Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  djajN Structured version   Visualization version   GIF version

Theorem djajN 35941
Description: Transfer lattice join to DVecA partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
djaj.k = (join‘𝐾)
djaj.h 𝐻 = (LHyp‘𝐾)
djaj.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
djaj.j 𝐽 = ((vA‘𝐾)‘𝑊)
Assertion
Ref Expression
djajN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋)𝐽(𝐼𝑌)))

Proof of Theorem djajN
StepHypRef Expression
1 hllat 34165 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
21ad2antrr 761 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝐾 ∈ Lat)
3 hlop 34164 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OP)
43ad2antrr 761 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OP)
5 eqid 2621 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
6 djaj.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
7 djaj.i . . . . . . . . . 10 𝐼 = ((DIsoA‘𝐾)‘𝑊)
85, 6, 7diadmclN 35841 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾))
98adantrr 752 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝑋 ∈ (Base‘𝐾))
10 eqid 2621 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
115, 10opoccl 33996 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑋 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
124, 9, 11syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾))
135, 6lhpbase 34799 . . . . . . . . 9 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1413ad2antlr 762 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝑊 ∈ (Base‘𝐾))
155, 10opoccl 33996 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
164, 14, 15syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))
17 djaj.k . . . . . . . 8 = (join‘𝐾)
185, 17latjcl 16983 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
192, 12, 16, 18syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
20 eqid 2621 . . . . . . 7 (meet‘𝐾) = (meet‘𝐾)
215, 20latmcl 16984 . . . . . 6 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
222, 19, 14, 21syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
235, 6, 7diadmclN 35841 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌 ∈ (Base‘𝐾))
2423adantrl 751 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝑌 ∈ (Base‘𝐾))
255, 10opoccl 33996 . . . . . . . 8 ((𝐾 ∈ OP ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
264, 24, 25syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾))
275, 17latjcl 16983 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
282, 26, 16, 27syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾))
295, 20latmcl 16984 . . . . . 6 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
302, 28, 14, 29syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
315, 20latmcl 16984 . . . . 5 ((𝐾 ∈ Lat ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
322, 22, 30, 31syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
33 eqid 2621 . . . . 5 (le‘𝐾) = (le‘𝐾)
345, 33, 20latmle2 17009 . . . . . 6 ((𝐾 ∈ Lat ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
352, 22, 30, 34syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
365, 33, 20latmle2 17009 . . . . . 6 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
372, 28, 14, 36syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
385, 33, 2, 32, 30, 14, 35, 37lattrd 16990 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)
395, 33, 6, 7diaeldm 35840 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼 ↔ ((((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)))
4039adantr 481 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼 ↔ ((((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))(le‘𝐾)𝑊)))
4132, 38, 40mpbir2and 956 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼)
42 eqid 2621 . . . 4 ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
43 eqid 2621 . . . 4 ((ocA‘𝐾)‘𝑊) = ((ocA‘𝐾)‘𝑊)
4417, 20, 10, 6, 42, 7, 43diaocN 35929 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
4541, 44syldan 487 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
46 hloml 34159 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OML)
4746ad2antrr 761 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OML)
485, 17latjcl 16983 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝑋 𝑌) ∈ (Base‘𝐾))
492, 9, 24, 48syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑋 𝑌) ∈ (Base‘𝐾))
5033, 6, 7diadmleN 35842 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋(le‘𝐾)𝑊)
5150adantrr 752 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝑋(le‘𝐾)𝑊)
5233, 6, 7diadmleN 35842 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌(le‘𝐾)𝑊)
5352adantrl 751 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝑌(le‘𝐾)𝑊)
545, 33, 17latjle12 16994 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑋(le‘𝐾)𝑊𝑌(le‘𝐾)𝑊) ↔ (𝑋 𝑌)(le‘𝐾)𝑊))
552, 9, 24, 14, 54syl13anc 1325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝑋(le‘𝐾)𝑊𝑌(le‘𝐾)𝑊) ↔ (𝑋 𝑌)(le‘𝐾)𝑊))
5651, 53, 55mpbi2and 955 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑋 𝑌)(le‘𝐾)𝑊)
