3016 . . 3 (𝑌 ∈ (𝑉 ∖ { 0 }) → ¬ 𝑌 = 0 )
22 iffalse 4448 . . 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 2857 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 399   = wceq 1538  wcel 2114  cdif 3905  ifcif 4439  {csn 4539  cotp 4547  cfv 6334  crio 7097  (class class class)co 7140  Basecbs 16474  0gc0g 16704  -gcsg 18096  LSpanclspn 19734  HLchlt 36604  LHypclh 37238  DVecHcdvh 38332  LCDualclcd 38840  mapdcmpd 38878  HDMap1chdma1 39045
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-ot 4548  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-1st 7675  df-2nd 7676  df-hdmap1 39047
This theorem is referenced by:  hdmap1eq  39055
  Copyright terms: Public domain W3C validator