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Theorem omlfh1N 36280
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 29328 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 36264 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
2 omlfh1.b . . . . . 6 𝐵 = (Base‘𝐾)
3 eqid 2826 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4 omlfh1.j . . . . . 6 = (join‘𝐾)
5 omlfh1.m . . . . . 6 = (meet‘𝐾)
62, 3, 4, 5latledi 17694 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
71, 6sylan 580 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
873adant3 1126 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
91adantr 481 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
10 simpr1 1188 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
11 simpr2 1189 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
12 simpr3 1190 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
132, 4latjcl 17656 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
149, 11, 12, 13syl3anc 1365 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
152, 5latmcom 17680 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
169, 10, 14, 15syl3anc 1365 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
17 omlol 36262 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
1817adantr 481 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
192, 5latmcl 17657 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
209, 10, 11, 19syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
212, 5latmcl 17657 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
229, 10, 12, 21syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
23 eqid 2826 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
242, 4, 5, 23oldmj1 36243 . . . . . . . 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 1365 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))))
262, 4, 5, 23oldmm1 36239 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
2718, 10, 11, 26syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
282, 4, 5, 23oldmm1 36239 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
2918, 10, 12, 28syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
3027, 29oveq12d 7168 . . . . . . 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 2861 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))
3216, 31oveq12d 7168 . . . . 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 1126 . . . 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 36263 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
3534adantr 481 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
362, 23opoccl 36216 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
3735, 10, 36syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
382, 23opoccl 36216 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
3935, 11, 38syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
402, 4latjcl 17656 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
419, 37, 39, 40syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
422, 23opoccl 36216 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
4335, 12, 42syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
442, 4latjcl 17656 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
459, 37, 43, 44syl3anc 1365 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
462, 5latmcl 17657 . . . . . . . 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 1365 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)
482, 5latmassOLD 36251 . . . . . . 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 1366 . . . . . 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 1126 . . . . 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 36272 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
53523adant3r3 1178 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
54 simpl 483 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
552, 4, 5, 23, 51cmtbr3N 36276 . . . . . . . . . . . . 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 1365 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑌) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5753, 56bitrd 280 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5857biimpa 477 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
5958adantrr 713 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
60593impa 1104 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
612, 23, 51cmt2N 36272 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
62613adant3r2 1177 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
632, 4, 5, 23, 51cmtbr3N 36276 . . . . . . . . . . . . 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 1365 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑍) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6562, 64bitrd 280 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6665biimpa 477 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6766adantrl 712 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
68673impa 1104 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6960, 68oveq12d 7168 . . . . . . 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 36256 . . . . . . . . 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 1366 . . . . . . . 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 1126 . . . . . . 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 36256 . . . . . . . . 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 1366 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
75743adant3 1126 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
7669, 72, 753eqtr4d 2871 . . . . . 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 7166 . . . . 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 17657 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
799, 39, 43, 78syl3anc 1365 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
802, 5latm12 36252 . . . . . . 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 1366 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
82813adant3 1126 . . . . 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 2865 . . . 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 36243 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8518, 11, 12, 84syl3anc 1365 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8685oveq2d 7166 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))))
87 eqid 2826 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
882, 23, 5, 87opnoncon 36230 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (𝑌 𝑍) ∈ 𝐵) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = (0.‘𝐾))
8935, 14, 88syl2anc 584 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = (0.‘𝐾))
9086, 89eqtr3d 2863 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (0.‘𝐾))
9190oveq2d 7166 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 (0.‘𝐾)))
922, 5, 87olm01 36258 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9318, 10, 92syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9491, 93eqtrd 2861 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
95943adant3 1126 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
9633, 83, 953eqtrd 2865 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾))
972, 4latjcl 17656 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
989, 20, 22, 97syl3anc 1365 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
992, 5latmcl 17657 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1009, 10, 14, 99syl3anc 1365 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1012, 3, 5, 23, 87omllaw3 36267 . . . . 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 1365 . . . 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 1126 . . 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 695 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍)))
105104eqcomd 2832 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107   class class class wbr 5063  cfv 6354  (class class class)co 7150  Basecbs 16478  lecple 16567  occoc 16568  joincjn 17549  meetcmee 17550  0.cp0 17642  Latclat 17650  OPcops 36194  cmccmtN 36195  OLcol 36196  OMLcoml 36197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-proset 17533  df-poset 17551  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-lat 17651  df-oposet 36198  df-cmtN 36199  df-ol 36200  df-oml 36201
This theorem is referenced by:  omlfh3N  36281  omlmod1i2N  36282
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