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Theorem omlfh1N 39240
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 31647 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 39224 . . . . 5 (𝐾 ∈ OML → 𝐾 ∈ Lat)
2 omlfh1.b . . . . . 6 𝐵 = (Base‘𝐾)
3 eqid 2735 . . . . . 6 (le‘𝐾) = (le‘𝐾)
4 omlfh1.j . . . . . 6 = (join‘𝐾)
5 omlfh1.m . . . . . 6 = (meet‘𝐾)
62, 3, 4, 5latledi 18535 . . . . 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 1131 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍))(le‘𝐾)(𝑋 (𝑌 𝑍)))
91adantr 480 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
10 simpr1 1193 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
11 simpr2 1194 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
12 simpr3 1195 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
132, 4latjcl 18497 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑌𝐵𝑍𝐵) → (𝑌 𝑍) ∈ 𝐵)
149, 11, 12, 13syl3anc 1370 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑌 𝑍) ∈ 𝐵)
152, 5latmcom 18521 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
169, 10, 14, 15syl3anc 1370 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((𝑌 𝑍) 𝑋))
17 omlol 39222 . . . . . . . . 9 (𝐾 ∈ OML → 𝐾 ∈ OL)
1817adantr 480 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
192, 5latmcl 18498 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
209, 10, 11, 19syl3anc 1370 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑌) ∈ 𝐵)
212, 5latmcl 18498 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑍𝐵) → (𝑋 𝑍) ∈ 𝐵)
229, 10, 12, 21syl3anc 1370 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 𝑍) ∈ 𝐵)
23 eqid 2735 . . . . . . . . 9 (oc‘𝐾) = (oc‘𝐾)
242, 4, 5, 23oldmj1 39203 . . . . . . . 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 1370 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍))) = (((oc‘𝐾)‘(𝑋 𝑌)) ((oc‘𝐾)‘(𝑋 𝑍))))
262, 4, 5, 23oldmm1 39199 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
2718, 10, 11, 26syl3anc 1370 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(𝑋 𝑌)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)))
282, 4, 5, 23oldmm1 39199 . . . . . . . . 9 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑋 𝑍)) = (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))
2918, 10, 12, 28syl3anc 1370 . . . . . . . 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 2775 . . . . . 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 1131 . . . 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 39223 . . . . . . . . . . 11 (𝐾 ∈ OML → 𝐾 ∈ OP)
3534adantr 480 . . . . . . . . . 10 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
362, 23opoccl 39176 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
3735, 10, 36syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
382, 23opoccl 39176 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
3935, 11, 38syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
402, 4latjcl 18497 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
419, 37, 39, 40syl3anc 1370 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
422, 23opoccl 39176 . . . . . . . . . 10 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
4335, 12, 42syl2anc 584 . . . . . . . . 9 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
442, 4latjcl 18497 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
459, 37, 43, 44syl3anc 1370 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
462, 5latmcl 18498 . . . . . . . 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 1370 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) ∈ 𝐵)
482, 5latmassOLD 39211 . . . . . . 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 1371 . . . . . 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 1131 . . . . 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 39232 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
53523adant3r3 1183 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌𝑋𝐶((oc‘𝐾)‘𝑌)))
54 simpl 482 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
552, 4, 5, 23, 51cmtbr3N 39236 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 1109 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 ((oc‘𝐾)‘𝑌)))
612, 23, 51cmt2N 39232 . . . . . . . . . . . . 13 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
62613adant3r2 1182 . . . . . . . . . . . 12 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍𝑋𝐶((oc‘𝐾)‘𝑍)))
632, 4, 5, 23, 51cmtbr3N 39236 . . . . . . . . . . . . 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 1370 . . . . . . . . . . . 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 1109 . . . . . . . 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 39216 . . . . . . . . 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 1371 . . . . . . . 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 1131 . . . . . . 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 39216 . . . . . . . . 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 1371 . . . . . . . 8 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
75743adant3 1131 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((𝑋 ((oc‘𝐾)‘𝑌)) (𝑋 ((oc‘𝐾)‘𝑍))))
7669, 72, 753eqtr4d 2785 . . . . . 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 18498 . . . . . . . 8 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
799, 39, 43, 78syl3anc 1370 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
802, 5latm12 39212 . . . . . . 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 1371 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑌 𝑍) (𝑋 (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
82813adant3 1131 . . . . 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 2779 . . . 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 39203 . . . . . . . . . 10 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(𝑌 𝑍)) = (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))
8518, 11, 12, 84syl3anc 1370 . . . . . . . . 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 2735 . . . . . . . . . 10 (0.‘𝐾) = (0.‘𝐾)
882, 23, 5, 87opnoncon 39190 . . . . . . . . 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 2777 . . . . . . 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 39218 . . . . . . 7 ((𝐾 ∈ OL ∧ 𝑋𝐵) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9318, 10, 92syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (0.‘𝐾)) = (0.‘𝐾))
9491, 93eqtrd 2775 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
95943adant3 1131 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 ((𝑌 𝑍) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (0.‘𝐾))
9633, 83, 953eqtrd 2779 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 (𝑌 𝑍)) ((oc‘𝐾)‘((𝑋 𝑌) (𝑋 𝑍)))) = (0.‘𝐾))
972, 4latjcl 18497 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑋 𝑌) ∈ 𝐵 ∧ (𝑋 𝑍) ∈ 𝐵) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
989, 20, 22, 97syl3anc 1370 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) ∈ 𝐵)
992, 5latmcl 18498 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑋𝐵 ∧ (𝑌 𝑍) ∈ 𝐵) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1009, 10, 14, 99syl3anc 1370 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) ∈ 𝐵)
1012, 3, 5, 23, 87omllaw3 39227 . . . . 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 1370 . . . 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 1131 . . 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 2741 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 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  0.cp0 18481  Latclat 18489  OPcops 39154  cmccmtN 39155  OLcol 39156  OMLcoml 39157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-oposet 39158  df-cmtN 39159  df-ol 39160  df-oml 39161
This theorem is referenced by:  omlfh3N  39241  omlmod1i2N  39242
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