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Theorem hdmapval 40785
Description: Value of map from vectors to functionals in the closed kernel dual space. This is the function sigma on line 27 above part 9 in [Baer] p. 48. We select a convenient fixed reference vector 𝐸 to be ⟨0, 1⟩ (corresponding to vector u on p. 48 line 7) whose span is the lattice isomorphism map of the fiducial atom 𝑃 = ((ocβ€˜πΎ)β€˜π‘Š) (see dvheveccl 40069). (π½β€˜πΈ) is a fixed reference functional determined by this vector (corresponding to u' on line 8; mapdhvmap 40726 shows in Baer's notation (Fu)* = Gu'). Baer's independent vectors v and w on line 7 correspond to our 𝑧 that the βˆ€π‘§ ∈ 𝑉 ranges over. The middle term (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©) provides isolation to allow 𝐸 and 𝑇 to assume the same value without conflict. Closure is shown by hdmapcl 40787. If a separate auxiliary vector is known, hdmapval2 40789 provides a version without quantification. (Contributed by NM, 15-May-2015.)
Hypotheses
Ref Expression
hdmapval.h 𝐻 = (LHypβ€˜πΎ)
hdmapfval.e 𝐸 = ⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩
hdmapfval.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
hdmapfval.v 𝑉 = (Baseβ€˜π‘ˆ)
hdmapfval.n 𝑁 = (LSpanβ€˜π‘ˆ)
hdmapfval.c 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
hdmapfval.d 𝐷 = (Baseβ€˜πΆ)
hdmapfval.j 𝐽 = ((HVMapβ€˜πΎ)β€˜π‘Š)
hdmapfval.i 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
hdmapfval.s 𝑆 = ((HDMapβ€˜πΎ)β€˜π‘Š)
hdmapfval.k (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
hdmapval.t (πœ‘ β†’ 𝑇 ∈ 𝑉)
Assertion
Ref Expression
hdmapval (πœ‘ β†’ (π‘†β€˜π‘‡) = (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
Distinct variable groups:   𝑦,𝑧,𝐾   𝑦,𝐷   𝑦,𝐸,𝑧   𝑦,𝐼,𝑧   𝑦,π‘ˆ,𝑧   𝑦,𝑉,𝑧   𝑦,π‘Š,𝑧   𝑦,𝑇,𝑧
Allowed substitution hints:   πœ‘(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐢(𝑦,𝑧)   𝐷(𝑧)   𝑆(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝐽(𝑦,𝑧)   𝑁(𝑦,𝑧)

Proof of Theorem hdmapval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 hdmapval.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 hdmapfval.e . . . 4 𝐸 = ⟨( I β†Ύ (Baseβ€˜πΎ)), ( I β†Ύ ((LTrnβ€˜πΎ)β€˜π‘Š))⟩
3 hdmapfval.u . . . 4 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
4 hdmapfval.v . . . 4 𝑉 = (Baseβ€˜π‘ˆ)
5 hdmapfval.n . . . 4 𝑁 = (LSpanβ€˜π‘ˆ)
6 hdmapfval.c . . . 4 𝐢 = ((LCDualβ€˜πΎ)β€˜π‘Š)
7 hdmapfval.d . . . 4 𝐷 = (Baseβ€˜πΆ)
8 hdmapfval.j . . . 4 𝐽 = ((HVMapβ€˜πΎ)β€˜π‘Š)
9 hdmapfval.i . . . 4 𝐼 = ((HDMap1β€˜πΎ)β€˜π‘Š)
10 hdmapfval.s . . . 4 𝑆 = ((HDMapβ€˜πΎ)β€˜π‘Š)
11 hdmapfval.k . . . 4 (πœ‘ β†’ (𝐾 ∈ 𝐴 ∧ π‘Š ∈ 𝐻))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hdmapfval 40784 . . 3 (πœ‘ β†’ 𝑆 = (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©)))))
1312fveq1d 6893 . 2 (πœ‘ β†’ (π‘†β€˜π‘‡) = ((𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©))))β€˜π‘‡))
14 hdmapval.t . . 3 (πœ‘ β†’ 𝑇 ∈ 𝑉)
15 riotaex 7371 . . 3 (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))) ∈ V
16 sneq 4638 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ {𝑑} = {𝑇})
1716fveq2d 6895 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (π‘β€˜{𝑑}) = (π‘β€˜{𝑇}))
1817uneq2d 4163 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) = ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})))
1918eleq2d 2819 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) ↔ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇}))))
2019notbid 317 . . . . . . 7 (𝑑 = 𝑇 β†’ (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) ↔ Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇}))))
21 oteq3 4884 . . . . . . . . 9 (𝑑 = 𝑇 β†’ βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ© = βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©)
2221fveq2d 6895 . . . . . . . 8 (𝑑 = 𝑇 β†’ (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©) = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))
2322eqeq2d 2743 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©) ↔ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©)))
2420, 23imbi12d 344 . . . . . 6 (𝑑 = 𝑇 β†’ ((Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©)) ↔ (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
2524ralbidv 3177 . . . . 5 (𝑑 = 𝑇 β†’ (βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©)) ↔ βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
2625riotabidv 7369 . . . 4 (𝑑 = 𝑇 β†’ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©))) = (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
27 eqid 2732 . . . 4 (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©)))) = (𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©))))
2826, 27fvmptg 6996 . . 3 ((𝑇 ∈ 𝑉 ∧ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))) ∈ V) β†’ ((𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©))))β€˜π‘‡) = (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
2914, 15, 28sylancl 586 . 2 (πœ‘ β†’ ((𝑑 ∈ 𝑉 ↦ (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑑})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‘βŸ©))))β€˜π‘‡) = (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
3013, 29eqtrd 2772 1 (πœ‘ β†’ (π‘†β€˜π‘‡) = (℩𝑦 ∈ 𝐷 βˆ€π‘§ ∈ 𝑉 (Β¬ 𝑧 ∈ ((π‘β€˜{𝐸}) βˆͺ (π‘β€˜{𝑇})) β†’ 𝑦 = (πΌβ€˜βŸ¨π‘§, (πΌβ€˜βŸ¨πΈ, (π½β€˜πΈ), π‘§βŸ©), π‘‡βŸ©))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆͺ cun 3946  {csn 4628  βŸ¨cop 4634  βŸ¨cotp 4636   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678  β€˜cfv 6543  β„©crio 7366  Basecbs 17146  LSpanclspn 20587  LHypclh 38941  LTrncltrn 39058  DVecHcdvh 40035  LCDualclcd 40543  HVMapchvm 40713  HDMap1chdma1 40748  HDMapchdma 40749
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-hdmap 40751
This theorem is referenced by:  hdmapcl  40787  hdmapval2lem  40788
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