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Theorem hdmap1vallem 37578
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 37577 . . 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 6410 . 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 6422 . . . 4 𝑄 ∈ V
19 riotaex 6839 . . . 4 (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))) ∈ V
2018, 19ifex 4327 . . 3 if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))) ∈ V
21 fveqeq2 6417 . . . . 5 (𝑥 = 𝑇 → ((2nd𝑥) = 0 ↔ (2nd𝑇) = 0 ))
22 fveq2 6408 . . . . . . . . . 10 (𝑥 = 𝑇 → (2nd𝑥) = (2nd𝑇))
2322sneqd 4382 . . . . . . . . 9 (𝑥 = 𝑇 → {(2nd𝑥)} = {(2nd𝑇)})
2423fveq2d 6412 . . . . . . . 8 (𝑥 = 𝑇 → (𝑁‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑇)}))
2524fveqeq2d 6416 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{})))
26 2fveq3 6413 . . . . . . . . . . . 12 (𝑥 = 𝑇 → (1st ‘(1st𝑥)) = (1st ‘(1st𝑇)))
2726, 22oveq12d 6892 . . . . . . . . . . 11 (𝑥 = 𝑇 → ((1st ‘(1st𝑥)) (2nd𝑥)) = ((1st ‘(1st𝑇)) (2nd𝑇)))
2827sneqd 4382 . . . . . . . . . 10 (𝑥 = 𝑇 → {((1st ‘(1st𝑥)) (2nd𝑥))} = {((1st ‘(1st𝑇)) (2nd𝑇))})
2928fveq2d 6412 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))}))
3029fveq2d 6412 . . . . . . . 8 (𝑥 = 𝑇 → (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})))
31 2fveq3 6413 . . . . . . . . . . 11 (𝑥 = 𝑇 → (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑇)))
3231oveq1d 6889 . . . . . . . . . 10 (𝑥 = 𝑇 → ((2nd ‘(1st𝑥))𝑅) = ((2nd ‘(1st𝑇))𝑅))
3332sneqd 4382 . . . . . . . . 9 (𝑥 = 𝑇 → {((2nd ‘(1st𝑥))𝑅)} = {((2nd ‘(1st𝑇))𝑅)})
3433fveq2d 6412 . . . . . . . 8 (𝑥 = 𝑇 → (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))
3530, 34eqeq12d 2821 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))
3625, 35anbi12d 618 . . . . . 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 6837 . . . . 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 4304 . . . 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 2806 . . . 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 6501 . . 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 576 . 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 2840 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 384   = wceq 1637  wcel 2156  Vcvv 3391  ifcif 4279  {csn 4370  cmpt 4923   × cxp 5309  cfv 6101  crio 6834  (class class class)co 6874  1st c1st 7396  2nd c2nd 7397  Basecbs 16068  0gc0g 16305  -gcsg 17629  LSpanclspn 19178  LHypclh 35764  DVecHcdvh 36859  LCDualclcd 37367  mapdcmpd 37405  HDMap1chdma1 37572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-hdmap1 37574
This theorem is referenced by:  hdmap1val  37579
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