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Theorem omlfh1N 39259
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 31637 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 39243 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
2 omlfh1.b . . . . . 6 𝐵 = (Base‘𝐾)
3 eqid 2737 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4 omlfh1.j . . . . . 6 = (join‘𝐾)
5 omlfh1.m . . . . . 6 = (meet‘𝐾)
62, 3, 4, 5latledi 18522 . . . . 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 1133 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
91adantr 480 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
10 simpr1 1195 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
11 simpr2 1196 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
12 simpr3 1197 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
132, 4latjcl 18484 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
149, 11, 12, 13syl3anc 1373 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
152, 5latmcom 18508 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
169, 10, 14, 15syl3anc 1373 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
17 omlol 39241 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
1817adantr 480 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
192, 5latmcl 18485 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
209, 10, 11, 19syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
212, 5latmcl 18485 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
229, 10, 12, 21syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
23 eqid 2737 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
242, 4, 5, 23oldmj1 39222 . . . . . . . 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 1373 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))))
262, 4, 5, 23oldmm1 39218 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
2718, 10, 11, 26syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
282, 4, 5, 23oldmm1 39218 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
2918, 10, 12, 28syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
3027, 29oveq12d 7449 . . . . . . 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 2777 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))
3216, 31oveq12d 7449 . . . . 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 1133 . . . 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 39242 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
3534adantr 480 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
362, 23opoccl 39195 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
3735, 10, 36syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
382, 23opoccl 39195 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
3935, 11, 38syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
402, 4latjcl 18484 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
419, 37, 39, 40syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
422, 23opoccl 39195 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
4335, 12, 42syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
442, 4latjcl 18484 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
459, 37, 43, 44syl3anc 1373 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
462, 5latmcl 18485 . . . . . . . 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 1373 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)
482, 5latmassOLD 39230 . . . . . . 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 1374 . . . . . 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 1133 . . . . 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 39251 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
53523adant3r3 1185 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
54 simpl 482 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
552, 4, 5, 23, 51cmtbr3N 39255 . . . . . . . . . . . . 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 1373 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑌) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5753, 56bitrd 279 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌))))
5857biimpa 476 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑌) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
5958adantrr 717 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
60593impa 1110 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
612, 23, 51cmt2N 39251 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
62613adant3r2 1184 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
632, 4, 5, 23, 51cmtbr3N 39255 . . . . . . . . . . . . 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 1373 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶((oc‘𝐾)‘𝑍) ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6562, 64bitrd 279 . . . . . . . . . . 11 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍))))
6665biimpa 476 . . . . . . . . . 10 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ 𝑋𝐶𝑍) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6766adantrl 716 . . . . . . . . 9 (((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
68673impa 1110 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 ((oc‘𝐾)‘𝑍)))
6960, 68oveq12d 7449 . . . . . . 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 39235 . . . . . . . . 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 1374 . . . . . . . 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 1133 . . . . . . 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 39235 . . . . . . . . 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 1374 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
75743adant3 1133 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
7669, 72, 753eqtr4d 2787 . . . . . 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 7447 . . . . 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 18485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
799, 39, 43, 78syl3anc 1373 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
802, 5latm12 39231 . . . . . . 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 1374 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
82813adant3 1133 . . . . 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 2781 . . . 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 39222 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8518, 11, 12, 84syl3anc 1373 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8685oveq2d 7447 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) ((oc‘𝐾)‘(𝑌 𝑍))) = ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))))
87 eqid 2737 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
882, 23, 5, 87opnoncon 39209 . . . . . . . . 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 2779 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (0.‘𝐾))
9190oveq2d 7447 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 (0.‘𝐾)))
922, 5, 87olm01 39237 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9318, 10, 92syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9491, 93eqtrd 2777 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
95943adant3 1133 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
9633, 83, 953eqtrd 2781 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾))
972, 4latjcl 18484 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
989, 20, 22, 97syl3anc 1373 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
992, 5latmcl 18485 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1009, 10, 14, 99syl3anc 1373 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1012, 3, 5, 23, 87omllaw3 39246 . . . . 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 1373 . . . 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 1133 . . 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 699 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍)) = (𝑋 (𝑌 𝑍)))
105104eqcomd 2743 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  0.cp0 18468  Latclat 18476  OPcops 39173  cmccmtN 39174  OLcol 39175  OMLcoml 39176
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18340  df-poset 18359  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-lat 18477  df-oposet 39177  df-cmtN 39178  df-ol 39179  df-oml 39180
This theorem is referenced by:  omlfh3N  39260  omlmod1i2N  39261
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