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Theorem ltrnco 35025
 Description: The composition of two translations is a translation. Part of proof of Lemma G of [Crawley] p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013.)
Hypotheses
Ref Expression
ltrnco.h 𝐻 = (LHyp‘𝐾)
ltrnco.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
ltrnco (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹𝐺) ∈ 𝑇)

Proof of Theorem ltrnco
Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1054 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 ltrnco.h . . . . 5 𝐻 = (LHyp‘𝐾)
3 eqid 2610 . . . . 5 ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4 ltrnco.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
52, 3, 4ltrnldil 34426 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
653adant3 1074 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
72, 3, 4ltrnldil 34426 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
873adant2 1073 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
92, 3ldilco 34420 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ 𝐺 ∈ ((LDil‘𝐾)‘𝑊)) → (𝐹𝐺) ∈ ((LDil‘𝐾)‘𝑊))
101, 6, 8, 9syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹𝐺) ∈ ((LDil‘𝐾)‘𝑊))
11 simp11 1084 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp2l 1080 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13 simp3l 1082 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
1412, 13jca 553 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊))
15 simp2r 1081 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
16 simp3r 1083 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
1715, 16jca 553 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊))
18 simp12 1085 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹𝑇)
19 simp13 1086 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐺𝑇)
20 eqid 2610 . . . . . 6 (le‘𝐾) = (le‘𝐾)
21 eqid 2610 . . . . . 6 (join‘𝐾) = (join‘𝐾)
22 eqid 2610 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
23 eqid 2610 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
2420, 21, 22, 23, 2, 4cdlemg41 35024 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊))
2511, 14, 17, 18, 19, 24syl122anc 1327 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊))
26253exp 1256 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊))))
2726ralrimivv 2953 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊)))
2820, 21, 22, 23, 2, 3, 4isltrn 34423 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝐹𝐺) ∈ 𝑇 ↔ ((𝐹𝐺) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊)))))
29283ad2ant1 1075 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → ((𝐹𝐺) ∈ 𝑇 ↔ ((𝐹𝐺) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹𝐺)‘𝑞))(meet‘𝐾)𝑊)))))
3010, 27, 29mpbir2and 959 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (𝐹𝐺) ∈ 𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  meetcmee 16768  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LDilcldil 34404  LTrncltrn 34405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464 This theorem is referenced by:  trlcocnv  35026  trlcoabs2N  35028  trlcoat  35029  trlconid  35031  trlcolem  35032  trlcone  35034  cdlemg44  35039  cdlemg46  35041  cdlemg47  35042  trljco  35046  tgrpgrplem  35055  tendoidcl  35075  tendococl  35078  tendoplcl2  35084  tendoplco2  35085  tendoplcl  35087  tendo0co2  35094  tendoicl  35102  cdlemh1  35121  cdlemh2  35122  cdlemh  35123  cdlemi2  35125  cdlemi  35126  cdlemk2  35138  cdlemk3  35139  cdlemk4  35140  cdlemk8  35144  cdlemk9  35145  cdlemk9bN  35146  cdlemkvcl  35148  cdlemk10  35149  cdlemk11  35155  cdlemk12  35156  cdlemk14  35160  cdlemk11u  35177  cdlemk12u  35178  cdlemk37  35220  cdlemkfid1N  35227  cdlemkid1  35228  cdlemk45  35253  cdlemk47  35255  cdlemk48  35256  cdlemk50  35258  cdlemk52  35260  cdlemk53a  35261  cdlemk54  35264  cdlemk55a  35265  cdlemk55u1  35271  cdlemk55u  35272  tendospcanN  35330  dvalveclem  35332  dialss  35353  dia2dimlem4  35374  dvhvaddcl  35402  diblss  35477  cdlemn3  35504  dihopelvalcpre  35555  dih1  35593  dihglbcpreN  35607  dihjatcclem3  35727  dihjatcclem4  35728
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