Theorem ltrnco 40684

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 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		ltrnco.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		eqid 2735	. . . . 5 ⊢ ((LDil‘𝐾)‘𝑊) = ((LDil‘𝐾)‘𝑊)
4		ltrnco.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	ltrnldil 40087	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
6	5	3adant3 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐹 ∈ ((LDil‘𝐾)‘𝑊))
7	2, 3, 4	ltrnldil 40087	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
8	7	3adant2 1131	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → 𝐺 ∈ ((LDil‘𝐾)‘𝑊))
9	2, 3	ldilco 40081	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ ((LDil‘𝐾)‘𝑊) ∧ 𝐺 ∈ ((LDil‘𝐾)‘𝑊)) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
10	1, 6, 8, 9	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ ((LDil‘𝐾)‘𝑊))
11		simp11 1204	. . . . 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 1200	. . . . . 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 1202	. . . . . 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 511	. . . . 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 1201	. . . . . 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 1203	. . . . . 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 511	. . . . 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 1205	. . . . 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 1206	. . . . 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 2735	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
21		eqid 2735	. . . . . 6 ⊢ (join‘𝐾) = (join‘𝐾)
22		eqid 2735	. . . . . 6 ⊢ (meet‘𝐾) = (meet‘𝐾)
23		eqid 2735	. . . . . 6 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
24	20, 21, 22, 23, 2, 4	cdlemg41 40683	. . . . 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 1381	. . . 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 1119	. . 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 3185	. 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 40084	. . 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 1133	. 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 713	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   class class class wbr 5119   ∘ ccom 5658  ‘cfv 6530  (class class class)co 7403  lecple 17276  joincjn 18321  meetcmee 18322  Atomscatm 39227  HLchlt 39314  LHypclh 39949  LDilcldil 40065  LTrncltrn 40066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-riotaBAD 38917

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-undef 8270  df-map 8840  df-proset 18304  df-poset 18323  df-plt 18338  df-lub 18354  df-glb 18355  df-join 18356  df-meet 18357  df-p0 18433  df-p1 18434  df-lat 18440  df-clat 18507  df-oposet 39140  df-ol 39142  df-oml 39143  df-covers 39230  df-ats 39231  df-atl 39262  df-cvlat 39286  df-hlat 39315  df-llines 39463  df-lplanes 39464  df-lvols 39465  df-lines 39466  df-psubsp 39468  df-pmap 39469  df-padd 39761  df-lhyp 39953  df-laut 39954  df-ldil 40069  df-ltrn 40070  df-trl 40124

This theorem is referenced by:  trlcocnv  40685  trlcoabs2N  40687  trlcoat  40688  trlconid  40690  trlcolem  40691  trlcone  40693  cdlemg44  40698  cdlemg46  40700  cdlemg47  40701  trljco  40705  tgrpgrplem  40714  tendoidcl  40734  tendococl  40737  tendoplcl2  40743  tendoplco2  40744  tendoplcl  40746  tendo0co2  40753  tendoicl  40761  cdlemh1  40780  cdlemh2  40781  cdlemh  40782  cdlemi2  40784  cdlemi  40785  cdlemk2  40797  cdlemk3  40798  cdlemk4  40799  cdlemk8  40803  cdlemk9  40804  cdlemk9bN  40805  cdlemkvcl  40807  cdlemk10  40808  cdlemk11  40814  cdlemk12  40815  cdlemk14  40819  cdlemk11u  40836  cdlemk12u  40837  cdlemk37  40879  cdlemkfid1N  40886  cdlemkid1  40887  cdlemk45  40912  cdlemk47  40914  cdlemk48  40915  cdlemk50  40917  cdlemk52  40919  cdlemk53a  40920  cdlemk54  40923  cdlemk55a  40924  cdlemk55u1  40930  cdlemk55u  40931  tendospcanN  40988  dvalveclem  40990  dialss  41011  dia2dimlem4  41032  dvhvaddcl  41060  diblss  41135  cdlemn3  41162  dihopelvalcpre  41213  dih1  41251  dihglbcpreN  41265  dihjatcclem3  41385  dihjatcclem4  41386