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Theorem omllaw 33548
 Description: The orthomodular law. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
omllaw.b 𝐵 = (Base‘𝐾)
omllaw.l = (le‘𝐾)
omllaw.j = (join‘𝐾)
omllaw.m = (meet‘𝐾)
omllaw.o = (oc‘𝐾)
Assertion
Ref Expression
omllaw ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))

Proof of Theorem omllaw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omllaw.b . . . . 5 𝐵 = (Base‘𝐾)
2 omllaw.l . . . . 5 = (le‘𝐾)
3 omllaw.j . . . . 5 = (join‘𝐾)
4 omllaw.m . . . . 5 = (meet‘𝐾)
5 omllaw.o . . . . 5 = (oc‘𝐾)
61, 2, 3, 4, 5isoml 33543 . . . 4 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
76simprbi 479 . . 3 (𝐾 ∈ OML → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))
8 breq1 4586 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
10 fveq2 6103 . . . . . . . 8 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1110oveq2d 6565 . . . . . . 7 (𝑥 = 𝑋 → (𝑦 ( 𝑥)) = (𝑦 ( 𝑋)))
129, 11oveq12d 6567 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 ( 𝑥))) = (𝑋 (𝑦 ( 𝑋))))
1312eqeq2d 2620 . . . . 5 (𝑥 = 𝑋 → (𝑦 = (𝑥 (𝑦 ( 𝑥))) ↔ 𝑦 = (𝑋 (𝑦 ( 𝑋)))))
148, 13imbi12d 333 . . . 4 (𝑥 = 𝑋 → ((𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))) ↔ (𝑋 𝑦𝑦 = (𝑋 (𝑦 ( 𝑋))))))
15 breq2 4587 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
16 id 22 . . . . . 6 (𝑦 = 𝑌𝑦 = 𝑌)
17 oveq1 6556 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 ( 𝑋)) = (𝑌 ( 𝑋)))
1817oveq2d 6565 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 ( 𝑋))) = (𝑋 (𝑌 ( 𝑋))))
1916, 18eqeq12d 2625 . . . . 5 (𝑦 = 𝑌 → (𝑦 = (𝑋 (𝑦 ( 𝑋))) ↔ 𝑌 = (𝑋 (𝑌 ( 𝑋)))))
2015, 19imbi12d 333 . . . 4 (𝑦 = 𝑌 → ((𝑋 𝑦𝑦 = (𝑋 (𝑦 ( 𝑋)))) ↔ (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
2114, 20rspc2v 3293 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
227, 21syl5com 31 . 2 (𝐾 ∈ OML → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋))))))
23223impib 1254 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  occoc 15776  joincjn 16767  meetcmee 16768  OLcol 33479  OMLcoml 33480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-oml 33484 This theorem is referenced by:  omllaw2N  33549  omllaw3  33550  omllaw4  33551
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