Theorem hgmapval 41870

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 41865. (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 41869	. . 3 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))
13	12	fveq1d 6909	. 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 7392	. . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))) ∈ V
16		fvoveq1 7454	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑀‘(𝑥 · 𝑣)) = (𝑀‘(𝑋 · 𝑣)))
17	16	eqeq1d 2737	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
18	17	ralbidv 3176	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)) ↔ ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
19	18	riotabidv 7390	. . . 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 2735	. . . 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 7014	. . 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 2775	1 ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ↦ cmpt 5231  ‘cfv 6563  ℩crio 7387  (class class class)co 7431  Basecbs 17245  Scalarcsca 17301   ·_𝑠 cvsca 17302  LHypclh 39967  DVecHcdvh 41061  LCDualclcd 41569  HDMapchdma 41775  HGMapchg 41866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-hgmap 41867

This theorem is referenced by:  hgmapcl  41872  hgmapvs  41874