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 41327
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 41253.) TODO: change 𝐼 = (π‘₯ ∈ V ↦... to (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘Œ > ) =... in e.g. mapdh8 41317 to shorten proofs with no $d on π‘₯. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmap1val.h 𝐻 = (LHypβ€˜πΎ)
hdmap1fval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1fval.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1fval.s βˆ’ = (-gβ€˜π‘ˆ)
hdmap1fval.o 0 = (0gβ€˜π‘ˆ)
hdmap1fval.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1fval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1fval.d 𝐷 = (Baseβ€˜πΆ)
hdmap1fval.r 𝑅 = (-gβ€˜πΆ)
hdmap1fval.q 𝑄 = (0gβ€˜πΆ)
hdmap1fval.j 𝐽 = (LSpanβ€˜πΆ)
hdmap1fval.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1fval.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1fval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
hdmap1val.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
hdmap1val.f (πœ‘ β†’ 𝐹 ∈ 𝐷)
hdmap1val.y (πœ‘ β†’ π‘Œ ∈ 𝑉)
Assertion
Ref Expression
hdmap1val (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
Distinct variable groups:   𝐢,β„Ž   𝐷,β„Ž   β„Ž,𝐽   β„Ž,𝑀   β„Ž,𝑁   π‘ˆ,β„Ž   β„Ž,𝑉   β„Ž,𝐹   β„Ž,𝑋   β„Ž,π‘Œ   πœ‘,β„Ž
Allowed substitution hints:   𝐴(β„Ž)   𝑄(β„Ž)   𝑅(β„Ž)   𝐻(β„Ž)   𝐼(β„Ž)   𝐾(β„Ž)   βˆ’ (β„Ž)   π‘Š(β„Ž)   0 (β„Ž)

Proof of Theorem hdmap1val
StepHypRef Expression
1 hdmap1val.h . . 3 𝐻 = (LHypβ€˜πΎ)
2 hdmap1fval.u . . 3 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hdmap1fval.v . . 3 𝑉 = (Baseβ€˜π‘ˆ)
4 hdmap1fval.s . . 3 βˆ’ = (-gβ€˜π‘ˆ)
5 hdmap1fval.o . . 3 0 = (0gβ€˜π‘ˆ)
6 hdmap1fval.n . . 3 𝑁 = (LSpanβ€˜π‘ˆ)
7 hdmap1fval.c . . 3 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
8 hdmap1fval.d . . 3 𝐷 = (Baseβ€˜πΆ)
9 hdmap1fval.r . . 3 𝑅 = (-gβ€˜πΆ)
10 hdmap1fval.q . . 3 𝑄 = (0gβ€˜πΆ)
11 hdmap1fval.j . . 3 𝐽 = (LSpanβ€˜πΆ)
12 hdmap1fval.m . . 3 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
13 hdmap1fval.i . . 3 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
14 hdmap1fval.k . . 3 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
15 df-ot 4633 . . . 4 βŸ¨π‘‹, 𝐹, π‘ŒβŸ© = βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ©
16 hdmap1val.x . . . . . 6 (πœ‘ β†’ 𝑋 ∈ 𝑉)
17 hdmap1val.f . . . . . 6 (πœ‘ β†’ 𝐹 ∈ 𝐷)
18 opelxp 5708 . . . . . 6 (βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷) ↔ (𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷))
1916, 17, 18sylanbrc 581 . . . . 5 (πœ‘ β†’ βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷))
20 hdmap1val.y . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑉)
21 opelxp 5708 . . . . 5 (βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↔ (βŸ¨π‘‹, 𝐹⟩ ∈ (𝑉 Γ— 𝐷) ∧ π‘Œ ∈ 𝑉))
2219, 20, 21sylanbrc 581 . . . 4 (πœ‘ β†’ βŸ¨βŸ¨π‘‹, 𝐹⟩, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
2315, 22eqeltrid 2829 . . 3 (πœ‘ β†’ βŸ¨π‘‹, 𝐹, π‘ŒβŸ© ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 41326 . 2 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})))))
25 ot3rdg 8007 . . . . 5 (π‘Œ ∈ 𝑉 β†’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = π‘Œ)
2620, 25syl 17 . . . 4 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = π‘Œ)
2726eqeq1d 2727 . . 3 (πœ‘ β†’ ((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 ↔ π‘Œ = 0 ))
2826sneqd 4636 . . . . . . 7 (πœ‘ β†’ {(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)} = {π‘Œ})
2928fveq2d 6896 . . . . . 6 (πœ‘ β†’ (π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)}) = (π‘β€˜{π‘Œ}))
3029fveqeq2d 6900 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ↔ (π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž})))
31 ot1stg 8005 . . . . . . . . . . 11 ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ π‘Œ ∈ 𝑉) β†’ (1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝑋)
3216, 17, 20, 31syl3anc 1368 . . . . . . . . . 10 (πœ‘ β†’ (1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝑋)
3332, 26oveq12d 7434 . . . . . . . . 9 (πœ‘ β†’ ((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = (𝑋 βˆ’ π‘Œ))
3433sneqd 4636 . . . . . . . 8 (πœ‘ β†’ {((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))} = {(𝑋 βˆ’ π‘Œ)})
3534fveq2d 6896 . . . . . . 7 (πœ‘ β†’ (π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))}) = (π‘β€˜{(𝑋 βˆ’ π‘Œ)}))
3635fveq2d 6896 . . . . . 6 (πœ‘ β†’ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})))
37 ot2ndg 8006 . . . . . . . . . 10 ((𝑋 ∈ 𝑉 ∧ 𝐹 ∈ 𝐷 ∧ π‘Œ ∈ 𝑉) β†’ (2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝐹)
3816, 17, 20, 37syl3anc 1368 . . . . . . . . 9 (πœ‘ β†’ (2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) = 𝐹)
3938oveq1d 7431 . . . . . . . 8 (πœ‘ β†’ ((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž) = (πΉπ‘…β„Ž))
4039sneqd 4636 . . . . . . 7 (πœ‘ β†’ {((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)} = {(πΉπ‘…β„Ž)})
4140fveq2d 6896 . . . . . 6 (πœ‘ β†’ (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}) = (π½β€˜{(πΉπ‘…β„Ž)}))
4236, 41eqeq12d 2741 . . . . 5 (πœ‘ β†’ ((π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}) ↔ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))
4330, 42anbi12d 630 . . . 4 (πœ‘ β†’ (((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})) ↔ ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)}))))
4443riotabidv 7374 . . 3 (πœ‘ β†’ (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)}))) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)}))))
4527, 44ifbieq2d 4550 . 2 (πœ‘ β†’ if((2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©)) βˆ’ (2nd β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))})) = (π½β€˜{((2nd β€˜(1st β€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©))π‘…β„Ž)})))) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
4624, 45eqtrd 2765 1 (πœ‘ β†’ (πΌβ€˜βŸ¨π‘‹, 𝐹, π‘ŒβŸ©) = if(π‘Œ = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{π‘Œ})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{(𝑋 βˆ’ π‘Œ)})) = (π½β€˜{(πΉπ‘…β„Ž)})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ifcif 4524  {csn 4624  βŸ¨cop 4630  βŸ¨cotp 4632   Γ— cxp 5670  β€˜cfv 6543  β„©crio 7371  (class class class)co 7416  1st c1st 7989  2nd c2nd 7990  Basecbs 17179  0gc0g 17420  -gcsg 18896  LSpanclspn 20859  LHypclh 39513  DVecHcdvh 40607  LCDualclcd 41115  mapdcmpd 41153  HDMap1chdma1 41320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-ot 4633  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-1st 7991  df-2nd 7992  df-hdmap1 41322
This theorem is referenced by:  hdmap1val0  41328  hdmap1val2  41329  hdmap1valc  41332
  Copyright terms: Public domain W3C validator