Theorem ltrnco 36873

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 1127	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		ltrnco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2778	. . . . 5 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4		ltrnco.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnldil 36276	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	5	3adant3 1123	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7	2, 3, 4	ltrnldil 36276	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
8	7	3adant2 1122	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
9	2, 3	ldilco 36270	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ 𝐺 ∈ ((LDil‘𝐾)‘𝑊)) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
10	1, 6, 8, 9	syl3anc 1439	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
11		simp11 1217	. . . . 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 1213	. . . . . 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 1215	. . . . . 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 507	. . . . 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 1214	. . . . . 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 1216	. . . . . 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 507	. . . . 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 1218	. . . . 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 1219	. . . . 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 2778	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
21		eqid 2778	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
22		eqid 2778	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
23		eqid 2778	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
24	20, 21, 22, 23, 2, 4	cdlemg41 36872	. . . . 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 1447	. . . 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 1109	. . 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 3152	. 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 36273	. . 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 1124	. 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 703	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090   class class class wbr 4886   ∘ ccom 5359  ‘cfv 6135  (class class class)co 6922  lecple 16345  joincjn 17330  meetcmee 17331  Atomscatm 35417  HLchlt 35504  LHypclh 36138  LDilcldil 36254  LTrncltrn 36255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-riotaBAD 35107

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-undef 7681  df-map 8142  df-proset 17314  df-poset 17332  df-plt 17344  df-lub 17360  df-glb 17361  df-join 17362  df-meet 17363  df-p0 17425  df-p1 17426  df-lat 17432  df-clat 17494  df-oposet 35330  df-ol 35332  df-oml 35333  df-covers 35420  df-ats 35421  df-atl 35452  df-cvlat 35476  df-hlat 35505  df-llines 35652  df-lplanes 35653  df-lvols 35654  df-lines 35655  df-psubsp 35657  df-pmap 35658  df-padd 35950  df-lhyp 36142  df-laut 36143  df-ldil 36258  df-ltrn 36259  df-trl 36313

This theorem is referenced by:  trlcocnv  36874  trlcoabs2N  36876  trlcoat  36877  trlconid  36879  trlcolem  36880  trlcone  36882  cdlemg44  36887  cdlemg46  36889  cdlemg47  36890  trljco  36894  tgrpgrplem  36903  tendoidcl  36923  tendococl  36926  tendoplcl2  36932  tendoplco2  36933  tendoplcl  36935  tendo0co2  36942  tendoicl  36950  cdlemh1  36969  cdlemh2  36970  cdlemh  36971  cdlemi2  36973  cdlemi  36974  cdlemk2  36986  cdlemk3  36987  cdlemk4  36988  cdlemk8  36992  cdlemk9  36993  cdlemk9bN  36994  cdlemkvcl  36996  cdlemk10  36997  cdlemk11  37003  cdlemk12  37004  cdlemk14  37008  cdlemk11u  37025  cdlemk12u  37026  cdlemk37  37068  cdlemkfid1N  37075  cdlemkid1  37076  cdlemk45  37101  cdlemk47  37103  cdlemk48  37104  cdlemk50  37106  cdlemk52  37108  cdlemk53a  37109  cdlemk54  37112  cdlemk55a  37113  cdlemk55u1  37119  cdlemk55u  37120  tendospcanN  37177  dvalveclem  37179  dialss  37200  dia2dimlem4  37221  dvhvaddcl  37249  diblss  37324  cdlemn3  37351  dihopelvalcpre  37402  dih1  37440  dihglbcpreN  37454  dihjatcclem3  37574  dihjatcclem4  37575