Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hgmapval Structured version   Visualization version   GIF version

Theorem hgmapval 41271
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 41266. (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 41270 . . 3 (πœ‘ β†’ 𝐼 = (π‘₯ ∈ 𝐡 ↦ (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(π‘₯ Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£)))))
1312fveq1d 6887 . 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 7365 . . 3 (℩𝑦 ∈ 𝐡 βˆ€π‘£ ∈ 𝑉 (π‘€β€˜(𝑋 Β· 𝑣)) = (𝑦 βˆ™ (π‘€β€˜π‘£))) ∈ V
16 fvoveq1 7428 . . . . . . 7 (π‘₯ = 𝑋 β†’ (π‘€β€˜(π‘₯ Β· 𝑣)) = (π‘€β€˜(𝑋 Β· 𝑣)))
1716eqeq1d 2728 . . . . . 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 3171 . . . . 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 7363 . . . 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 2726 . . . 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 6990 . . 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 585 . 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 2766 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   ↦ cmpt 5224  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  Basecbs 17153  Scalarcsca 17209   ·𝑠 cvsca 17210  LHypclh 39368  DVecHcdvh 40462  LCDualclcd 40970  HDMapchdma 41176  HGMapchg 41267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-hgmap 41268
This theorem is referenced by:  hgmapcl  41273  hgmapvs  41275
  Copyright terms: Public domain W3C validator