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

Theorem cmtcomlemN 35322
Description: Lemma for cmtcomN 35323. (cmcmlem 29001 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
cmtcom.b 𝐵 = (Base‘𝐾)
cmtcom.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
cmtcomlemN ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))

Proof of Theorem cmtcomlemN
StepHypRef Expression
1 omllat 35316 . . . . . . . . . . . 12 (𝐾 ∈ OML → 𝐾 ∈ Lat)
213ad2ant1 1167 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ Lat)
3 omlop 35315 . . . . . . . . . . . . 13 (𝐾 ∈ OML → 𝐾 ∈ OP)
4 cmtcom.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐾)
5 eqid 2825 . . . . . . . . . . . . . 14 (oc‘𝐾) = (oc‘𝐾)
64, 5opoccl 35268 . . . . . . . . . . . . 13 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
73, 6sylan 575 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
873adant3 1166 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
9 simp3 1172 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
10 eqid 2825 . . . . . . . . . . . 12 (le‘𝐾) = (le‘𝐾)
11 eqid 2825 . . . . . . . . . . . 12 (join‘𝐾) = (join‘𝐾)
124, 10, 11latlej2 17421 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵𝑌𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
132, 8, 9, 12syl3anc 1494 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))
144, 11latjcl 17411 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
152, 8, 9, 14syl3anc 1494 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵)
16 eqid 2825 . . . . . . . . . . . 12 (meet‘𝐾) = (meet‘𝐾)
174, 10, 16latleeqm2 17440 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑌𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵) → (𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ↔ ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌))
182, 9, 15, 17syl3anc 1494 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌(le‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ↔ ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌))
1913, 18mpbid 224 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌) = 𝑌)
2019oveq2d 6926 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)𝑌))
21 omlol 35314 . . . . . . . . . 10 (𝐾 ∈ OML → 𝐾 ∈ OL)
22213ad2ant1 1167 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OL)
2333ad2ant1 1167 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OP)
244, 5opoccl 35268 . . . . . . . . . . 11 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
2523, 9, 24syl2anc 579 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
264, 11latjcl 17411 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
272, 8, 25, 26syl3anc 1494 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵)
284, 16latmassOLD 35303 . . . . . . . . 9 ((𝐾 ∈ OL ∧ ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵𝑌𝐵)) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)))
2922, 27, 15, 9, 28syl13anc 1495 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)((((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)(meet‘𝐾)𝑌)))
304, 11, 16, 5oldmm1 35291 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
3121, 30syl3an1 1206 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌)) = (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))
3231oveq1d 6925 . . . . . . . 8 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)𝑌))
3320, 29, 323eqtr4rd 2872 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌))
3433adantr 474 . . . . . 6 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌))
354, 11, 16, 5oldmj4 35298 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
3621, 35syl3an1 1206 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))) = (𝑋(meet‘𝐾)𝑌))
374, 11, 16, 5oldmj2 35296 . . . . . . . . . . . . 13 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = (𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))
3821, 37syl3an1 1206 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = (𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))
3936, 38oveq12d 6928 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))) = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌))))
4039eqeq2d 2835 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))) ↔ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
4140biimpar 471 . . . . . . . . 9 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → 𝑋 = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))))
4241fveq2d 6441 . . . . . . . 8 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘𝑋) = ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))))
434, 11, 16, 5oldmj4 35298 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌) ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
4422, 27, 15, 43syl3anc 1494 . . . . . . . . 9 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
4544adantr 474 . . . . . . . 8 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((oc‘𝐾)‘(((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌)))(join‘𝐾)((oc‘𝐾)‘(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))) = ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)))
4642, 45eqtr2d 2862 . . . . . . 7 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌)) = ((oc‘𝐾)‘𝑋))
4746oveq1d 6925 . . . . . 6 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((((oc‘𝐾)‘𝑋)(join‘𝐾)((oc‘𝐾)‘𝑌))(meet‘𝐾)(((oc‘𝐾)‘𝑋)(join‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌))
4834, 47eqtrd 2861 . . . . 5 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → (((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌) = (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌))
4948oveq2d 6926 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)))
50 simp1 1170 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → 𝐾 ∈ OML)
514, 16latmcl 17412 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
521, 51syl3an1 1206 . . . . . . 7 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) ∈ 𝐵)
5350, 52, 93jca 1162 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑌𝐵))
544, 10, 16latmle2 17437 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
551, 54syl3an1 1206 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌)
564, 10, 11, 16, 5omllaw2N 35318 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋(meet‘𝐾)𝑌) ∈ 𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(le‘𝐾)𝑌 → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌))
5753, 55, 56sylc 65 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌)
5857adantr 474 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘(𝑋(meet‘𝐾)𝑌))(meet‘𝐾)𝑌)) = 𝑌)
594, 16latmcom 17435 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
601, 59syl3an1 1206 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)𝑋))
614, 16latmcom 17435 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
622, 8, 9, 61syl3anc 1494 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌) = (𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))
6360, 62oveq12d 6928 . . . . 5 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)) = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
6463adantr 474 . . . 4 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(((oc‘𝐾)‘𝑋)(meet‘𝐾)𝑌)) = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
6549, 58, 643eqtr3d 2869 . . 3 (((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))) → 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋))))
6665ex 403 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌))) → 𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
67 cmtcom.c . . 3 𝐶 = (cm‘𝐾)
684, 11, 16, 5, 67cmtvalN 35285 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋 = ((𝑋(meet‘𝐾)𝑌)(join‘𝐾)(𝑋(meet‘𝐾)((oc‘𝐾)‘𝑌)))))
694, 11, 16, 5, 67cmtvalN 35285 . . 3 ((𝐾 ∈ OML ∧ 𝑌𝐵𝑋𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
70693com23 1160 . 2 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑌𝐶𝑋𝑌 = ((𝑌(meet‘𝐾)𝑋)(join‘𝐾)(𝑌(meet‘𝐾)((oc‘𝐾)‘𝑋)))))
7166, 68, 703imtr4d 286 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑌𝐶𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164   class class class wbr 4875  cfv 6127  (class class class)co 6910  Basecbs 16229  lecple 16319  occoc 16320  joincjn 17304  meetcmee 17305  Latclat 17405  OPcops 35246  cmccmtN 35247  OLcol 35248  OMLcoml 35249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-proset 17288  df-poset 17306  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-lat 17406  df-oposet 35250  df-cmtN 35251  df-ol 35252  df-oml 35253
This theorem is referenced by:  cmtcomN  35323
  Copyright terms: Public domain W3C validator