Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dia0 Structured version   Visualization version   GIF version
Theorem dia0 39911
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 38218 . . . . 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 38161 . . . . 5 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
62, 5syl 17 . . . 4 (𝐾 ∈ HL β†’ 0 ∈ 𝐡)
76adantr 481 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐡)
8 dia0.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
93, 8lhpbase 38857 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
10 eqid 2732 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
113, 10, 4atl0le 38162 . . . 4 ((𝐾 ∈ AtLat ∧ π‘Š ∈ 𝐡) β†’ 0 (leβ€˜πΎ)π‘Š)
122, 9, 11syl2an 596 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 (leβ€˜πΎ)π‘Š)
13 eqid 2732 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
14 eqid 2732 . . . 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 39891 . . 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 835 . 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 724 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ 𝐾 ∈ AtLat)
193, 8, 13, 14trlcl 39023 . . . . 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 38163 . . . . 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 584 . . . 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 39037 . . . 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 281 . . 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 3439 . 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 39009 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
26 rabsn 4724 . . 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 2776 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 396   = wceq 1541   ∈ wcel 2106  {crab 3432  {csn 4627   class class class wbr 5147   I cid 5572   β†Ύ cres 5677  β€˜cfv 6540  Basecbs 17140  lecple 17200  0.cp0 18372  AtLatcal 38122  HLchlt 38208  LHypclh 38843  LTrncltrn 38960  trLctrl 39017  DIsoAcdia 39887
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-p1 18375  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-lhyp 38847  df-laut 38848  df-ldil 38963  df-ltrn 38964  df-trl 39018  df-disoa 39888
This theorem is referenced by:  dib0  40023
  Copyright terms: Public domain W3C validator