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Theorem idltrn 39515
Description: The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
idltrn.b 𝐵 = (Base‘𝐾)
idltrn.h 𝐻 = (LHyp‘𝐾)
idltrn.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
idltrn ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)

Proof of Theorem idltrn
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idltrn.b . . 3 𝐵 = (Base‘𝐾)
2 idltrn.h . . 3 𝐻 = (LHyp‘𝐾)
3 eqid 2724 . . 3 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
41, 2, 3idldil 39479 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊))
5 simpll 764 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
6 simplrr 775 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
7 simprr 770 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
8 eqid 2724 . . . . . . 7 (le‘𝐾) = (le‘𝐾)
9 eqid 2724 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
10 eqid 2724 . . . . . . 7 (0.‘𝐾) = (0.‘𝐾)
11 eqid 2724 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
128, 9, 10, 11, 2lhpmat 39395 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾))
135, 6, 7, 12syl12anc 834 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(meet‘𝐾)𝑊) = (0.‘𝐾))
141, 11atbase 38653 . . . . . . . . 9 (𝑞 ∈ (Atoms‘𝐾) → 𝑞𝐵)
15 fvresi 7164 . . . . . . . . 9 (𝑞𝐵 → (( I ↾ 𝐵)‘𝑞) = 𝑞)
166, 14, 153syl 18 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑞) = 𝑞)
1716oveq2d 7418 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = (𝑞(join‘𝐾)𝑞))
18 simplll 772 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
19 eqid 2724 . . . . . . . . 9 (join‘𝐾) = (join‘𝐾)
2019, 11hlatjidm 38733 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑞 ∈ (Atoms‘𝐾)) → (𝑞(join‘𝐾)𝑞) = 𝑞)
2118, 6, 20syl2anc 583 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)𝑞) = 𝑞)
2217, 21eqtrd 2764 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞)) = 𝑞)
2322oveq1d 7417 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊) = (𝑞(meet‘𝐾)𝑊))
24 simplrl 774 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
251, 11atbase 38653 . . . . . . . . . 10 (𝑝 ∈ (Atoms‘𝐾) → 𝑝𝐵)
26 fvresi 7164 . . . . . . . . . 10 (𝑝𝐵 → (( I ↾ 𝐵)‘𝑝) = 𝑝)
2724, 25, 263syl 18 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (( I ↾ 𝐵)‘𝑝) = 𝑝)
2827oveq2d 7418 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = (𝑝(join‘𝐾)𝑝))
2919, 11hlatjidm 38733 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
3018, 24, 29syl2anc 583 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
3128, 30eqtrd 2764 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝)) = 𝑝)
3231oveq1d 7417 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (𝑝(meet‘𝐾)𝑊))
33 simprl 768 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
348, 9, 10, 11, 2lhpmat 39395 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾))
355, 24, 33, 34syl12anc 834 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝(meet‘𝐾)𝑊) = (0.‘𝐾))
3632, 35eqtrd 2764 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = (0.‘𝐾))
3713, 23, 363eqtr4rd 2775 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊))
3837ex 412 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))
3938ralrimivva 3192 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))
40 idltrn.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
418, 19, 9, 11, 2, 3, 40isltrn 39484 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (( I ↾ 𝐵) ∈ 𝑇 ↔ (( I ↾ 𝐵) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)(( I ↾ 𝐵)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)(( I ↾ 𝐵)‘𝑞))(meet‘𝐾)𝑊)))))
424, 39, 41mpbir2and 710 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053   class class class wbr 5139   I cid 5564  cres 5669  cfv 6534  (class class class)co 7402  Basecbs 17145  lecple 17205  joincjn 18268  meetcmee 18269  0.cp0 18380  Atomscatm 38627  HLchlt 38714  LHypclh 39349  LDilcldil 39465  LTrncltrn 39466
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 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-lat 18389  df-covers 38630  df-ats 38631  df-atl 38662  df-cvlat 38686  df-hlat 38715  df-lhyp 39353  df-laut 39354  df-ldil 39469  df-ltrn 39470
This theorem is referenced by:  trlid0  39541  tgrpgrplem  40114  tendoid  40138  tendo0cl  40155  cdlemkid2  40289  cdlemkid3N  40298  cdlemkid4  40299  cdlemkid5  40300  cdlemk35s-id  40303  dva0g  40392  dian0  40404  dia0  40417  dvhgrp  40472  dvh0g  40476  dvheveccl  40477  dvhopN  40481  dihmeetlem4preN  40671
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