Theorem ltrnco 41883

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 1154	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		ltrnco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2760	. . . . 5 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4		ltrnco.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnldil 41286	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	5	3adant3 1150	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7	2, 3, 4	ltrnldil 41286	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
8	7	3adant2 1149	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
9	2, 3	ldilco 41280	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ 𝐺 ∈ ((LDil‘𝐾)‘𝑊)) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
10	1, 6, 8, 9	syl3anc 1398	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
11		simp11 1222	. . . . 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 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 ∈ (Atoms‘𝐾))
13		simp3l 1220	. . . . . 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 521	. . . . 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 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 ∈ (Atoms‘𝐾))
16		simp3r 1221	. . . . . 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 521	. . . . 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 1223	. . . . 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 1224	. . . . 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 2760	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
21		eqid 2760	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
22		eqid 2760	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
23		eqid 2760	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
24	20, 21, 22, 23, 2, 4	cdlemg41 41882	. . . . 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 1406	. . . 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 1137	. . 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 3203	. 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 41283	. . 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 1151	. 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 726	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   class class class wbr 5102   ∘ ccom 5654  ‘cfv 6541  (class class class)co 7422  lecple 17442  joincjn 18492  meetcmee 18493  Atomscatm 40427  HLchlt 40514  LHypclh 41148  LDilcldil 41264  LTrncltrn 41265

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-riotaBAD 40117

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-undef 8295  df-map 8854  df-proset 18475  df-poset 18494  df-plt 18509  df-lub 18525  df-glb 18526  df-join 18527  df-meet 18528  df-p0 18604  df-p1 18605  df-lat 18613  df-clat 18680  df-oposet 40340  df-ol 40342  df-oml 40343  df-covers 40430  df-ats 40431  df-atl 40462  df-cvlat 40486  df-hlat 40515  df-llines 40662  df-lplanes 40663  df-lvols 40664  df-lines 40665  df-psubsp 40667  df-pmap 40668  df-padd 40960  df-lhyp 41152  df-laut 41153  df-ldil 41268  df-ltrn 41269  df-trl 41323

This theorem is used by:  trlcocnv  41884  trlcoabs2N  41886  trlcoat  41887  trlconid  41889  trlcolem  41890  trlcone  41892  cdlemg44  41897  cdlemg46  41899  cdlemg47  41900  trljco  41904  tgrpgrplem  41913  tendoidcl  41933  tendococl  41936  tendoplcl2  41942  tendoplco2  41943  tendoplcl  41945  tendo0co2  41952  tendoicl  41960  cdlemh1  41979  cdlemh2  41980  cdlemh  41981  cdlemi2  41983  cdlemi  41984  cdlemk2  41996  cdlemk3  41997  cdlemk4  41998  cdlemk8  42002  cdlemk9  42003  cdlemk9bN  42004  cdlemkvcl  42006  cdlemk10  42007  cdlemk11  42013  cdlemk12  42014  cdlemk14  42018  cdlemk11u  42035  cdlemk12u  42036  cdlemk37  42078  cdlemkfid1N  42085  cdlemkid1  42086  cdlemk45  42111  cdlemk47  42113  cdlemk48  42114  cdlemk50  42116  cdlemk52  42118  cdlemk53a  42119  cdlemk54  42122  cdlemk55a  42123  cdlemk55u1  42129  cdlemk55u  42130  tendospcanN  42187  dvalveclem  42189  dialss  42210  dia2dimlem4  42231  dvhvaddcl  42259  diblss  42334  cdlemn3  42361  dihopelvalcpre  42412  dih1  42450  dihglbcpreN  42464  dihjatcclem3  42584  dihjatcclem4  42585