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Theorem hgmapval 40353
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 40348. (Contributed by NM, 25-Mar-2015.)
Hypotheses
Ref Expression
hgmapval.h 𝐻 = (LHypβ€˜πΎ)
hgmapfval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hgmapfval.v 𝑉 = (Baseβ€˜π‘ˆ)
hgmapfval.t Β· = ( ·𝑠 β€˜π‘ˆ)
hgmapfval.r 𝑅 = (Scalarβ€˜π‘ˆ)
hgmapfval.b 𝐡 = (Baseβ€˜π‘…)
hgmapfval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hgmapfval.s βˆ™ = ( ·𝑠 β€˜πΆ)
hgmapfval.m 𝑀 = ((HDMapβ€˜πΎ)β€˜π‘Š)
hgmapfval.i 𝐼 = ((HGMapβ€˜πΎ)β€˜π‘Š)
hgmapfval.k (πœ‘ β†’ (𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻))
hgmapval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
Assertion
Ref Expression
hgmapval (πœ‘ β†’ (πΌβ€˜π‘‹) = (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
Distinct variable groups:   𝑦,𝑣,𝐾   𝑣,𝐡,𝑦   𝑣,𝑀,𝑦   𝑣,π‘ˆ,𝑦   𝑣,𝑉   𝑣,π‘Š,𝑦   𝑣,𝑋,𝑦
Allowed substitution hints:   πœ‘(𝑦,𝑣)   𝐢(𝑦,𝑣)   𝑅(𝑦,𝑣)   βˆ™ (𝑦,𝑣)   Β· (𝑦,𝑣)   𝐻(𝑦,𝑣)   𝐼(𝑦,𝑣)   𝑉(𝑦)   π‘Œ(𝑦,𝑣)

Proof of Theorem hgmapval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 hgmapval.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 hgmapfval.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
3 hgmapfval.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
4 hgmapfval.t . . . 4 Β· = ( ·𝑠 β€˜π‘ˆ)
5 hgmapfval.r . . . 4 𝑅 = (Scalarβ€˜π‘ˆ)
6 hgmapfval.b . . . 4 𝐡 = (Baseβ€˜π‘…)
7 hgmapfval.c . . . 4 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
8 hgmapfval.s . . . 4 βˆ™ = ( ·𝑠 β€˜πΆ)
9 hgmapfval.m . . . 4 𝑀 = ((HDMapβ€˜πΎ)β€˜π‘Š)
10 hgmapfval.i . . . 4 𝐼 = ((HGMapβ€˜πΎ)β€˜π‘Š)
11 hgmapfval.k . . . 4 (πœ‘ β†’ (𝐾 ∈ π‘Œ ∧ π‘Š ∈ 𝐻))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hgmapfval 40352 . . 3 (πœ‘ β†’ 𝐼 = (π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£)))))
1312fveq1d 6845 . 2 (πœ‘ β†’ (πΌβ€˜π‘‹) = ((π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))β€˜π‘‹))
14 hgmapval.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
15 riotaex 7318 . . 3 (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))) ∈ V
16 fvoveq1 7381 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘€β€˜(π‘₯ Β· 𝑣)) = (π‘€β€˜(𝑋 Β· 𝑣)))
1716eqeq1d 2739 . . . . . 6 (π‘₯ = 𝑋 β†’ ((π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£)) ↔ (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
1817ralbidv 3175 . . . . 5 (π‘₯ = 𝑋 β†’ (βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£)) ↔ βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
1918riotabidv 7316 . . . 4 (π‘₯ = 𝑋 β†’ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))) = (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
20 eqid 2737 . . . 4 (π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£)))) = (π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
2119, 20fvmptg 6947 . . 3 ((𝑋 ∈ 𝐡 ∧ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))) ∈ V) β†’ ((π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))β€˜π‘‹) = (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
2214, 15, 21sylancl 587 . 2 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))β€˜π‘‹) = (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
2313, 22eqtrd 2777 1 (πœ‘ β†’ (πΌβ€˜π‘‹) = (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3446   ↦ cmpt 5189  β€˜cfv 6497  β„©crio 7313  (class class class)co 7358  Basecbs 17084  Scalarcsca 17137   ·𝑠 cvsca 17138  LHypclh 38450  DVecHcdvh 39544  LCDualclcd 40052  HDMapchdma 40258  HGMapchg 40349
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-hgmap 40350
This theorem is referenced by:  hgmapcl  40355  hgmapvs  40357
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