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Theorem omlfh1N 35067
Description: Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 28808 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlfh1.b 𝐵 = (Base‘𝐾)
omlfh1.j = (join‘𝐾)
omlfh1.m = (meet‘𝐾)
omlfh1.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
omlfh1N ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Proof of Theorem omlfh1N
StepHypRef Expression
1 omllat 35051 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
2 omlfh1.b . . . . . 6 𝐵 = (Base‘𝐾)
3 eqid 2761 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4 omlfh1.j . . . . . 6 = (join‘𝐾)
5 omlfh1.m . . . . . 6 = (meet‘𝐾)
62, 3, 4, 5latledi 17311 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
71, 6sylan 489 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
873adant3 1127 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
91adantr 472 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
10 simpr1 1234 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
11 simpr2 1236 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
12 simpr3 1238 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
132, 4latjcl 17273 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
149, 11, 12, 13syl3anc 1477 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
152, 5latmcom 17297 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
169, 10, 14, 15syl3anc 1477 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
17 omlol 35049 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
1817adantr 472 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
192, 5latmcl 17274 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
209, 10, 11, 19syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
212, 5latmcl 17274 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
229, 10, 12, 21syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
23 eqid 2761 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
242, 4, 5, 23oldmj1 35030 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))))
2518, 20, 22, 24syl3anc 1477 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))))
262, 4, 5, 23oldmm1 35026 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
2718, 10, 11, 26syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
282, 4, 5, 23oldmm1 35026 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
2918, 10, 12, 28syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
3027, 29oveq12d 6833 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))
3125, 30eqtrd 2795 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))
3216, 31oveq12d 6833 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
33323adant3 1127 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
34 omlop 35050 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
3534adantr 472 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
362, 23opoccl 35003 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
3735, 10, 36syl2anc 696 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
382, 23opoccl 35003 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
3935, 11, 38syl2anc 696 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
402, 4latjcl 17273 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
419, 37, 39, 40syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
422, 23opoccl 35003 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
4335, 12, 42syl2anc 696 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
442, 4latjcl 17273 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
459, 37, 43, 44syl3anc 1477 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
462, 5latmcl 17274 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)
479, 41, 45, 46syl3anc 1477 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)
482, 5latmassOLD 35038 . . . . . . 7 ((𝐾 ∈ OL ∧ ((𝑌 𝑍) ∈ 𝐵𝑋𝐵 ∧ ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)) → (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑌 𝑍) (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))))
4918, 14, 10, 47, 48syl13anc 1479 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑌 𝑍) (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))))
50493adant3 1127 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑌 𝑍) (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))))
51 omlfh1.c . . . . . . . . . . . . . 14 𝐶 = (cm‘𝐾)
522, 23, 51cmt2N 35059 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
53523adant3r3 1200 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
54 simpl 474 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
552, 4, 5, 23, 51cmtbr3N 35063 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (𝑋𝐶((oc‘𝐾)‘𝑌) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5654, 10, 39, 55syl3anc 1477 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑌) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5753, 56bitrd 268 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5857biimpa 502 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
5958adantrr 755 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
60593impa 1101 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
612, 23, 51cmt2N 35059 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
62613adant3r2 1199 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
632, 4, 5, 23, 51cmtbr3N 35063 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (𝑋𝐶((oc‘𝐾)‘𝑍) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6454, 10, 43, 63syl3anc 1477 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑍) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6562, 64bitrd 268 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6665biimpa 502 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6766adantrl 754 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
68673impa 1101 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6960, 68oveq12d 6833 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
702, 5latmmdiN 35043 . . . . . . . . 9 ((𝐾 ∈ OL ∧ (𝑋𝐵 ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)) → (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
7118, 10, 41, 45, 70syl13anc 1479 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
72713adant3 1127 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
732, 5latmmdiN 35043 . . . . . . . . 9 ((𝐾 ∈ OL ∧ (𝑋𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
7418, 10, 39, 43, 73syl13anc 1479 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
75743adant3 1127 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
7669, 72, 753eqtr4d 2805 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))))
7776oveq2d 6831 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑌 𝑍) (𝑋 ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))) = ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
782, 5latmcl 17274 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
799, 39, 43, 78syl3anc 1477 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
802, 5latm12 35039 . . . . . . 7 ((𝐾 ∈ OL ∧ ((𝑌 𝑍) ∈ 𝐵𝑋𝐵 ∧ (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
8118, 14, 10, 79, 80syl13anc 1479 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
82813adant3 1127 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
8350, 77, 823eqtrd 2799 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (((𝑌 𝑍) 𝑋) ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
842, 4, 5, 23oldmj1 35030 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8518, 11, 12, 84syl3anc 1477 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8685oveq2d 6831 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))))
87 eqid 2761 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
882, 23, 5, 87opnoncon 35017 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = (0.‘𝐾))
8935, 14, 88syl2anc 696 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = (0.‘𝐾))
9086, 89eqtr3d 2797 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (0.‘𝐾))
9190oveq2d 6831 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 (0.‘𝐾)))
922, 5, 87olm01 35045 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9318, 10, 92syl2anc 696 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9491, 93eqtrd 2795 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
95943adant3 1127 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
9633, 83, 953eqtrd 2799 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾))
972, 4latjcl 17273 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
989, 20, 22, 97syl3anc 1477 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
992, 5latmcl 17274 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1009, 10, 14, 99syl3anc 1477 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1012, 3, 5, 23, 87omllaw3 35054 . . . . 5 ((𝐾 ∈ OML ∧ ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵 ∧ (𝑋 (𝑌 𝑍)) ∈ 𝐵) → ((((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍))))
10254, 98, 100, 101syl3anc 1477 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍))))
1031023adant3 1127 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)) ∧ ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍))))
1048, 96, 103mp2and 717 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍)))
105104eqcomd 2767 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2140   class class class wbr 4805  cfv 6050  (class class class)co 6815  Basecbs 16080  lecple 16171  occoc 16172  joincjn 17166  meetcmee 17167  0.cp0 17259  Latclat 17267  OPcops 34981  cmccmtN 34982  OLcol 34983  OMLcoml 34984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-preset 17150  df-poset 17168  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-lat 17268  df-oposet 34985  df-cmtN 34986  df-ol 34987  df-oml 34988
This theorem is referenced by:  omlfh3N  35068  omlmod1i2N  35069
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