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Theorem dia1N 39519
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 2737 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
3 dia1.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3lhpbase 38464 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ (Baseβ€˜πΎ))
54adantl 483 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘Š ∈ (Baseβ€˜πΎ))
6 hllat 37828 . . . 4 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
7 eqid 2737 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
82, 7latref 18331 . . . 4 ((𝐾 ∈ Lat ∧ π‘Š ∈ (Baseβ€˜πΎ)) β†’ π‘Š(leβ€˜πΎ)π‘Š)
96, 4, 8syl2an 597 . . 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 2737 . . . 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 39498 . . 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 38650 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ)π‘Š)
1615ralrimiva 3144 . . 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 3437 . . 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 2780 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜π‘Š) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   class class class wbr 5106  β€˜cfv 6497  Basecbs 17084  lecple 17141  Latclat 18321  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  trLctrl 38624  DIsoAcdia 39494
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-pow 5321  ax-pr 5385  ax-un 7673
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-pw 4563  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-oprab 7362  df-mpo 7363  df-map 8768  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625  df-disoa 39495
This theorem is referenced by:  dia1elN  39520
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