Theorem hgmapval 40423

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 40418. (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 40422	. . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))
13	12	fveq1d 6849	. 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 7322	. . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V
16		fvoveq1 7385	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
17	16	eqeq1d 2733	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
18	17	ralbidv 3170	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
19	18	riotabidv 7320	. . . 4 ⊢ (𝑥 = 𝑋 → (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
20		eqid 2731	. . . 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))))
21	19, 20	fvmptg 6951	. . 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))))
22	14, 15, 21	sylancl 586	. 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))))
23	13, 22	eqtrd 2771	1 ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3446   ↦ cmpt 5193  ‘cfv 6501  ℩crio 7317  (class class class)co 7362  Basecbs 17094  Scalarcsca 17150   ·_𝑠 cvsca 17151  LHypclh 38520  DVecHcdvh 39614  LCDualclcd 40122  HDMapchdma 40328  HGMapchg 40419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-hgmap 40420

This theorem is referenced by:  hgmapcl  40425  hgmapvs  40427