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Theorem dia0 39518
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 37825 . . . . 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 37768 . . . . 5 (𝐾 ∈ AtLat β†’ 0 ∈ 𝐡)
62, 5syl 17 . . . 4 (𝐾 ∈ HL β†’ 0 ∈ 𝐡)
76adantr 482 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 ∈ 𝐡)
8 dia0.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
93, 8lhpbase 38464 . . . 4 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
10 eqid 2737 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
113, 10, 4atl0le 37769 . . . 4 ((𝐾 ∈ AtLat ∧ π‘Š ∈ 𝐡) β†’ 0 (leβ€˜πΎ)π‘Š)
122, 9, 11syl2an 597 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 0 (leβ€˜πΎ)π‘Š)
13 eqid 2737 . . . 4 ((LTrnβ€˜πΎ)β€˜π‘Š) = ((LTrnβ€˜πΎ)β€˜π‘Š)
14 eqid 2737 . . . 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 39498 . . 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 836 . 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 725 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)) β†’ 𝐾 ∈ AtLat)
193, 8, 13, 14trlcl 38630 . . . . 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 37770 . . . . 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 585 . . . 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 38644 . . . 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 3415 . 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 38616 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ ((LTrnβ€˜πΎ)β€˜π‘Š))
26 rabsn 4683 . . 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 2781 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 397   = wceq 1542   ∈ wcel 2107  {crab 3408  {csn 4587   class class class wbr 5106   I cid 5531   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  lecple 17141  0.cp0 18313  AtLatcal 37729  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-clat 18389  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:  dib0  39630
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