Theorem hvmapffval 39759

Description: Map from nonzero vectors to nonzero functionals in the closed kernel dual space. (Contributed by NM, 23-Mar-2015.)

Hypothesis

Ref	Expression
hvmapval.h	⊢ 𝐻 = (LHyp‘𝐾)

Assertion

Ref	Expression
hvmapffval	⊢ (𝐾 ∈ 𝑋 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))

Distinct variable groups:   𝑤,𝐻   𝑡,𝑗,𝑣,𝑥,𝑤,𝐾

Allowed substitution hints:   𝐻(𝑥,𝑣,𝑡,𝑗)   𝑋(𝑥,𝑤,𝑣,𝑡,𝑗)

Proof of Theorem hvmapffval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3449	. 2 ⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ V)
2		fveq2 6768	. . . . 5 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾))
3		hvmapval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	eqtr4di 2796	. . . 4 ⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻)
5		fveq2 6768	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾))
6	5	fveq1d 6770	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤))
7	6	fveq2d 6772	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤)))
8	6	fveq2d 6772	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (0g‘((DVecH‘𝑘)‘𝑤)) = (0g‘((DVecH‘𝐾)‘𝑤)))
9	8	sneqd 4575	. . . . . 6 ⊢ (𝑘 = 𝐾 → {(0g‘((DVecH‘𝑘)‘𝑤))} = {(0g‘((DVecH‘𝐾)‘𝑤))})
10	7, 9	difeq12d 4059	. . . . 5 ⊢ (𝑘 = 𝐾 → ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) = ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}))
11	6	fveq2d 6772	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (Scalar‘((DVecH‘𝑘)‘𝑤)) = (Scalar‘((DVecH‘𝐾)‘𝑤)))
12	11	fveq2d 6772	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤))) = (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤))))
13		fveq2 6768	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (ocH‘𝑘) = (ocH‘𝐾))
14	13	fveq1d 6770	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ((ocH‘𝑘)‘𝑤) = ((ocH‘𝐾)‘𝑤))
15	14	fveq1d 6770	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘{𝑥}) = (((ocH‘𝐾)‘𝑤)‘{𝑥}))
16	6	fveq2d 6772	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (+g‘((DVecH‘𝑘)‘𝑤)) = (+g‘((DVecH‘𝐾)‘𝑤)))
17		eqidd 2739	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → 𝑡 = 𝑡)
18	6	fveq2d 6772	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → ( ·𝑠 ‘((DVecH‘𝑘)‘𝑤)) = ( ·𝑠 ‘((DVecH‘𝐾)‘𝑤)))
19	18	oveqd 7286	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥) = (𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))
20	16, 17, 19	oveq123d 7290	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)) = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))
21	20	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))
22	15, 21	rexeqbidv 3336	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)) ↔ ∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))
23	12, 22	riotaeqbidv 7229	. . . . . 6 ⊢ (𝑘 = 𝐾 → (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥))) = (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))
24	7, 23	mpteq12dv 5166	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)))) = (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))
25	10, 24	mpteq12dv 5166	. . . 4 ⊢ (𝑘 = 𝐾 → (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥))))) = (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥))))))
26	4, 25	mpteq12dv 5166	. . 3 ⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))
27		df-hvmap 39758	. . 3 ⊢ HVMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖ {(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝑘)‘𝑤))𝑥)))))))
28	26, 27, 3	mptfvmpt 7098	. 2 ⊢ (𝐾 ∈ V → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))
29	1, 28	syl 17	1 ⊢ (𝐾 ∈ 𝑋 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖ {(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈ (Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠 ‘((DVecH‘𝐾)‘𝑤))𝑥)))))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3431   ∖ cdif 3885  {csn 4563   ↦ cmpt 5158  ‘cfv 6428  ℩crio 7225  (class class class)co 7269  Basecbs 16901  +_gcplusg 16951  Scalarcsca 16954   ·_𝑠 cvsca 16955  0_gc0g 17139  LHypclh 37985  DVecHcdvh 39079  ocHcoch 39348  HVMapchvm 39757

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-hvmap 39758

This theorem is referenced by:  hvmapfval  39760