Theorem ltrnco 41382

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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)

Proof of Theorem ltrnco

Dummy variables 𝑞 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1152	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		ltrnco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2769	. . . . 5 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4		ltrnco.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnldil 40785	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	5	3adant3 1148	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7	2, 3, 4	ltrnldil 40785	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
8	7	3adant2 1147	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
9	2, 3	ldilco 40779	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ 𝐺 ∈ ((LDil‘𝐾)‘𝑊)) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
10	1, 6, 8, 9	syl3anc 1396	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
11		simp11 1220	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simp2l 1216	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
13		simp3l 1218	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑝(le‘𝐾)𝑊)
14	12, 13	jca 520	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊))
15		simp2r 1217	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝑞 ∈ (Atoms‘𝐾))
16		simp3r 1219	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ¬ 𝑞(le‘𝐾)𝑊)
17	15, 16	jca 520	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊))
18		simp12 1221	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐹 ∈ 𝑇)
19		simp13 1222	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → 𝐺 ∈ 𝑇)
20		eqid 2769	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
21		eqid 2769	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
22		eqid 2769	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
23		eqid 2769	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
24	20, 21, 22, 23, 2, 4	cdlemg41 41381	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ ¬ 𝑞(le‘𝐾)𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊))
25	11, 14, 17, 18, 19, 24	syl122anc 1404	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊)) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊))
26	25	3exp 1135	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊))))
27	26	ralrimivv 3212	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊)))
28	20, 21, 22, 23, 2, 3, 4	isltrn 40782	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝐹 ∘ 𝐺) ∈ 𝑇 ↔ ((𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊)))))
29	28	3ad2ant1 1149	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝐹 ∘ 𝐺) ∈ 𝑇 ↔ ((𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊) ∧ ∀𝑝 ∈ (Atoms‘𝐾)∀𝑞 ∈ (Atoms‘𝐾)((¬ 𝑝(le‘𝐾)𝑊 ∧ ¬ 𝑞(le‘𝐾)𝑊) → ((𝑝(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑝))(meet‘𝐾)𝑊) = ((𝑞(join‘𝐾)((𝐹 ∘ 𝐺)‘𝑞))(meet‘𝐾)𝑊)))))
30	10, 27, 29	mpbir2and 725	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411  lecple 17316  joincjn 18366  meetcmee 18367  Atomscatm 39926  HLchlt 40013  LHypclh 40647  LDilcldil 40763  LTrncltrn 40764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-riotaBAD 39616

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-undef 8268  df-map 8825  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-p1 18479  df-lat 18487  df-clat 18554  df-oposet 39839  df-ol 39841  df-oml 39842  df-covers 39929  df-ats 39930  df-atl 39961  df-cvlat 39985  df-hlat 40014  df-llines 40161  df-lplanes 40162  df-lvols 40163  df-lines 40164  df-psubsp 40166  df-pmap 40167  df-padd 40459  df-lhyp 40651  df-laut 40652  df-ldil 40767  df-ltrn 40768  df-trl 40822

This theorem is referenced by:  trlcocnv  41383  trlcoabs2N  41385  trlcoat  41386  trlconid  41388  trlcolem  41389  trlcone  41391  cdlemg44  41396  cdlemg46  41398  cdlemg47  41399  trljco  41403  tgrpgrplem  41412  tendoidcl  41432  tendococl  41435  tendoplcl2  41441  tendoplco2  41442  tendoplcl  41444  tendo0co2  41451  tendoicl  41459  cdlemh1  41478  cdlemh2  41479  cdlemh  41480  cdlemi2  41482  cdlemi  41483  cdlemk2  41495  cdlemk3  41496  cdlemk4  41497  cdlemk8  41501  cdlemk9  41502  cdlemk9bN  41503  cdlemkvcl  41505  cdlemk10  41506  cdlemk11  41512  cdlemk12  41513  cdlemk14  41517  cdlemk11u  41534  cdlemk12u  41535  cdlemk37  41577  cdlemkfid1N  41584  cdlemkid1  41585  cdlemk45  41610  cdlemk47  41612  cdlemk48  41613  cdlemk50  41615  cdlemk52  41617  cdlemk53a  41618  cdlemk54  41621  cdlemk55a  41622  cdlemk55u1  41628  cdlemk55u  41629  tendospcanN  41686  dvalveclem  41688  dialss  41709  dia2dimlem4  41730  dvhvaddcl  41758  diblss  41833  cdlemn3  41860  dihopelvalcpre  41911  dih1  41949  dihglbcpreN  41963  dihjatcclem3  42083  dihjatcclem4  42084