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Theorem trlnidatb 39352
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 39348? Why do both this and ltrnideq 39350 need trlnidat 39348? (Contributed by NM, 4-Jun-2013.)
Hypotheses
Ref Expression
trlnidatb.b 𝐡 = (Baseβ€˜πΎ)
trlnidatb.a 𝐴 = (Atomsβ€˜πΎ)
trlnidatb.h 𝐻 = (LHypβ€˜πΎ)
trlnidatb.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
trlnidatb.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trlnidatb (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 β‰  ( I β†Ύ 𝐡) ↔ (π‘…β€˜πΉ) ∈ 𝐴))
Proof of Theorem trlnidatb
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 trlnidatb.b . . . 4 𝐡 = (Baseβ€˜πΎ)
2 trlnidatb.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
3 trlnidatb.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 trlnidatb.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 trlnidatb.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5trlnidat 39348 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) β†’ (π‘…β€˜πΉ) ∈ 𝐴)
763expia 1120 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 β‰  ( I β†Ύ 𝐡) β†’ (π‘…β€˜πΉ) ∈ 𝐴))
8 eqid 2731 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
98, 2, 3lhpexnle 39181 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝(leβ€˜πΎ)π‘Š)
109adantr 480 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ 𝐴 Β¬ 𝑝(leβ€˜πΎ)π‘Š)
111, 8, 2, 3, 4ltrnideq 39350 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ (πΉβ€˜π‘) = 𝑝))
12113expa 1117 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ (πΉβ€˜π‘) = 𝑝))
13 simp1l 1196 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (πΉβ€˜π‘) = 𝑝) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
14 simp2 1136 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (πΉβ€˜π‘) = 𝑝) β†’ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š))
15 simp1r 1197 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (πΉβ€˜π‘) = 𝑝) β†’ 𝐹 ∈ 𝑇)
16 simp3 1137 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (πΉβ€˜π‘) = 𝑝) β†’ (πΉβ€˜π‘) = 𝑝)
17 eqid 2731 . . . . . . . . 9 (0.β€˜πΎ) = (0.β€˜πΎ)
188, 17, 2, 3, 4, 5trl0 39345 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘) = 𝑝)) β†’ (π‘…β€˜πΉ) = (0.β€˜πΎ))
1913, 14, 15, 16, 18syl112anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š) ∧ (πΉβ€˜π‘) = 𝑝) β†’ (π‘…β€˜πΉ) = (0.β€˜πΎ))
20193expia 1120 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘) = 𝑝 β†’ (π‘…β€˜πΉ) = (0.β€˜πΎ)))
21 simplll 772 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ 𝐾 ∈ HL)
22 hlatl 38534 . . . . . . . 8 (𝐾 ∈ HL β†’ 𝐾 ∈ AtLat)
2317, 2atn0 38482 . . . . . . . . 9 ((𝐾 ∈ AtLat ∧ (π‘…β€˜πΉ) ∈ 𝐴) β†’ (π‘…β€˜πΉ) β‰  (0.β€˜πΎ))
2423ex 412 . . . . . . . 8 (𝐾 ∈ AtLat β†’ ((π‘…β€˜πΉ) ∈ 𝐴 β†’ (π‘…β€˜πΉ) β‰  (0.β€˜πΎ)))
2521, 22, 243syl 18 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((π‘…β€˜πΉ) ∈ 𝐴 β†’ (π‘…β€˜πΉ) β‰  (0.β€˜πΎ)))
2625necon2bd 2955 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((π‘…β€˜πΉ) = (0.β€˜πΎ) β†’ Β¬ (π‘…β€˜πΉ) ∈ 𝐴))
2720, 26syld 47 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘) = 𝑝 β†’ Β¬ (π‘…β€˜πΉ) ∈ 𝐴))
2812, 27sylbid 239 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝐹 = ( I β†Ύ 𝐡) β†’ Β¬ (π‘…β€˜πΉ) ∈ 𝐴))
2910, 28rexlimddv 3160 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 = ( I β†Ύ 𝐡) β†’ Β¬ (π‘…β€˜πΉ) ∈ 𝐴))
3029necon2ad 2954 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((π‘…β€˜πΉ) ∈ 𝐴 β†’ 𝐹 β‰  ( I β†Ύ 𝐡)))
317, 30impbid 211 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 β‰  ( I β†Ύ 𝐡) ↔ (π‘…β€˜πΉ) ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069   class class class wbr 5149   I cid 5574   β†Ύ cres 5679  β€˜cfv 6544  Basecbs 17149  lecple 17209  0.cp0 18381  Atomscatm 38437  AtLatcal 38438  HLchlt 38524  LHypclh 39159  LTrncltrn 39276  trLctrl 39333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8825  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334
This theorem is referenced by:  trlid0b  39353  cdlemfnid  39739  trlconid  39900  dih1dimb2  40416
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