575, 33, 17, 20, 10omlspjN 34063 . . . . 5 ((𝐾 ∈ OML ∧ ((𝑋 𝑌) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ (𝑋 𝑌)(le‘𝐾)𝑊) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋 𝑌))
5847, 49, 14, 56, 57syl121anc 1328 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = (𝑋 𝑌))
595, 17latjidm 17006 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾)) → (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
602, 16, 59syl2anc 692 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘𝑊))
6160oveq2d 6626 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝑋 𝑌) (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊))) = ((𝑋 𝑌) ((oc‘𝐾)‘𝑊)))
625, 17latjass 17027 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((𝑋 𝑌) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘𝑊) ∈ (Base‘𝐾))) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊)) ((oc‘𝐾)‘𝑊)) = ((𝑋 𝑌) (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊))))
632, 49, 16, 16, 62syl13anc 1325 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊)) ((oc‘𝐾)‘𝑊)) = ((𝑋 𝑌) (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊))))
64 hlol 34163 . . . . . . . . . . 11 (𝐾 ∈ HL → 𝐾 ∈ OL)
6564ad2antrr 761 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → 𝐾 ∈ OL)
665, 17, 20, 10oldmm2 34020 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 𝑌))(meet‘𝐾)𝑊)) = ((𝑋 𝑌) ((oc‘𝐾)‘𝑊)))
6765, 49, 14, 66syl3anc 1323 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 𝑌))(meet‘𝐾)𝑊)) = ((𝑋 𝑌) ((oc‘𝐾)‘𝑊)))
685, 17, 20, 10oldmj1 34023 . . . . . . . . . . . . . 14 ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))
6965, 9, 24, 68syl3anc 1323 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)))
705, 33, 20latleeqm1 17011 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
712, 9, 14, 70syl3anc 1323 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑋(le‘𝐾)𝑊 ↔ (𝑋(meet‘𝐾)𝑊) = 𝑋))
7251, 71mpbid 222 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑋(meet‘𝐾)𝑊) = 𝑋)
7372fveq2d 6157 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘𝑋))
745, 17, 20, 10oldmm1 34019 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OL ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)))
7565, 9, 14, 74syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)))
7673, 75eqtr3d 2657 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑋) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)))
775, 33, 20latleeqm1 17011 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Lat ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
782, 24, 14, 77syl3anc 1323 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑌(le‘𝐾)𝑊 ↔ (𝑌(meet‘𝐾)𝑊) = 𝑌))
7953, 78mpbid 222 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑌(meet‘𝐾)𝑊) = 𝑌)
8079fveq2d 6157 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘𝑌))
815, 17, 20, 10oldmm1 34019 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ OL ∧ 𝑌 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))
8265, 24, 14, 81syl3anc 1323 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑌(meet‘𝐾)𝑊)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))
8380, 82eqtr3d 2657 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘𝑌) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))
8476, 83oveq12d 6628 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)((oc‘𝐾)‘𝑌)) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))))
8569, 84eqtrd 2655 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(𝑋 𝑌)) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))))
8685oveq1d 6625 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘(𝑋 𝑌))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊))
875, 20latmmdir 34037 . . . . . . . . . . . 12 ((𝐾 ∈ OL ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
8865, 19, 28, 14, 87syl13anc 1325 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊)))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
8986, 88eqtrd 2655 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((oc‘𝐾)‘(𝑋 𝑌))(meet‘𝐾)𝑊) = (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
9089fveq2d 6157 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((oc‘𝐾)‘(((oc‘𝐾)‘(𝑋 𝑌))(meet‘𝐾)𝑊)) = ((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
9167, 90eqtr3d 2657 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝑋 𝑌) ((oc‘𝐾)‘𝑊)) = ((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
9291oveq1d 6625 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊)) ((oc‘𝐾)‘𝑊)) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊)))
9363, 92eqtr3d 2657 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝑋 𝑌) (((oc‘𝐾)‘𝑊) ((oc‘𝐾)‘𝑊))) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊)))
9461, 93eqtr3d 2657 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝑋 𝑌) ((oc‘𝐾)‘𝑊)) = (((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊)))
9594oveq1d 6625 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((𝑋 𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) = ((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
