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Theorem hdmap1val 37608
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 37534.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 37598 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 4326 . . . 4 𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌
16 hdmap1val.x . . . . . 6 (𝜑𝑋𝑉)
17 hdmap1val.f . . . . . 6 (𝜑𝐹𝐷)
18 opelxp 5285 . . . . . 6 (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋𝑉𝐹𝐷))
1916, 17, 18sylanbrc 572 . . . . 5 (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20 hdmap1val.y . . . . 5 (𝜑𝑌𝑉)
21 opelxp 5285 . . . . 5 (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌𝑉))
2219, 20, 21sylanbrc 572 . . . 4 (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
2315, 22syl5eqel 2854 . . 3 (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 37607 . 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 7335 . . . . 5 (𝑌𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2620, 25syl 17 . . . 4 (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2726eqeq1d 2773 . . 3 (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0𝑌 = 0 ))
2826sneqd 4329 . . . . . . . 8 (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
2928fveq2d 6337 . . . . . . 7 (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
3029fveq2d 6337 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝑀‘(𝑁‘{𝑌})))
3130eqeq1d 2773 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{})))
32 ot1stg 7333 . . . . . . . . . . 11 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3316, 17, 20, 32syl3anc 1476 . . . . . . . . . 10 (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3433, 26oveq12d 6814 . . . . . . . . 9 (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 𝑌))
3534sneqd 4329 . . . . . . . 8 (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 𝑌)})
3635fveq2d 6337 . . . . . . 7 (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 𝑌)}))
3736fveq2d 6337 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 𝑌)})))
38 ot2ndg 7334 . . . . . . . . . 10 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
3916, 17, 20, 38syl3anc 1476 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
4039oveq1d 6811 . . . . . . . 8 (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅) = (𝐹𝑅))
4140sneqd 4329 . . . . . . 7 (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)} = {(𝐹𝑅)})
4241fveq2d 6337 . . . . . 6 (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) = (𝐽‘{(𝐹𝑅)}))
4337, 42eqeq12d 2786 . . . . 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)})))
4431, 43anbi12d 616 . . . 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)}))))
4544riotabidv 6759 . . 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)}))))
4627, 45ifbieq2d 4251 . 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)})))))
4724, 46eqtrd 2805 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 382   = wceq 1631  wcel 2145  ifcif 4226  {csn 4317  cop 4323  cotp 4325   × cxp 5248  cfv 6030  crio 6756  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  Basecbs 16064  0gc0g 16308  -gcsg 17632  LSpanclspn 19184  LHypclh 35793  DVecHcdvh 36888  LCDualclcd 37396  mapdcmpd 37434  HDMap1chdma1 37601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-ot 4326  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-1st 7319  df-2nd 7320  df-hdmap1 37603
This theorem is referenced by:  hdmap1val0  37609  hdmap1val2  37610  hdmap1valc  37613
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