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Theorem dia0 40426
Description: The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
dia0.b 𝐡 = (Baseβ€˜πΎ)
dia0.z 0 = (0.β€˜πΎ)
dia0.h 𝐻 = (LHypβ€˜πΎ)
dia0.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dia0 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {( I β†Ύ 𝐡)})
Proof of Theorem dia0
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 hlatl 38733 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
3 dia0.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
4 dia0.z . . . . . 6 0 = (0.β€˜πΎ)
53, 4atl0cl 38676 . . . . 5 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
62, 5syl 17 . . . 4 (𝐾 ∈ HL β†’ 0 ∈ 𝐡)
76adantr 480 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐡)
8 dia0.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
93, 8lhpbase 39372 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
10 eqid 2724 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
113, 10, 4atl0le 38677 . . . 4 ((𝐾 ∈ AtLat ∧ π‘Š ∈ 𝐡) β†’ 0 (leβ€˜πΎ)π‘Š)
122, 9, 11syl2an 595 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 (leβ€˜πΎ)π‘Š)
13 eqid 2724 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
14 eqid 2724 . . . 4 ((trLβ€˜πΎ)β€˜π‘Š) = ((trLβ€˜πΎ)β€˜π‘Š)
15 dia0.i . . . 4 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
163, 10, 8, 13, 14, 15diaval 40406 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( 0 ∈ 𝐡 ∧ 0 (leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜ 0 ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 })
171, 7, 12, 16syl12anc 834 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 })
182ad2antrr 723 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ 𝐾 ∈ AtLat)
193, 8, 13, 14trlcl 39538 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ∈ 𝐡)
203, 10, 4atlle0 38678 . . . . 5 ((𝐾 ∈ AtLat ∧ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) ∈ 𝐡) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
2118, 19, 20syl2anc 583 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
223, 4, 8, 13, 14trlid0b 39552 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ (𝑓 = ( I β†Ύ 𝐡) ↔ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“) = 0 ))
2321, 22bitr4d 282 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ ((((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 ↔ 𝑓 = ( I β†Ύ 𝐡)))
2423rabbidva 3431 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ (((trLβ€˜πΎ)β€˜π‘Š)β€˜π‘“)(leβ€˜πΎ) 0 } = {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)})
253, 8, 13idltrn 39524 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
26 rabsn 4718 . . 3 (( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)} = {( I β†Ύ 𝐡)})
2725, 26syl 17 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ {𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š) ∣ 𝑓 = ( I β†Ύ 𝐡)} = {( I β†Ύ 𝐡)})
2817, 24, 273eqtrd 2768 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (πΌβ€˜ 0 ) = {( I β†Ύ 𝐡)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3424  {csn 4621   class class class wbr 5139   I cid 5564   β†Ύ cres 5669  β€˜cfv 6534  Basecbs 17149  lecple 17209  0.cp0 18384  AtLatcal 38637  HLchlt 38723  LHypclh 39358  LTrncltrn 39475  trLctrl 39532  DIsoAcdia 40402
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-proset 18256  df-poset 18274  df-plt 18291  df-lub 18307  df-glb 18308  df-join 18309  df-meet 18310  df-p0 18386  df-p1 18387  df-lat 18393  df-clat 18460  df-oposet 38549  df-ol 38551  df-oml 38552  df-covers 38639  df-ats 38640  df-atl 38671  df-cvlat 38695  df-hlat 38724  df-lhyp 39362  df-laut 39363  df-ldil 39478  df-ltrn 39479  df-trl 39533  df-disoa 40403
This theorem is referenced by:  dib0  40538
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