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Theorem hdmap1val 39087
 Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (Restatement of mapdhval 39013.) TODO: change 𝐼 = (𝑥 ∈ V ↦... to (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌 > ) =... in e.g. mapdh8 39077 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 4537 . . . 4 𝑋, 𝐹, 𝑌⟩ = ⟨⟨𝑋, 𝐹⟩, 𝑌
16 hdmap1val.x . . . . . 6 (𝜑𝑋𝑉)
17 hdmap1val.f . . . . . 6 (𝜑𝐹𝐷)
18 opelxp 5559 . . . . . 6 (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ↔ (𝑋𝑉𝐹𝐷))
1916, 17, 18sylanbrc 586 . . . . 5 (𝜑 → ⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷))
20 hdmap1val.y . . . . 5 (𝜑𝑌𝑉)
21 opelxp 5559 . . . . 5 (⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉) ↔ (⟨𝑋, 𝐹⟩ ∈ (𝑉 × 𝐷) ∧ 𝑌𝑉))
2219, 20, 21sylanbrc 586 . . . 4 (𝜑 → ⟨⟨𝑋, 𝐹⟩, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
2315, 22eqeltrid 2897 . . 3 (𝜑 → ⟨𝑋, 𝐹, 𝑌⟩ ∈ ((𝑉 × 𝐷) × 𝑉))
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 23hdmap1vallem 39086 . 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 7691 . . . . 5 (𝑌𝑉 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2620, 25syl 17 . . . 4 (𝜑 → (2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 𝑌)
2726eqeq1d 2803 . . 3 (𝜑 → ((2nd ‘⟨𝑋, 𝐹, 𝑌⟩) = 0𝑌 = 0 ))
2826sneqd 4540 . . . . . . 7 (𝜑 → {(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)} = {𝑌})
2928fveq2d 6653 . . . . . 6 (𝜑 → (𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)}) = (𝑁‘{𝑌}))
3029fveqeq2d 6657 . . . . 5 (𝜑 → ((𝑀‘(𝑁‘{(2nd ‘⟨𝑋, 𝐹, 𝑌⟩)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{})))
31 ot1stg 7689 . . . . . . . . . . 11 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3216, 17, 20, 31syl3anc 1368 . . . . . . . . . 10 (𝜑 → (1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝑋)
3332, 26oveq12d 7157 . . . . . . . . 9 (𝜑 → ((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩)) = (𝑋 𝑌))
3433sneqd 4540 . . . . . . . 8 (𝜑 → {((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))} = {(𝑋 𝑌)})
3534fveq2d 6653 . . . . . . 7 (𝜑 → (𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))}) = (𝑁‘{(𝑋 𝑌)}))
3635fveq2d 6653 . . . . . 6 (𝜑 → (𝑀‘(𝑁‘{((1st ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) (2nd ‘⟨𝑋, 𝐹, 𝑌⟩))})) = (𝑀‘(𝑁‘{(𝑋 𝑌)})))
37 ot2ndg 7690 . . . . . . . . . 10 ((𝑋𝑉𝐹𝐷𝑌𝑉) → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
3816, 17, 20, 37syl3anc 1368 . . . . . . . . 9 (𝜑 → (2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩)) = 𝐹)
3938oveq1d 7154 . . . . . . . 8 (𝜑 → ((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅) = (𝐹𝑅))
4039sneqd 4540 . . . . . . 7 (𝜑 → {((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)} = {(𝐹𝑅)})
4140fveq2d 6653 . . . . . 6 (𝜑 → (𝐽‘{((2nd ‘(1st ‘⟨𝑋, 𝐹, 𝑌⟩))𝑅)}) = (𝐽‘{(𝐹𝑅)}))
4236, 41eqeq12d 2817 . . . . 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 633 . . . 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 7099 . . 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 4453 . 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 2836 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 399   = wceq 1538   ∈ wcel 2112  ifcif 4428  {csn 4528  ⟨cop 4534  ⟨cotp 4536   × cxp 5521  ‘cfv 6328  ℩crio 7096  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Basecbs 16478  0gc0g 16708  -gcsg 18100  LSpanclspn 19739  LHypclh 37273  DVecHcdvh 38367  LCDualclcd 38875  mapdcmpd 38913  HDMap1chdma1 39080 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-ot 4537  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-1st 7675  df-2nd 7676  df-hdmap1 39082 This theorem is referenced by:  hdmap1val0  39088  hdmap1val2  39089  hdmap1valc  39092
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