9658, 95eqtr3d 2657 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝑋 𝑌) = ((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))
9796fveq2d 6157 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = (𝐼‘((((oc‘𝐾)‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))
98 simpl 473 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
996, 7diaclN 35854 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ ran 𝐼)
10099adantrr 752 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼𝑋) ∈ ran 𝐼)
1016, 42, 7diaelrnN 35849 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼𝑋) ∈ ran 𝐼) → (𝐼𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
102100, 101syldan 487 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼𝑋) ⊆ ((LTrn‘𝐾)‘𝑊))
1036, 7diaclN 35854 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼𝑌) ∈ ran 𝐼)
104103adantrl 751 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼𝑌) ∈ ran 𝐼)
1056, 42, 7diaelrnN 35849 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼𝑌) ∈ ran 𝐼) → (𝐼𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
106104, 105syldan 487 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))
107 djaj.j . . . . 5 𝐽 = ((vA‘𝐾)‘𝑊)
1086, 42, 7, 43, 107djavalN 35939 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐼𝑋) ⊆ ((LTrn‘𝐾)‘𝑊) ∧ (𝐼𝑌) ⊆ ((LTrn‘𝐾)‘𝑊))) → ((𝐼𝑋)𝐽(𝐼𝑌)) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌)))))
10998, 102, 106, 108syl12anc 1321 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝐼𝑋)𝐽(𝐼𝑌)) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌)))))
1105, 33, 20latmle2 17009 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
1112, 19, 14, 110syl3anc 1323 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)
1125, 33, 6, 7diaeldm 35840 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
113112adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
11422, 111, 113mpbir2and 956 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
1155, 33, 6, 7diaeldm 35840 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
116115adantr 481 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ↔ (((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(le‘𝐾)𝑊)))
11730, 37, 116mpbir2and 956 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)
11820, 6, 7diameetN 35860 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼 ∧ ((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊) ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((𝐼‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
11998, 114, 117, 118syl12anc 1321 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((𝐼‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))))
12017, 20, 10, 6, 42, 7, 43diaocN 35929 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)))
121120adantrr 752 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)))
12217, 20, 10, 6, 42, 7, 43diaocN 35929 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌 ∈ dom 𝐼) → (𝐼‘((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌)))
123122adantrl 751 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) = (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌)))
124121, 123ineq12d 3798 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝐼‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)) ∩ (𝐼‘((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌))))
125119, 124eqtrd 2655 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊))) = ((((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌))))
126125fveq2d 6157 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))) = (((ocA‘𝐾)‘𝑊)‘((((ocA‘𝐾)‘𝑊)‘(𝐼𝑋)) ∩ (((ocA‘𝐾)‘𝑊)‘(𝐼𝑌)))))
127109, 126eqtr4d 2658 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → ((𝐼𝑋)𝐽(𝐼𝑌)) = (((ocA‘𝐾)‘𝑊)‘(𝐼‘(((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)(meet‘𝐾)((((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑊))(meet‘𝐾)𝑊)))))
12845, 97, 1273eqtr4d 2665 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋)𝐽(𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  cin 3558  wss 3559   class class class wbr 4618  dom cdm 5079  ran crn 5080  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  occoc 15881  joincjn 16876  meetcmee 16877  Latclat 16977  OPcops 33974  OLcol 33976  OMLcoml 33977  HLchlt 34152  LHypclh 34785  LTrncltrn 34902  DIsoAcdia 35832  ocAcocaN 35923  vAcdjaN 35935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-riotaBAD 33754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-undef 7351  df-map 7811  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-p1 16972  df-lat 16978  df-clat 17040  df-oposet 33978  df-cmtN 33979  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-llines 34299  df-lplanes 34300  df-lvols 34301  df-lines 34302  df-psubsp 34304  df-pmap 34305  df-padd 34597  df-lhyp 34789  df-laut 34790  df-ldil 34905  df-ltrn 34906  df-trl 34961  df-disoa 35833  df-docaN 35924  df-djaN 35936
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator