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Theorem dia1N 40526
Description: The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dia1.h 𝐻 = (LHypβ€˜πΎ)
dia1.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dia1.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dia1N ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = 𝑇)

Proof of Theorem dia1N
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 eqid 2728 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 dia1.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3lhpbase 39471 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
54adantl 481 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š ∈ (Baseβ€˜πΎ))
6 hllat 38835 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
7 eqid 2728 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
82, 7latref 18432 . . . 4 ((𝐾 ∈ Lat ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ π‘Š(leβ€˜πΎ)π‘Š)
96, 4, 8syl2an 595 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š(leβ€˜πΎ)π‘Š)
10 dia1.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
11 eqid 2728 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
12 dia1.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
132, 7, 3, 10, 11, 12diaval 40505 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Š ∈ (Baseβ€˜πΎ) ∧ π‘Š(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜π‘Š) = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
141, 5, 9, 13syl12anc 836 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
157, 3, 10, 11trlle 39657 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
1615ralrimiva 3143 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
17 rabid2 3461 . . 3 (𝑇 = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š} ↔ βˆ€π‘“ ∈ 𝑇 (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
1816, 17sylibr 233 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑇 = {𝑓 ∈ 𝑇 ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š})
1914, 18eqtr4d 2771 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  {crab 3429   class class class wbr 5148  β€˜cfv 6548  Basecbs 17179  lecple 17239  Latclat 18422  HLchlt 38822  LHypclh 39457  LTrncltrn 39574  trLctrl 39631  DIsoAcdia 40501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823  df-lhyp 39461  df-laut 39462  df-ldil 39577  df-ltrn 39578  df-trl 39632  df-disoa 40502
This theorem is referenced by:  dia1elN  40527
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