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Theorem hdmap1vallem 37608
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 37607 . . 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 6335 . 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))
18 fvex 6343 . . . . 5 (0g𝐶) ∈ V
1910, 18eqeltri 2846 . . . 4 𝑄 ∈ V
20 riotaex 6759 . . . 4 (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))) ∈ V
2119, 20ifex 4296 . . 3 if((2nd𝑇) = 0 , 𝑄, (𝐷 ((𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))) ∈ V
22 fveq2 6333 . . . . . 6 (𝑥 = 𝑇 → (2nd𝑥) = (2nd𝑇))
2322eqeq1d 2773 . . . . 5 (𝑥 = 𝑇 → ((2nd𝑥) = 0 ↔ (2nd𝑇) = 0 ))
2422sneqd 4329 . . . . . . . . . 10 (𝑥 = 𝑇 → {(2nd𝑥)} = {(2nd𝑇)})
2524fveq2d 6337 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑁‘{(2nd𝑥)}) = (𝑁‘{(2nd𝑇)}))
2625fveq2d 6337 . . . . . . . 8 (𝑥 = 𝑇 → (𝑀‘(𝑁‘{(2nd𝑥)})) = (𝑀‘(𝑁‘{(2nd𝑇)})))
2726eqeq1d 2773 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{(2nd𝑥)})) = (𝐽‘{}) ↔ (𝑀‘(𝑁‘{(2nd𝑇)})) = (𝐽‘{})))
28 fveq2 6333 . . . . . . . . . . . . 13 (𝑥 = 𝑇 → (1st𝑥) = (1st𝑇))
2928fveq2d 6337 . . . . . . . . . . . 12 (𝑥 = 𝑇 → (1st ‘(1st𝑥)) = (1st ‘(1st𝑇)))
3029, 22oveq12d 6812 . . . . . . . . . . 11 (𝑥 = 𝑇 → ((1st ‘(1st𝑥)) (2nd𝑥)) = ((1st ‘(1st𝑇)) (2nd𝑇)))
3130sneqd 4329 . . . . . . . . . 10 (𝑥 = 𝑇 → {((1st ‘(1st𝑥)) (2nd𝑥))} = {((1st ‘(1st𝑇)) (2nd𝑇))})
3231fveq2d 6337 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))}) = (𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))}))
3332fveq2d 6337 . . . . . . . 8 (𝑥 = 𝑇 → (𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})))
3428fveq2d 6337 . . . . . . . . . . 11 (𝑥 = 𝑇 → (2nd ‘(1st𝑥)) = (2nd ‘(1st𝑇)))
3534oveq1d 6809 . . . . . . . . . 10 (𝑥 = 𝑇 → ((2nd ‘(1st𝑥))𝑅) = ((2nd ‘(1st𝑇))𝑅))
3635sneqd 4329 . . . . . . . . 9 (𝑥 = 𝑇 → {((2nd ‘(1st𝑥))𝑅)} = {((2nd ‘(1st𝑇))𝑅)})
3736fveq2d 6337 . . . . . . . 8 (𝑥 = 𝑇 → (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)}))
3833, 37eqeq12d 2786 . . . . . . 7 (𝑥 = 𝑇 → ((𝑀‘(𝑁‘{((1st ‘(1st𝑥)) (2nd𝑥))})) = (𝐽‘{((2nd ‘(1st𝑥))𝑅)}) ↔ (𝑀‘(𝑁‘{((1st ‘(1st𝑇)) (2nd𝑇))})) = (𝐽‘{((2nd ‘(1st𝑇))𝑅)})))
3927, 38anbi12d 610 . . . . . 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)}))))
4039riotabidv 6757 . . . . 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)}))))
4123, 40ifbieq2d 4251 . . . 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)})))))
42 eqid 2771 . . . 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)})))))
4341, 42fvmptg 6423 . . 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)})))))
4417, 21, 43sylancl 568 . 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)})))))
4516, 44eqtrd 2805 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 382   = wceq 1631  wcel 2145  Vcvv 3351  ifcif 4226  {csn 4317  cmpt 4864   × cxp 5248  cfv 6032  crio 6754  (class class class)co 6794  1st c1st 7314  2nd c2nd 7315  Basecbs 16065  0gc0g 16309  -gcsg 17633  LSpanclspn 19185  LHypclh 35793  DVecHcdvh 36889  LCDualclcd 37397  mapdcmpd 37435  HDMap1chdma1 37602
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 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  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 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-hdmap1 37604
This theorem is referenced by:  hdmap1val  37609
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