Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1val Structured version   Visualization version   GIF version

Theorem hdmap1val 40261
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 40187.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 40251 to shorten proofs with no $d on 𝑥. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h 𝐻 = (LHyp‘𝐾)
hdmap1fval.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap1fval.v 𝑉 = (Base‘𝑈)
hdmap1fval.s = (-g𝑈)
hdmap1fval.o 0 = (0g𝑈)
hdmap1fval.n 𝑁 = (LSpan‘𝑈)
hdmap1fval.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap1fval.d 𝐷 = (Base‘𝐶)
hdmap1fval.r 𝑅 = (-g𝐶)
hdmap1fval.q 𝑄 = (0g𝐶)
hdmap1fval.j 𝐽 = (LSpan‘𝐶)
hdmap1fval.m 𝑀 = ((mapd‘𝐾)‘𝑊)
hdmap1fval.i 𝐼 = ((HDMap1‘𝐾)‘𝑊)
hdmap1fval.k (𝜑 → (𝐾𝐴𝑊𝐻))
hdmap1val.x (𝜑𝑋𝑉)
hdmap1val.f (𝜑𝐹𝐷)
hdmap1val.y (𝜑𝑌𝑉)
Assertion
Ref Expression
hdmap1val (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
Distinct variable groups:   𝐶,   𝐷,   ,𝐽   ,𝑀   ,𝑁   𝑈,   ,𝑉   ,𝐹   ,𝑋   ,𝑌   𝜑,
Allowed substitution hints:   𝐴()   𝑄()   𝑅()   𝐻()   𝐼()   𝐾()   ()   𝑊()   0 ()

Proof of Theorem hdmap1val
StepHypRef Expression
1 hdmap1val.h . . 3 𝐻 = (LHyp‘𝐾)
2 hdmap1fval.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hdmap1fval.v . . 3 𝑉 = (Base‘𝑈)
4 hdmap1fval.s . . 3 = (-g𝑈)
5 hdmap1fval.o . . 3 0 = (0g𝑈)
6 hdmap1fval.n . . 3 𝑁 = (LSpan‘𝑈)
7 hdmap1fval.c . . 3 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hdmap1fval.d . . 3 𝐷 = (Base‘𝐶)
9 hdmap1fval.r . . 3 𝑅 = (-g𝐶)
10 hdmap1fval.q . . 3 𝑄 = (0g𝐶)
11 hdmap1fval.j . . 3 𝐽 = (LSpan‘𝐶)
12 hdmap1fval.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
13 hdmap1fval.i . . 3 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14 hdmap1fval.k . . 3 (𝜑 → (𝐾𝐴𝑊𝐻))
15 df-ot 4595 . . . 4 𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌
16 hdmap1val.x . . . . . 6 (𝜑𝑋𝑉)
17 hdmap1val.f . . . . . 6 (𝜑𝐹𝐷)
18 opelxp 5669 . . . . . 6 (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋𝑉𝐹𝐷))
1916, 17, 18sylanbrc 583 . . . . 5 (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20 hdmap1val.y . . . . 5 (𝜑𝑌𝑉)
21 opelxp 5669 . . . . 5 (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌𝑉))
2219, 20, 21sylanbrc 583 . . . 4 (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
2315, 22eqeltrid 2842 . . 3 (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 40260 . 2 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))))
25 ot3rdg 7937 . . . . 5 (𝑌𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2620, 25syl 17 . . . 4 (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2726eqeq1d 2738 . . 3 (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0𝑌 = 0 ))
2826sneqd 4598 . . . . . . 7 (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
2928fveq2d 6846 . . . . . 6 (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
3029fveqeq2d 6850 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{})))
31 ot1stg 7935 . . . . . . . . . . 11 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3216, 17, 20, 31syl3anc 1371 . . . . . . . . . 10 (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3332, 26oveq12d 7375 . . . . . . . . 9 (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 𝑌))
3433sneqd 4598 . . . . . . . 8 (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 𝑌)})
3534fveq2d 6846 . . . . . . 7 (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 𝑌)}))
3635fveq2d 6846 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 𝑌)})))
37 ot2ndg 7936 . . . . . . . . . 10 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
3816, 17, 20, 37syl3anc 1371 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
3938oveq1d 7372 . . . . . . . 8 (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅) = (𝐹𝑅))
4039sneqd 4598 . . . . . . 7 (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)} = {(𝐹𝑅)})
4140fveq2d 6846 . . . . . 6 (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) = (𝐽‘{(𝐹𝑅)}))
4236, 41eqeq12d 2752 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) ↔ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))
4330, 42anbi12d 631 . . . 4 (𝜑 → (((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
4443riotabidv 7315 . . 3 (𝜑 → (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)}))))
4527, 44ifbieq2d 4512 . 2 (𝜑 → if((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)})))) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
4624, 45eqtrd 2776 1 (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = if(𝑌 = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{(𝑋 𝑌)})) = (𝐽‘{(𝐹𝑅)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ifcif 4486  {csn 4586  cop 4592  cotp 4594   × cxp 5631  cfv 6496  crio 7312  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  0gc0g 17321  -gcsg 18750  LSpanclspn 20432  LHypclh 38447  DVecHcdvh 39541  LCDualclcd 40049  mapdcmpd 40087  HDMap1chdma1 40254
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-ot 4595  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-1st 7921  df-2nd 7922  df-hdmap1 40256
This theorem is referenced by:  hdmap1val0  40262  hdmap1val2  40263  hdmap1valc  40266
  Copyright terms: Public domain W3C validator