Theorem hgmapval 37179

Description: Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 37174. (Contributed by NM, 25-Mar-2015.)

Hypotheses

Ref	Expression
hgmapval.h	⊢ 𝐻 = (LHyp‘𝐾)
hgmapfval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmapfval.v	⊢ 𝑉 = (Base‘𝑈)
hgmapfval.t	⊢ · = ( ·𝑠 ‘𝑈)
hgmapfval.r	⊢ 𝑅 = (Scalar‘𝑈)
hgmapfval.b	⊢ 𝐵 = (Base‘𝑅)
hgmapfval.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmapfval.s	⊢ ∙ = ( ·𝑠 ‘𝐶)
hgmapfval.m	⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
hgmapfval.i	⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊)
hgmapfval.k	⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻))
hgmapval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
hgmapval	⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))

Distinct variable groups:   𝑦,𝑣,𝐾   𝑣,𝐵,𝑦   𝑣,𝑀,𝑦   𝑣,𝑈,𝑦   𝑣,𝑉   𝑣,𝑊,𝑦   𝑣,𝑋,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑣)   𝐶(𝑦,𝑣)   𝑅(𝑦,𝑣)   ∙ (𝑦,𝑣)   · (𝑦,𝑣)   𝐻(𝑦,𝑣)   𝐼(𝑦,𝑣)   𝑉(𝑦)   𝑌(𝑦,𝑣)

Proof of Theorem hgmapval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgmapval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hgmapfval.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hgmapfval.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
4		hgmapfval.t	. . . 4 ⊢ · = ( ·𝑠 ‘𝑈)
5		hgmapfval.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
6		hgmapfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
7		hgmapfval.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
8		hgmapfval.s	. . . 4 ⊢ ∙ = ( ·𝑠 ‘𝐶)
9		hgmapfval.m	. . . 4 ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)
10		hgmapfval.i	. . . 4 ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊)
11		hgmapfval.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hgmapfval 37178	. . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))
13	12	fveq1d 6193	. 2 ⊢ (𝜑 → (𝐼‘𝑋) = ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋))
14		hgmapval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		riotaex 6615	. . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V
16		oveq1 6657	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑣) = (𝑋 · 𝑣))
17	16	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
18	17	eqeq1d 2624	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
19	18	ralbidv 2986	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
20	19	riotabidv 6613	. . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
21		eqid 2622	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))) = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
22	20, 21	fvmptg 6280	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V) → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
23	14, 15, 22	sylancl 694	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
24	13, 23	eqtrd 2656	1 ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  LHypclh 35270  DVecHcdvh 36367  LCDualclcd 36875  HDMapchdma 37082  HGMapchg 37175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-hgmap 37176

This theorem is referenced by:  hgmapcl  37181  hgmapvs  37183