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Theorem hdmap1vallem 41799
Description: Value of preliminary map from vectors to functionals in the closed kernel dual space. (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.t (𝜑𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))
Assertion
Ref Expression
hdmap1vallem (𝜑 → (𝐼𝑇) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))
Distinct variable groups:   𝐶,   𝐷,   ,𝐽   ,𝑀   ,𝑁   𝑈,   ,𝑉   𝑇,
Allowed substitution hints:   𝜑()   𝐴()   𝑄()   𝑅()   𝐻()   𝐼()   𝐾()   ()   𝑊()   0 ()

Proof of Theorem hdmap1vallem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 hdmap1val.h . . . 4 𝐻 = (LHyp‘𝐾)
2 hdmap1fval.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 hdmap1fval.v . . . 4 𝑉 = (Base‘𝑈)
4 hdmap1fval.s . . . 4 = (-g𝑈)
5 hdmap1fval.o . . . 4 0 = (0g𝑈)
6 hdmap1fval.n . . . 4 𝑁 = (LSpan‘𝑈)
7 hdmap1fval.c . . . 4 𝐶 = ((LCDual‘𝐾)‘𝑊)
8 hdmap1fval.d . . . 4 𝐷 = (Base‘𝐶)
9 hdmap1fval.r . . . 4 𝑅 = (-g𝐶)
10 hdmap1fval.q . . . 4 𝑄 = (0g𝐶)
11 hdmap1fval.j . . . 4 𝐽 = (LSpan‘𝐶)
12 hdmap1fval.m . . . 4 𝑀 = ((mapd‘𝐾)‘𝑊)
13 hdmap1fval.i . . . 4 𝐼 = ((HDMap1‘𝐾)‘𝑊)
14 hdmap1fval.k . . . 4 (𝜑 → (𝐾𝐴𝑊𝐻))
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmap1fval 41798 . . 3 (𝜑𝐼 = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))))
1615fveq1d 6908 . 2 (𝜑 → (𝐼𝑇) = ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))‘𝑇))
17 hdmap1val.t . . 3 (𝜑𝑇 ∈ ((𝑉 × 𝐷) × 𝑉))
1810fvexi 6920 . . . 4 𝑄 ∈ V
19 riotaex 7392 . . . 4 (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))) ∈ V
2018, 19ifex 4576 . . 3 if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))) ∈ V
21 fveqeq2 6915 . . . . 5 (𝑥 = 𝑇 → ((2nd𝑥) = 0 ↔ (2nd𝑇) = 0 ))
22 fveq2 6906 . . . . . . . . . 10 (𝑥 = 𝑇 → (2nd𝑥) = (2nd𝑇))
2322sneqd 4638 . . . . . . . . 9 (𝑥 = 𝑇 → {(2nd𝑥)} = {(2nd𝑇)})
2423fveq2d 6910 . . . . . . . 8 (𝑥 = 𝑇 → (𝑁‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑇)}))
2524fveqeq2d 6914 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{})))
26 2fveq3 6911 . . . . . . . . . . . 12 (𝑥 = 𝑇 → (1st ‘(1st𝑥)) = (1st ‘(1st𝑇)))
2726, 22oveq12d 7449 . . . . . . . . . . 11 (𝑥 = 𝑇 → ((1st ‘(1st𝑥)) (2nd𝑥)) = ((1st ‘(1st𝑇)) (2nd𝑇)))
2827sneqd 4638 . . . . . . . . . 10 (𝑥 = 𝑇 → {((1st ‘(1st𝑥)) (2nd𝑥))} = {((1st ‘(1st𝑇)) (2nd𝑇))})
2928fveq2d 6910 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))}))
3029fveq2d 6910 . . . . . . . 8 (𝑥 = 𝑇 → (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})))
31 2fveq3 6911 . . . . . . . . . . 11 (𝑥 = 𝑇 → (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑇)))
3231oveq1d 7446 . . . . . . . . . 10 (𝑥 = 𝑇 → ((2nd ‘(1st𝑥))𝑅) = ((2nd ‘(1st𝑇))𝑅))
3332sneqd 4638 . . . . . . . . 9 (𝑥 = 𝑇 → {((2nd ‘(1st𝑥))𝑅)} = {((2nd ‘(1st𝑇))𝑅)})
3433fveq2d 6910 . . . . . . . 8 (𝑥 = 𝑇 → (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))
3530, 34eqeq12d 2753 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))
3625, 35anbi12d 632 . . . . . 6 (𝑥 = 𝑇 → (((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})) ↔ ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))))
3736riotabidv 7390 . . . . 5 (𝑥 = 𝑇 → (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))) = (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))))
3821, 37ifbieq2d 4552 . . . 4 (𝑥 = 𝑇 → if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))
39 eqid 2737 . . . 4 (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}))))) = (𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))
4038, 39fvmptg 7014 . . 3 ((𝑇 ∈ ((𝑉 × 𝐷) × 𝑉) ∧ if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))) ∈ V) → ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))‘𝑇) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))
4117, 20, 40sylancl 586 . 2 (𝜑 → ((𝑥 ∈ ((𝑉 × 𝐷) × 𝑉) ↦ if((2nd𝑥) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)})))))‘𝑇) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))
4216, 41eqtrd 2777 1 (𝜑 → (𝐼𝑇) = if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  ifcif 4525  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  crio 7387  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  0gc0g 17484  -gcsg 18953  LSpanclspn 20969  LHypclh 39986  DVecHcdvh 41080  LCDualclcd 41588  mapdcmpd 41626  HDMap1chdma1 41793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-hdmap1 41795
This theorem is referenced by:  hdmap1val  41800
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