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Theorem omllaw 35052
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 35047 . . . 4 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
76simprbi 483 . . 3 (𝐾 ∈ OML → ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥)))))
8 breq1 4808 . . . . 5 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
9 id 22 . . . . . . 7 (𝑥 = 𝑋𝑥 = 𝑋)
10 fveq2 6354 . . . . . . . 8 (𝑥 = 𝑋 → ( 𝑥) = ( 𝑋))
1110oveq2d 6831 . . . . . . 7 (𝑥 = 𝑋 → (𝑦 ( 𝑥)) = (𝑦 ( 𝑋)))
129, 11oveq12d 6833 . . . . . 6 (𝑥 = 𝑋 → (𝑥 (𝑦 ( 𝑥))) = (𝑋 (𝑦 ( 𝑋))))
1312eqeq2d 2771 . . . . 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 4809 . . . . 5 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
16 id 22 . . . . . 6 (𝑦 = 𝑌𝑦 = 𝑌)
17 oveq1 6822 . . . . . . 7 (𝑦 = 𝑌 → (𝑦 ( 𝑋)) = (𝑌 ( 𝑋)))
1817oveq2d 6831 . . . . . 6 (𝑦 = 𝑌 → (𝑋 (𝑦 ( 𝑋))) = (𝑋 (𝑌 ( 𝑋))))
1916, 18eqeq12d 2776 . . . . 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 3462 . . 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 1109 1 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌𝑌 = (𝑋 (𝑌 ( 𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wral 3051   class class class wbr 4805  cfv 6050  (class class class)co 6815  Basecbs 16080  lecple 16171  occoc 16172  joincjn 17166  meetcmee 17167  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-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
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-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058  df-ov 6818  df-oml 34988
This theorem is referenced by:  omllaw2N  35053  omllaw3  35054  omllaw4  35055
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