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Theorem hdmap1vallem 40310
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 = (LHypβ€˜πΎ)
hdmap1fval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmap1fval.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmap1fval.s βˆ’ = (-gβ€˜π‘ˆ)
hdmap1fval.o 0 = (0gβ€˜π‘ˆ)
hdmap1fval.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmap1fval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmap1fval.d 𝐷 = (Baseβ€˜πΆ)
hdmap1fval.r 𝑅 = (-gβ€˜πΆ)
hdmap1fval.q 𝑄 = (0gβ€˜πΆ)
hdmap1fval.j 𝐽 = (LSpanβ€˜πΆ)
hdmap1fval.m 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
hdmap1fval.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmap1fval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
hdmap1val.t (πœ‘ β†’ 𝑇 ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
Assertion
Ref Expression
hdmap1vallem (πœ‘ β†’ (πΌβ€˜π‘‡) = if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))))
Distinct variable groups:   𝐢,β„Ž   𝐷,β„Ž   β„Ž,𝐽   β„Ž,𝑀   β„Ž,𝑁   π‘ˆ,β„Ž   β„Ž,𝑉   𝑇,β„Ž
Allowed substitution hints:   πœ‘(β„Ž)   𝐴(β„Ž)   𝑄(β„Ž)   𝑅(β„Ž)   𝐻(β„Ž)   𝐼(β„Ž)   𝐾(β„Ž)   βˆ’ (β„Ž)   π‘Š(β„Ž)   0 (β„Ž)

Proof of Theorem hdmap1vallem
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 hdmap1val.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 hdmap1fval.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hdmap1fval.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
4 hdmap1fval.s . . . 4 βˆ’ = (-gβ€˜π‘ˆ)
5 hdmap1fval.o . . . 4 0 = (0gβ€˜π‘ˆ)
6 hdmap1fval.n . . . 4 𝑁 = (LSpanβ€˜π‘ˆ)
7 hdmap1fval.c . . . 4 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
8 hdmap1fval.d . . . 4 𝐷 = (Baseβ€˜πΆ)
9 hdmap1fval.r . . . 4 𝑅 = (-gβ€˜πΆ)
10 hdmap1fval.q . . . 4 𝑄 = (0gβ€˜πΆ)
11 hdmap1fval.j . . . 4 𝐽 = (LSpanβ€˜πΆ)
12 hdmap1fval.m . . . 4 𝑀 = ((mapdβ€˜πΎ)β€˜π‘Š)
13 hdmap1fval.i . . . 4 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
14 hdmap1fval.k . . . 4 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14hdmap1fval 40309 . . 3 (πœ‘ β†’ 𝐼 = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))))
1615fveq1d 6848 . 2 (πœ‘ β†’ (πΌβ€˜π‘‡) = ((π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))β€˜π‘‡))
17 hdmap1val.t . . 3 (πœ‘ β†’ 𝑇 ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉))
1810fvexi 6860 . . . 4 𝑄 ∈ V
19 riotaex 7321 . . . 4 (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)}))) ∈ V
2018, 19ifex 4540 . . 3 if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))) ∈ V
21 fveqeq2 6855 . . . . 5 (π‘₯ = 𝑇 β†’ ((2nd β€˜π‘₯) = 0 ↔ (2nd β€˜π‘‡) = 0 ))
22 fveq2 6846 . . . . . . . . . 10 (π‘₯ = 𝑇 β†’ (2nd β€˜π‘₯) = (2nd β€˜π‘‡))
2322sneqd 4602 . . . . . . . . 9 (π‘₯ = 𝑇 β†’ {(2nd β€˜π‘₯)} = {(2nd β€˜π‘‡)})
2423fveq2d 6850 . . . . . . . 8 (π‘₯ = 𝑇 β†’ (π‘β€˜{(2nd β€˜π‘₯)}) = (π‘β€˜{(2nd β€˜π‘‡)}))
2524fveqeq2d 6854 . . . . . . 7 (π‘₯ = 𝑇 β†’ ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ↔ (π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž})))
26 2fveq3 6851 . . . . . . . . . . . 12 (π‘₯ = 𝑇 β†’ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘‡)))
2726, 22oveq12d 7379 . . . . . . . . . . 11 (π‘₯ = 𝑇 β†’ ((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯)) = ((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡)))
2827sneqd 4602 . . . . . . . . . 10 (π‘₯ = 𝑇 β†’ {((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))} = {((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})
2928fveq2d 6850 . . . . . . . . 9 (π‘₯ = 𝑇 β†’ (π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))}) = (π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))}))
3029fveq2d 6850 . . . . . . . 8 (π‘₯ = 𝑇 β†’ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})))
31 2fveq3 6851 . . . . . . . . . . 11 (π‘₯ = 𝑇 β†’ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘‡)))
3231oveq1d 7376 . . . . . . . . . 10 (π‘₯ = 𝑇 β†’ ((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž) = ((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž))
3332sneqd 4602 . . . . . . . . 9 (π‘₯ = 𝑇 β†’ {((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)} = {((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})
3433fveq2d 6850 . . . . . . . 8 (π‘₯ = 𝑇 β†’ (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)}))
3530, 34eqeq12d 2749 . . . . . . 7 (π‘₯ = 𝑇 β†’ ((π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}) ↔ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))
3625, 35anbi12d 632 . . . . . 6 (π‘₯ = 𝑇 β†’ (((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})) ↔ ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)}))))
3736riotabidv 7319 . . . . 5 (π‘₯ = 𝑇 β†’ (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))) = (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)}))))
3821, 37ifbieq2d 4516 . . . 4 (π‘₯ = 𝑇 β†’ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))) = if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))))
39 eqid 2733 . . . 4 (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)}))))) = (π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))
4038, 39fvmptg 6950 . . 3 ((𝑇 ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ∧ if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))) ∈ V) β†’ ((π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))β€˜π‘‡) = if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))))
4117, 20, 40sylancl 587 . 2 (πœ‘ β†’ ((π‘₯ ∈ ((𝑉 Γ— 𝐷) Γ— 𝑉) ↦ if((2nd β€˜π‘₯) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘₯)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘₯)) βˆ’ (2nd β€˜π‘₯))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘₯))π‘…β„Ž)})))))β€˜π‘‡) = if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))))
4216, 41eqtrd 2773 1 (πœ‘ β†’ (πΌβ€˜π‘‡) = if((2nd β€˜π‘‡) = 0 , 𝑄, (β„©β„Ž ∈ 𝐷 ((π‘€β€˜(π‘β€˜{(2nd β€˜π‘‡)})) = (π½β€˜{β„Ž}) ∧ (π‘€β€˜(π‘β€˜{((1st β€˜(1st β€˜π‘‡)) βˆ’ (2nd β€˜π‘‡))})) = (π½β€˜{((2nd β€˜(1st β€˜π‘‡))π‘…β„Ž)})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447  ifcif 4490  {csn 4590   ↦ cmpt 5192   Γ— cxp 5635  β€˜cfv 6500  β„©crio 7316  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  Basecbs 17091  0gc0g 17329  -gcsg 18758  LSpanclspn 20476  LHypclh 38497  DVecHcdvh 39591  LCDualclcd 40099  mapdcmpd 40137  HDMap1chdma1 40304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-hdmap1 40306
This theorem is referenced by:  hdmap1val  40311
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